Issue 
J. Space Weather Space Clim.
Volume 9, 2019



Article Number  A20  
Number of page(s)  27  
DOI  https://doi.org/10.1051/swsc/2019017  
Published online  14 June 2019 
Research Article
Timeofday/timeofyear response functions of planetary geomagnetic indices
^{1}
Department of Meteorology, University of Reading, Whiteknights Campus, Earley Gate, PO Box 243, Reading RG6 6BB, UK
^{2}
Institut de Physique du Globe de Strasbourg, UMR7516, Université de Strasbourg/EOST, CNRS, 5 rue René Descartes, Strasbourg Cedex 67084, France
^{3}
Rutherford Appleton Laboratory, Science and Technology Facilities Council, Harwell Campus, Didcot OX11 0QX, UK
^{*} Corresponding author: m.lockwood@reading.ac.uk
Received:
11
January
2019
Accepted:
1
April
2019
Aims: To elucidate differences between commonlyused midlatitude geomagnetic indices and study quantitatively the differences in their responses to solar forcing as a function of Universal Time (UT), timeofyear (F), and solarterrestrial activity level. To identify the strengths, weaknesses and applicability of each index and investigate ways to correct for any weaknesses without damaging their strengths.
Methods: We model how the location of a geomagnetic observatory influences its sensitivity to solar forcing. This modelling for a single station can then be applied to indices that employ analytic algorithms to combine data from different stations and thereby we derive the patterns of response of the indices as a function of UT, F and activity level. The model allows for effects of solar zenith angle on ionospheric conductivity and of the station’s proximity to the midnightsector auroral oval: it employs coefficients that are derived iteratively by comparing data from the current aa index stations (Hartland and Canberra) to simultaneous values of the am index, constructed from chains of stations in both hemispheres. This is done separately for eight overlapping bands of activity level, as quantified by the am index. Initial estimates were obtained by assuming the am response is independent of both F and UT and the coefficients so derived were then used to compute a corrected FUT response pattern for am. This cycle was repeated until it resulted in changes in predicted values that were below the adopted uncertainty level (0.001%). The ideal response pattern of an index would be uniform and linear (i.e., independent of both UT and F and the same at all activity levels). We quantify the response uniformity using the percentage variation at any activity level, V = 100 (σ_{S}/〈S〉), where S is the index’s sensitivity at that activity level and σ_{S} is the standard deviation of S: both S and σ_{S} were computed using the eight UT ranges of the 3hourly indices and 20 equalwidth ranges of F. As an overall metric of index performance, we take an occurrenceweighted mean of V, V_{av}, over the eight activitylevel bins. This metric would ideally be zero and a large value shows that the index compilation is introducing large spurious UT and/or F variations into the data. We also study index performance by comparisons with the SME and SML indices, compiled from a very large number of stations, and with an optimum solar wind “coupling function”, derived from simultaneous interplanetary observations.
Results: It is shown that a station’s response patterns depend strongly on the level of geomagnetic activity because at low activity levels the effect of solar zenith angle on ionospheric conductivity dominates over the effect of station proximity to the midnightsector auroral oval, whereas the converse applies at high activity levels. The metric V_{av} for the twostation aa index is modelled to be 8.95%, whereas for the multistation am index it is 0.65%. The ap (and hence Kp) index cannot be analyzed directly this way because its construction employs tabular conversions, but the very low V_{av} for am allows us to use 〈ap〉/〈am〉 to evaluate the UTF response patterns for ap. This yields V_{av} = 11.20% for ap. The same empirical test applied to the classical aa index and the new “homogenous” aa index, aa_{H} (derived from aa using the station sensitivity model), yields V_{av} of, respectively, 10.62% (i.e., slightly higher than the modelled value) and 5.54%. The ap index value of V_{av} is shown to be high because it exaggerates the average semiannual variation and has an annual variation giving a lower average response in northern hemisphere winter. It also contains a strong artefact UT variation. We derive an algorithm for correcting for this uneven response which gives a corrected ap value, ap_{C}, for which V_{av} is reduced to 1.78%. The unevenness of the ap response arises from the dominance of European stations in the network used and the fact that all data are referred to a European station (Niemegk). However, in other contexts, this is a strength of ap, because averaging similar data gives increased sensitivity and more accurate values on annual timescales, for which the UTF response pattern is averaged out.
Key words: indices / geomagnetism / substorm / space environment / space climate
© M. Lockwood et al., Published by EDP Sciences 2015
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Geomagnetic indices are widely used to quantify the level of activity in the terrestrial magnetosphereionospherethermosphere system, and have sometimes been used without consideration being given to their response characteristics and whether or not they are appropriate to the application in question. In this paper, we study the response characteristics of three widelyused 3hourly global geomagnetic indices compiled from observations by midlatitude observatories; indices which primarily respond to the substorm current wedge (see review by Lockwood, 2013). These are the “mondial” (meaning “global”) am index (and its components an and as in the northern and southern hemispheres, respectively), the “planetary” ap index (equivalent to Kp) and the aa index (and its hemispheric components, aa_{N} and aa_{S}). For each of these indices, we study the response as a function of the timeofday (i.e., Universal Time, UT), the time of year (quantified by the fraction of the calendar year, F) and the level of geomagnetic activity.
Maps showing the stations currently contributing to these indices are presented in Figure 1. Lists of stations, their locations and the intervals over which each was used, are given for each index in Appendix A. The am index is designed to give good and even longitudinal coverage in both hemispheres, currently employing 14 stations in the northern hemisphere and 10 in the southern. There have been relatively minor changes to the network of stations deployed and the use of areaweighted averaging over longitude sectors has minimised the effect of these changes. It is available continuously from 1 January 1959 to the present day. The ap (and Kp) index is currently made from data from 11 stations in the northern hemisphere and 2 in the southern. It is available continuously since 1 January 1932, although the number and distribution of stations has varied considerably: initially there were 10 stations all in the northern hemisphere, the first southern hemisphere station being added in 1958. The ap and Kp indices have always been dominated by data from European stations. The aa index uses just two stations, one in each hemisphere, and although suppressing the resulting spurious annual variation in the response rather well, aa shows a large spurious diurnal variation (e.g., Lockwood et al., 2018b). The aa index is continuously available for the longest interval (1 January 1868 to the present day) and has been relatively homogeneous in its construction. It is this longevity that gives aa its importance. Recent work has shown that it is greatly improved by corrections to allow for the effects of secular change in the intrinsic geomagnetic field and the locationdependent sensitivity of the stations deployed, yielding the “homogeneous aa index” aa_{H} (Lockwood et al., 2018a, b). Note that above we discuss the 3hourly indices am, an, as, ap, aa, aa_{N}, aa_{S} and aa_{H}: the same considerations apply to their respective daily means, Am, An, As, Ap, Aa, Aa_{N}, Aa_{S} and Aa_{H}, and their 8point (24hour) running (boxcar) means Am*, An*, As*, Ap*, Aa*, Aa_{N}*, Aa_{S}* and Aa_{H}* (Allen, 1982).
Fig. 1 Maps of networks of stations currently contributing to (a) the Kp (and hence ap) index, (b) the am index and (c) the aa index. In each map, the light grey bands are typical locations of the auroral oval and dark grey bands are ideal middle geomagnetic latitudes for stations to give a Kindex value, being close enough to give a large signal, but far enough away that the response is monotonic because, for all but the very largest disturbances, the auroral oval approaches the station as the activity level increases. Details of these stations, and others used in the past, are given in Appendix A. Images courtesy of the International Service of Geomagnetic Indices (ISGI). 
Figure 2a shows an example of a 7day interval of 3hourly geomagnetic index data and Figure 2b shows the corresponding 24hour (8point) running means for the same interval. The interval is 27 October 2003 to 2 November and so includes the “Halloween storms” that generated considerable GIC events (Kappenman, 2005). This example is chosen here because it gave the highest daily means in the interval 1995–2017. The values given for the am, ap, aa and aa_{H} indices in Figure 2a (and their corresponding 8point means, Am*, Ap*, Aa*, and Aa_{H}* in Fig. 2b) are the published values but the ap and Ap* values have been multiplied a factor f = 〈am〉_{all}/〈ap〉_{all}, the ratio of overall means of am and ap, which allows for the difference in the scaling of ap and the other indices. Figure 2b shows the most clearcut difference between the indices is that the scaled ap (and hence Ap*) values during the storm are consistently larger than the corresponding values for am, aa, and aa_{H} whereas before and after they are proportionally much more similar. This demonstrates that differences between the responses of the indices can depend on the level of geomagnetic activity. Given the normalisation using f, this means there must be other times when f × ap is lower than am, aa and aa_{H}. This paper investigates how systematic this difference is with timeof year. In comparison, the Aa*, Aa_{H}* and Am* values are much more similar. The 3hourly values shown in Figure 2a show considerable pointtopoint variation in each index and pointtopoint variation in the relationships between the various indices. There is no evidence for a systematic difference between the indices with UT (by which points in Fig. 2a are colourcoded to aid comparisons). Such differences would be convolved with random effects, such as the timing of perturbations within the 3hour intervals over which the range indices are measured, and so are likely to emerge only in statistical surveys that also allow for the effects of timeof year and activity level.
Fig. 2 Variations in geomagnetic range indices for 27 October 2003 to 2 November, showing the “Halloween storms”: (a) 3hourly values am, f × ap, aa and aa_{H} (b) their 24hour (8point) running (“boxcar”) means Am*, f × Ap*, Aa* and Aa_{H}*. The ap and Ap* values have been multiplied by f = 〈am〉_{all}/〈ap〉_{all}, the ratio of overall means of am and ap for 1995–2017, to allow for the difference between the scaling of ap and that for other indices. Circles, triangles, squares and diamonds are for am (Am*), f × ap (f × Ap*), aa (Aa*), and aa_{H} (Aa_{H}*), respectively. Points are colourcoded by the UT of observation in (a) and vertical grey lines are at UT = 0. 
1.1 K indices
K values, on which all these indices are based, were introduced by Bartels et al. (1939). They are derived for each magnetometer station from the range of variation observed in each of eight 3hour intervals (0–3 UT, 3–6 UT, up to 21–24 UT). Originally scaled manually, Kvalue derivation was increasingly automated as magnetometer data recording moved from analogue to digital (Riddick & Stuart, 1984). The range of the irregular variations (i.e., after subtraction of the regular daily variation) in either of the horizontal components (X northward or Y eastward, whichever gives the larger value), ΔH_{ X } or ΔH_{ Y }, is ranked into one of 10 classes using quasilogarithmic band limits that are specific to the observatory and to which a K value of 0–9 is assigned. The original idea of this procedure was that the scale of threshold values used to convert the continuous ΔH_{ X } or ΔH_{ Y } range values into the quantized K values is adjusted for each station to allow for its location and characteristics such that the K value is a standardized measure of the geomagnetic activity level, irrespective of where it is measured. In practice, the range limits for all K bands for a given station are all set by just one number, L, the lower limit of the K = 9 band, the lower limit for the K = 0 band being always set to zero (Menvielle & Berthelier, 1991). This is because the same relative scale is used at all stations with the thresholds for the K bands given in Table 1.
Bands of range values used to generate quantized Kindices for a station with a lower limit of the K = 9 band of L. ΔH_{ X or Y } is the range between extreme values in the 3hour intervals of the northward or westward horizontal component, whichever is the larger. The right hand column gives the quantized a_{K} values ascribed to the Klevels using the “K2aK” or “midclass amplitudes” scale.
The very high correlations between the range indices and auroral electrojet indices such as AE and AL (e.g., Adebesin, 2016; Lockwood et al., 2019a) indicate that geomagnetic activity at midlatitude observatories is dominated by the ionospheric segment of the substorm current wedge, i.e., the main westward auroral electrojet (e.g., Saba et al., 1997; Finch et al., 2008; Lockwood, 2013). (See further discussion in Sect. 1.5). As a result, the value of L used for a station is set by its closest proximity to the midnight Magnetic Local Time (MLT) sector of a nominal auroral oval which is where the range response of a station is greatest (Clauer & McPherron, 1974; Finch et al. 2008; Chambodut et al., 2013). The L value used is decreased with decreasing magnetic latitude because the range observed decreases with increasing distance from the auroral oval (e.g., Rostoker, 1972). In practice, this is quantified by the geocentric angle, δ, between the station and the point of closest approach of the nominal auroral oval (which occurs near midnight MLT) which is taken to be along a corrected geomagnetic latitude Λ_{CG} of 69°. Station Kindices are converted to a_{K} values (in nT) using a standard scale called “midclass amplitudes”, for which the range threshold for the K = 9 band is L = 500 nT: the conversion table for implementing this scale is referred to as K2aK and is given in the third column of Table 1. All the planetary and hemispheric indices discussed in this paper are based on these observatory Kvalues.
1.2 The aa index
The simplest of the indices that we study on this paper is the aa index, which was devised by Mayaud (1971, 1972, 1980) to give a continuous, wellcalibrated and homogeneous record of geomagnetic activity that extends back to 1868. This index uses just two stations at similar geomagnetic latitudes, one in each hemisphere. The northern and southern hemisphere aa indices, aa_{N} and aa_{S}, are the a_{K} values from the station in that hemisphere multiplied by a station scaling factor. For the “classic” aa index [the official aa index generated by École et Observatoire des Sciences de la Terre (EOST), and available from International Service of Geomagnetic Indices (ISGI), at http://isgi.unistra.fr/ and data centres around the world] the station scaling factors are constant with time; however, recently Lockwood et al. (2018a) have shown that anomalies in the secular variations of aa_{N} and aa_{S} are removed if timedependent factors, calculated from models of the global intrinsic (“main”) field, are used. The hemispheric aa indices are then averaged together to give aa = (aa_{N} + aa_{S})/2. This averaging, to a large extent, gives cancellation of the seasonal variation in the geomagnetic response to solar forcing that is found in aa_{N} and aa_{S} individually. Lockwood et al. (2018b) have demonstrated how good cancellation is effectively achieved for the annual variation but that, because the best available stations are roughly 10 h apart in local time (instead of the ideal 12 h), the diurnal variations at the two stations do not cancel as completely as the annual variation. Recently, Lockwood et al. (2018b) have used a model of the stations’ response (and made the correction to allow for the effects of the secular change in the geomagnetic field) to generate the “homogeneous” version of the aa index, aa_{H}, so named because it largely eliminates wellknown hemispheric asymmetries between the mean values and distributions of aa_{N} and aa_{S}.
1.3 The am index
The stations used to compile the am index (Mayaud, 1980) are situated at subauroral latitudes close to corrected geomagnetic latitude Λ_{CG} = 50°. They are grouped into longitude sectors, with five such groups in the Northern hemisphere, and four in the Southern. The K indices for stations in a longitude sector are averaged together and the result is converted into a sector a_{K} value using the standard K2aK scale. Weighted averages of these sector a_{K} values are then generated in each hemisphere giving an and as, the weighting factors accounting for the differences in the longitude extents of the sectors. The index am is equal to (an + as)/2. Note that, like aa, am is compiled using only mathematical operations. We here employ all available am data up to the end of 2017 and that after the end of 2014 these data are classed as “provisional” which means they have passed initial quality checks and can be used, but not yet been through the final review that defines them as “definitive”. We here apply additional checks to the data for 2015–2017 by testing for, and removing, any outliers in the scatter plots (more than 3σ from the mean) with the SuperMAG SME index or the Auroral Electrojet AE index (see Sect. 1.5).
1.4 The ap (Kp) index
The ap index is currently made using K index data from 11 northern and two southern hemisphere stations between corrected geomagnetic latitudes Λ_{CG} between 44° and 60°. The K indices are first converted into standardised K_{S} values to account for the timeofyear and UT response characteristics of the observing site and to, as far as possible, normalise them to the values seen simultaneously by Niemegk, which was chosen as the reference station. The Kp index is the arithmetic mean of the 3hour standardized K_{S} values for the observatories employed. The 3hourly Kp values are converted into ap values using a standard table that is constructed such that ap may be regarded as the range of the mostdisturbed of the two horizontal field components, expressed in units of 2 nT, at a station at dipole latitude of 50°.
The standardization from K to K_{S} is achieved using conversion tables for each observatory that were defined for the original stations by Bartels (1949, 1957). These give a multiplication factor K_{S}/K that depends on the station location, UT, timeofyear, F, and the K value and hence application of these factors is a nonlinear operation. The present conversion tables used are for three seasons and many were generated using selected data from 1943 to 1948 only. The three seasons are: (1) the months around winter solstice (January, February, November and December); (2) the months around the equinoxes (March, April, September and October); and (3) the months around summer solstice (May, June, July and August). The network of stations used to compile the ap and Kp indices has varied, and the intervals over which each station was used are given in Appendix A.
In theory, if the K to K_{S} conversions were always ideal, the distribution of ap stations would be of no consequence as the various K_{S} values would be all be different measurements of the same thing. However, to some extent the tables will fall short of the ideal so the index response pattern will also depend, to some extent, on the distribution of stations. Even if the K to K_{S} conversions were ideal, converting the data to what Niemegk would have seen causes the index to take on the UTF response characteristics of the Niemegk site. Note that we here quote ap (and hence ap_{C}, Ap_{C} * and [Ap_{C} *]_{max}) in the most widely used form – namely as an index without units: the standard ap values are an index in units of 2 nT and hence the values in units of nT would be double those given in this paper (Menvielle & Berthelier, 1991).
1.5 Relationship of midlatitude Kbased indices to auroral electrojet indices
In this section, we compare the midlatitude range indices (am, ap and aa) with the SME and SML indices from the northern hemisphere SuperMAG magnetometers (Newell & Gjerloev, 2011) and the standard AE and AL auroral electrojet indices from the ring of 12 northern hemisphere stations at auroral latitudes (Davis & Sugiura, 1966). The AE indices are based on 1minute data on the quietdaysubtracted horizontal magnetic component from the 12 stations. The upper and lower envelopes of the distribution of values at any one time define AU and AL, respectively, and AE is the width of the envelope between these maximum and minimum values, AE = AU − AL. The AL values are negative and driven by the westward auroral electrojet that is the ionospheric segment of the substorm current wedge and so are usually recorded by a station in the midnight MLT sector and AL responds mainly to the enhancement of the electrojet during substorm expansion phases (see review by Lockwood, 2013). On the other hand, AU responds primarily to the eastward electrojet associated with westward convective flow in the afternoon MLT sector and is enhanced during both the growth and expansion phases of substorms. The SuperMAG SME and SML indices are computed in exactly the same way as AE and AL but using 1minute data from a much greater number of stations, which varies between 93 and 118 over the interval studied in this section (1996–2017). These indices capture better the extreme deflections which define them. The SuperMAG network is global but has an excess of northern hemisphere stations that are particularly clustered in the European and American longitude sectors and SME and SML are generated from northern hemisphere stations only. However, because SME and SML, like AE and AL, use extreme rather than average values, the clustering is not an issue and the network is extensive enough for these indices to be considered almost free of spurious diurnal variations. However, the use of only northern hemisphere stations will mean these indices, like AE and AL, will have an annual variation due to seasonal effects. The am, ap and aa indices are all based on the range of variation in 3hour intervals, so it makes sense to compare them with the largest SME value (SME_{max}) and the minimum SML value (SML_{min}) in the same 3hour intervals. However, as shown by Table 2, strong correlations are also found with the average SME and SML for the coincident 3hour intervals (〈SME〉_{ τ=3h} and 〈SML〉_{ τ=3h}), although not quite as strong as for SME_{max} and SML_{min}. Correlation coefficients are typically between 0.8 and 0.9 and always stronger for the 24hour smoothed values than the 3hourly values and for the SuperMAG indices than for the traditional AE indices. In most cases, the am index yields the best correlations, but those for aa are very similar: correlations for ap are invariably lower. Note that the correlations for AL are slightly weaker than those for AE, as are those for SML compared to SME. This indicates that the midlatitude range indices do respond to the directly driven system, as detected by AU and SMU, as well as the strong influence of the substorm unloading system, as detected by AL and SML.
Linear correlation coefficients between midlatitude range indices and the SuperMAG and auroral electrojet indices: r and r* are for 3hourly values and the 8point running means, respectively. The data used are for 1996–2017 (inclusive). This yields 61368 3hourly samples (am, ap and aa) and 6136124 runningmean samples (Am*, Ap* and Aa*). For all correlations, the large number of samples ensures that the correlation significance level, derived by comparison with the AR1 red noise model, is 100% to within at least three decimal places for all cases. The maximum SME and minimum SML in each 3hour intervals is SME_{max} and SML_{min} respectively.
Figure 3 illustrates examples of the relationships of (a) am, (b) ap, and (c) aa to SME_{max} for the years 1996–2017 (inclusive). The general features are the same in corresponding plots for 〈SME〉_{ τ=3h}, −SML_{min}, −〈SML〉_{ τ=3h}, AE_{max}, 〈AE〉_{ τ=3h}, −AL_{min}, or −〈AL〉_{ τ=3h}. In each panel a scatter plot of the threehourly values is given by grey dots and of their 8point running means (24hour averages) by orange dots. Note the grey dots show the quantization of both aa and ap levels, whereas am (like Am*, Ap* and Aa*) is essentially continuous. Black dots are means in 1percentile bins of the midlatitude range index along the xaxis (giving 614 samples in each bin) and the horizontal and vertical error bars are plus and minus one standard deviation for that bin. The mauve lines are fourthorder polynomial fits to the 3hourly values. The RMS deviation of the observed 3hourly SME_{max} values from the fitted polynomial values, Δ_{RMS} are given in each panel, as are the linear correlation coefficients r and r* for the 3hourly values and the 8point running means, respectively. Note that the polynomial fit order m = 4 that was used was the largest that could be employed (desirable as it preserves the nonlinearity of the variation) that gave a good fit to the tail of the index distribution; the latter criterion being tested by evaluating the Δ_{RMS} ^{2}/(n – m − 1) value for the n = 614 samples in the top percentile of the index dataset and checking it was not significantly larger (at the 1 − σ level) than the minimum value for any m. The cyan lines are linear fits to the 3hourly values for quiettimes (SME_{max} < 750 nT), and are plotted to gauge the deviations from linearity of the data for larger SME_{max}. Figure 3 shows that all three midlatitude range indices have a nonlinear response, with values being overestimates, relative to SME_{max} values, at the highest activity levels. However, the tendency is much stronger for ap than for am and aa: Figure 3a and c shows that the deviation from linearity is only present in the top 1% of samples for am and aa; however, for ap the deviation is persistent for the top 35% and is significant (greater than 1 standard deviation) for the top 17% of ap samples. The scatter (quantified by the RMS deviation of samples from the fourthorder polynomial fit, Δ_{RMS}) is smallest for am but largest for aa, but the linear correlations r and r* for aa are higher than for ap because the polynomial fit deviates from linearity much less for aa than for ap. Indeed, for 24hour running means, the correlation for Aa* is even very slightly higher than that for Am*.
Fig. 3 Scatter plots for 1996–2017 (inclusive) of the midlatitude range indices with the maximum of the SME index, SME_{max}, seen in the same 3hour intervals by the SuperMAG global magnetometer network. (a) (grey) 3hourly SME_{max} values as a function of 3hourly am and (orange) 24hour running means of SME_{max} as a function of corresponding running means of am, Am*. Black dots are means in 1percentile ranges of Am* (giving 614 samples in each bin) and the horizontal and vertical error bars are ±1 standard deviation. The mauve line is a fourthorder polynomial fit to the 3hourly values. The RMS deviation of the observed 3hourly SME_{max} values from the fitted polynomial value for the corresponding am, Δ_{RMS} is given, as is the correlation coefficient r between 3hourly SME_{max} and am values. (b) The same as (a) for 3hourly ap and its 24hour running mean, Ap*. (c) The same as (a) for 3hourly aa and its 24hour running mean, Aa*. In each panel the cyan line is a linear fit to the 3hourly values for SME_{max} < 750 nT, and is plotted to gauge the deviation from linearity of the data for larger SME_{max}. 
1.6 Relationship of midlatitude Kbased indices to interplanetary conditions
In Section 2 we employ the concept of the sensitivity of station to “solar forcing”. This sensitivity depends only on the station’s location, all other site factors (such as instrumentation sensitivity and ground conductivity) being accounted for by the instrument calibration. This sensitivity gives the station’s response to all solar forcings, including photon and particle ionization and heating effects (Aksnes et al., 2002; Ieda et al., 2014) as well as the enhanced electric fields and the associated expansion/contraction of the auroral oval due to solar windmagnetosphere coupling (Lockwood et al., 1990; Cowley & Lockwood, 1992; Milan et al., 2012). With this definition, not all of this solar forcing comes from the solar wind, because of the effects of solar EUV and Xray ionizing and heating radiations on ionospheric conductivities. Hence “solar wind forcing” is a part of, but not all of, “solar forcing”.
To quantify solar wind forcing, a number of “coupling functions” have been proposed as predictors and explainers of geomagnetic disturbance (see review in Lockwood et al., 2019a). These are combinations of parameters characterizing the nearEarth planetary environment, combined with various coefficients and exponents that are free fit parameters that are derived empirically to get the best fit to the observed geomagnetic activity response. The recent work by Lockwood et al. (2019a) highlights a serious problem with most previous coupling function studies that have generally neglected the effect of gaps in the interplanetary data series on the grounds that they occur at random and so their effects will average out. Lockwood et al. use MonteCarlo analysis by inserting synthetic data gaps at random into near continuous data to show that this is far from being a valid assumption and that the effect of data gaps is to add considerable noise into solar wind – geomagnetic activity correlation studies. This is true irrespective of the method used to deal with the data gaps (for example interpolation, piecewise removal of geomagnetic data, or simply ignoring their effects). This raises the potential for “overfitting” which is a serious problem in multiple regression analysis of geophysical time series that have internal noise: it is a recognized pitfall in areas where quasichaotic behaviors give large internal noise such as climate science (e.g., Knutti et al., 2006) and population growth (e.g., Knape & de Valpine, 2011) but had not been considered in solarwind/magnetosphere coupling studies. Overfitting occurs if a fit has too many degrees of freedom which allows it to fit to the noise in the training subset, and hence is not robust in general. Including all of the factors with their own weighting factors and/or exponents can result in extremely good fits that can, nevertheless, give details that are statistically meaningless as each additional fit parameter reduces the statistical significance of the correlation. Such fits can have limited, and in extreme cases, zero predictive capability because they have fitted noise rather than the signal. The addition of noise by neglecting the effect of gaps in interplanetary data (which before 1995 were common and often of long duration) means that overfitting is a serious problem. Another complication is that the relative performance of different coupling functions depends strongly on the data averaging timescale and the averaging timescale used to generate the best fit exponents (Finch & Lockwood, 2007).
As a result of these considerations, we here use just one coupling function, P_{ α }, that has just a single free fit parameter, the coupling exponent, α, which estimates the power input into the magnetosphere, using the theory of Vasyliunas et al. (1982). This theory is based on the fact that the dominant energy flux in the solar wind is the bulk flow kinetic energy flux of the particles (and not the Poynting flux assumed by the muchused but flawed epsilon parameter). The fraction of this energy flux that is converted into Poynting flux by currents flowing in the bow shock, magnetosheath and tail magnetopause is taken to be the (necessarily dimensionless) factor M_{A} ^{−2α }, where M_{A} is the Alfvén Mach number of the solar wind and α is an unknown coupling exponent. The value of α could vary from zero (which would mean all incident power could enter the magnetosphere if the orientation of the interplanetary magnetic field, IMF, were favorable) and a large positive value (which would mean a vanishingly small fraction could enter the magnetosphere). Typically, values of α between 0.35 and 0.5 have been derived from the fits to data which, for a typical M_{A} of 10, means that 5–10% of solar wind power is available to enter the magnetosphere. The coupling function P_{ α } contains terms in solar wind mean ion mass, number density and velocity and the IMF orientation, but the exponents for each are all selfconsistent, all being set by the single fit parameter α and the theory. (Note that, in some studies the exponent of the IMF orientation factor has been treated as a separate fit variable, but we employ the procedure recommended by Vasyliunas et al. that allow it to be computed selfconsistently from the data once the optimum α has been determined). Finch & Lockwood (2007) found that the optimum coupling exponent α used in generating P_{ α } depended on averaging timescale, a result that implied that there was at least one other mechanism at work and that P_{ α } was failing to capture all of the relevant physics. However, Lockwood et al. (2019a) have shown that this variation in α was an artefact caused by data gaps and that when steps are taken to minimize the effect of such data gaps, α is effectively constant on all timescales from 1 min up to 1 year.
Figure 4 shows scatter plots of ap (top), the new homogeneous aa index, aa_{H} (Lockwood et al., 2018a, b) (middle), and am (bottom) as a function of the power input into the magnetosphere, P_{ α } (normalized by dividing by its average for the whole interval, 〈P_{ α }〉_{all}, in order to cancel some constants). The left hand panels are for daily means and the right hand panels for annual means. We use daily means because at shorter timescales the lag introduced by substorm growth phases becomes a significant factor (Lockwood et al., 2019a). For aa, aa_{H} and am the best fit coupling exponent (at all timescales) is α = 0.44, whereas for ap it is α = 0.48 (Lockwood et al., 2019a). The correlation coefficients for the daily means Ap, Aa_{H} and Am with daily means of P_{ α }/〈P_{ α }〉_{all} are very high, being 0.866, 0.893 and 0.923, respectively. The rootmeansquare (RMS) fit residual for a linear regression of all the data, ε, was also computed, giving ε/〈Ap〉_{all} = 0.570 for Ap, ε/〈Aa〉_{all} = 0.401 for Aa and ε/〈Am〉_{all} = 0.342 for Am. Hence both these metrics give best agreement for am, and worst agreement for ap. We tested the significance of the difference between the correlations using the MengZ test (which allows for intercorrelations between the datasets (Meng et al., 1992)) and found that the pvalue for the null hypothesis that they are actually the same was undetectably small. Hence the agreement with the coupling function is significantly better for am than for aa and that for aa is significantly better than that for ap on this 1day averaging timescale. For the traditional aa index, the correlation was 0.883, which is slightly lower than for aa_{H} but the MengZ test gives that the pvalue for the null hypothesis that these two correlations are the same is only 5 × 10^{−5}. Thus the improvement of aa_{H} over aa is small but statistically highly significant.
Fig. 4 Scatter plots of geomagnetic indices as a function of normalized power input into the magnetosphere computed from nearEarth solar wind observation, P_{ α }/〈P_{ α }〉_{all}, where the average is over the full period considered (1995–2017, inclusive). The left hand panels are for daily means, the right hand panels for annual means. The top panels are for the ap index, the middle for aa index, the bottom for am index. For the daily data, linear regression fits are shown for: (red line) 91 days around the June solstice; (blue line) 91 days around the December solstice; and (orange line) 91 days around either equinox). For annual means the cyan lines are linear regression fits for all data. The number of valid daily P_{ α } data points is N = 8375 (an availability of 99.7%) and for annual means is N = 23. The bestfit coupling exponent used to generate P_{ α } is α = 0.44 for am and aa and α = 0.48 for ap. The linear correlation coefficients, r, and the Root Mean Square (RMS) linear fit residual ε (as a ratio of the overall mean value of the index) are given in each panel. 
For the annual means shown in the right hand panels, the ranking order of the correlations is reversed. In this case, the correlations for ap, aa_{H} (and aa) and am are 0.992, 0.988 and 0.987. These correlations are exceptionally high and differences are small. The MengZ test gives pvalues against the null hypothesis that they are the same of 0.314 for ap and aa_{H} and of 0.371 for ap and am. Hence the difference between the correlations for ap and aa_{H} is just significant at the 1 − σ level but it is not quite significant at the 1 − σ level for ap and am. The RMS residuals (as a ratio of the overall mean value) are lowest for aa, but higher for am than for ap. In conclusion, both metrics show that am outperforms ap in daily averages but is outperformed by ap in annual averages.
The linear regression lines shown indicate why this is the case. For annual means (righthand panels) the cyan lines are linear regression fits to all data but for the dailyaveraged data (lefthand panels) the data have been linearly regressed in three subsets: (1) 91 days around the June solstice (giving the red line); (2) 91 days around the December solstice (blue line); and (3) 91 days around either equinox (orange line). In Figure 4a the regression lines for the three seasons are different, with the Ap values for equinox at a given P_{ α }/〈P_{ α }〉_{all} being larger, whereas for northern hemisphere winter they are smaller. For Aa and Am these differences are much smaller. The annual and semianual variations in the response of aa and am are real and in a later publication we will show that they are highly significant features of magnetospheric behaviour, however they are both exaggerated in ap. On the other hand, when we average out these 6 and 12 month periodicities by taking annual means, ap performs better than the other two indices.
Figure 5 explains the seasonal variations of the correlations shown in the left hand panels of Figure 4 by showing the semiannual variations in the mean values of the geomagnetic indices and their best fit of the corresponding variation in P_{ α }. The power input P_{ α } shows a clear semiannual variation with peaks at the equinoxes, the September peak being somewhat more pronounced than the March one. There is a slight difference between the solstices with the June minimum being slightly deeper than the December one. This variation results almost completely from the sin^{4} (θ/2) term in P_{ α } (where θ is the IMF clock angle in the Geocentric Solar Magnetospheric, GSM, frame of reference) and is caused by the RussellMcPherron effect of the Earth’s dipole tilt on IMF orientation in the GSM frame (Russell & McPherron, 1973). Figures 4 and 5 use daily means to avoid complications associated with the lag of the geomagnetic response due to the variable length of substorm growth phases. Lockwood et al. (2016) show that for averaging timescales greater than 12 h the FUT response pattern due to the RusellMcPherron effect becomes “axial” in form, i.e., showing the equinoctial peaks but no UT variation. Note that on shorter timescales the FUT pattern of geomagnetic response becomes “equinoctial” in form (best demonstrated by the am index, but can also be detected in aa_{H}, Lockwood et al., 2018b) whereas P_{ α } shows the RussellMcPherron form (Lockwood et al., 2016) which confirms that the geomagnetic indices are not responding to the “directlydriven” system. This difference in the FUT patterns is therefore associated with the “storagerelease” system to which midlatitude range indices primarily respond (Finch et al., 2008; Lockwood, 2013). Hence the origins of the equinoctial pattern must be associated with the variable lags cause by the durations of substorm growth phases.
Fig. 5 The annual and semiannual variations in geomagnetic indices and estimated power input into the magnetosphere, P_{ α }, for coincident data from 1995 to 2017, inclusive. In each panel the coloured line shows mean values of daily means of the geomagnetic index in 30 equalwidth bins of timeofyear, F, smoothed with a 3point running mean. The black line is the bestfit variation of the nearcontinuous P_{ α } data for the same interval processed the same way. (a) is for the Ap index; (b) is for the Aa_{H} index, and (c) is for the Am index. In each panel, two goodness of fit metrics are given: the correlation coefficient r and the Root Mean Square (RMS) fit residual, ε, as a ratio of the overall mean value of the index. 
Figure 5 shows that the index that most closely reflects the semiannual variation in P_{ α } is Aa_{H}, for which the fits are, surprisingly, slightly better than for Am. The fits are not as good for Ap, which exaggerates the September equinox peak and has a deep minimum in December. Hence Figure 5 explains the seasonal differences in the regression fits in Figure 4 for Ap, and why they are smaller for Aa_{H} nor Am.
2 Analysis
Finch (2008) employed the concept of the locationdependent magnetometer station sensitivity, s, defined simply for any given type of singlestation geomagnetic activity measure g by
(1)where I_{S} is a measure of the input solar forcing (which includes the effects of both induced currents in nearEarth space driven by solar windmagnetosphere coupling and of conductivity due to ionizing EUV and Xray radiations from the Sun or particle precipitations). Finch defined s to be a function of only the instrument coordinates because instrument and local site characteristics are accounted for by other intercalibration procedures. By taking ratios of g seen simultaneously at many pairs of different stations, the I_{S} factor is cancelled and the ratios of the station sensitivities is known. Note that this concept is the same as that introduced by Bartels (1949) and that is still used today in the compilation of the ap (and kp) index: this is because Bartels’ lookup tables used to convert K data from a given station (XXX) into what Niemegk (NGK) would have seen (the K_{S} value) are tables of average empirical values of s_{NGK}/s_{XXX} (as a function of UT, F and activity level). If the data from different stations are combined into a geomagnetic index using linear mathematics, then the sensitivities are similarly combined. For example, if the g data from N stations are averaged together with weighting functions ω to give a planetary index G,
(2)where S is the sensitivity of the index which is the weighted mean of the station sensitivities, s_{i}. From comparisons of the ratios for many pairs of stations, Finch (2008) derived a functional form for computing the sensitivity of a station as a function of its geographic coordinates, date, timeofyear and timeofday:
A and B are constants, χ is the solar zenith angle (a function of location, UT and F), T is the MLT of the station (in hours – also a function of location, UT and F and, on long timescales, of the intrinsic geomagnetic field), F is the fraction of the year and F = F_{1} at the spring equinox (taken to be 100/365.25 for the northern hemisphere and 283/365.25 for the southern hemisphere). Lastly, m is a normalising factor that ensures that the average value of s, over all UT and all timesofyear (F) and activity levels, is unity for a given station and year: it is used to retain calibrations that allow for instrument characteristics and local site effects.
The BiotSavart law states that the field disturbance at an observatory O, ΔB_{o}, is proportional to the integral over space of J_{np}/r ^{2} for all points P, where J_{np} is the current density at P normal to the line OP and r is the distance OP. The inversesquare dependence on r means that there can be a range of contributions to the observed signal at a midlatitude station from large variations in J_{np} in the auroral oval (i.e., at larger r) to smaller fluctuations in J_{np} more local to the observatory (at smaller r). It is well known that substorm electric fields can penetrate the shielding provided by the region2 fieldaligned currents (that connect to the ring current) and so give “bays” in midlatitude magnetometer records (e.g., Caan et al., 1978; Kikuchi et al., 2000). Hence the conductivities of the ionosphere over the observatory can influence the local J_{np} and hence ΔB_{o}. On the other hand, the large and variable currents flowing along the midnightsector auroral oval during substorms will also have an effect that will depend strongly on how close this sector of the auroral oval is to the observatory. The first term on the right of equation (3) allows for the effect of solar zenith angle χ on the ionospheric conductivity over the station due to solar EUV and Xray radiation and thus depends on the station’s geographic coordinates, the UT and the timeofyear, F. If the Sun is below the horizon, χ is set to (π/2): hence the coefficient A controls the extent to which the effect of dayside conductivity at a given χ is enhanced over residual nightside values. Note that there are small changes to the precise formulation of Finch (2008), who used a cos^{0.5}(χ) dependence, as predicted by Chapman productionlayer theory and as also used in a great many prior applications. However, Ieda et al. (2014) have shown that a conductivity dependence on cos^{0.7}(χ) fits better with observations and is also predicted by theory when the upward gradient of the neutral atmospheric scale height is accounted for. Using the conductivity over the observatory is an approximate parameterisation as there will be contributions to the total ΔB_{o} that arise from currents that are between the observatory and the auroral oval.
The second term on the righthandside of equation (3) is the station’s sensitivity due to its distance from the location of peak response, which is at an MLT of T* in the midnight sector. The sine term in equation (4) is used to model the known earlier onset of enhanced substorm activity in summer. Equation (4) yields T* of 1 h MLT and 22 h MLT for the winter and summer solstices, respectively. This is based on the survey of midlatitude station responses to substorm expansion phases by Finch (2008) and agrees well with the results of Liou et al. (2001), who found substorm onset was typically at T = 22 h in summer but 23.5 h in winter. Similar behaviour was deduced by Wang et al. (2007). We note that we are most interested in the MLT where auroral electrojet currents have peak effect on midlatitude K indices: this is close to, but not the same as, the MLT of substorm onset (Clauer & McPherron, 1974; Chu et al., 2014).
In this paper, the solar zenith angle at a given station is computed as a function of time (year, fraction of year, F, and UT) using an ephemeris that gives the solar declination at that time. The MLT for a given UT is computed for that date using the IGRF15 model of the geomagnetic field (Thébault et al., 2015).
Finch (2008) assumed that the factors A and B were constants and had considerable success in modelling the average response of different stations and indices. However, there are reasons to also think that the relative importance of the two terms in equation (3) might change systematically with the level of geomagnetic activity. Firstly, particle precipitation fluxes are higher during enhanced activity over a wide range of locations (including midlatitudes (e.g., Shiokawa et al., 2005)), which could mean that photoninduced conductivity is less important, and hence the dependence on cos^{0.7}(χ) is weaker: as a result, the factor A would be reduced at higher activity levels. Secondly, the auroral oval expands equatorward when activity is enhanced, making the second factor in equation (3) (associated with the spatial proximity of the auroral electrojet) more important. The factor B sets the amplitude of the diurnal variation seen by the station because of the variation in its proximity to the peak of the substorm current wedge. For these reasons the factors A and B are here both treated as functions of geomagnetic activity level.
Lockwood et al. (2018b) quantified the factors A and B by assuming that the large number of stations used to derive the am index, and their even longitudinal spacing, results in the sensitivity of am index, S_{am}, being always unity, independent of both UT and timeofyear, F. We here use a more exact and iterative procedure, but get results which are very similar to those found by Lockwood et al. (2018b). In general
(5)where the station of sensitivity s_{K} gives a Kindex value that transforms to a_{K} using the standard K2aK scale. We here quantify the geomagnetic activity level using the am index and divide it into eight (generally overlapping) activity ranges: 0 ≤ am < 10 nT, 10 ≤ am < 20 nT, 20 ≤ am < 40 nT, 30 ≤ am < 50 nT, 40 ≤ am < 60 nT, 50 ≤ am < 90 nT, 60 ≤ am < 110 nT, and am ≥ 70 nT for which the years 1959–2017 give N_{ b } = 58183, 51083, 40894, 22691, 13157, 8302, 10869, and 6060 samples, respectively, and the mean am values are 5.32, 13.96, 27.50, 37.71, 47.76, 63.56, 73.90, and 109.14 nT. The distribution of am values and these band limits are shown in Figure 6.
Fig. 6 Cumulative probability distribution (c.d.f, mauve line) and histogram of number of am samples in bins Δam = 1 nT wide (N, shown by the black line as N/N_{max}, where N_{max} is the maximum value of N) for all am data in the years 1959–2017 (inclusive). The grey bars give the eight overlapping am bands employed in this paper: 0 ≤ am < 10 nT, 10 ≤ am < 20 nT, 20 ≤ am < 40 nT, 30 ≤ am < 50 nT, 40 ≤ am < 60 nT, 50 ≤ am < 90 nT, 60 ≤ am < 110 nT, and am ≥ 70 nT which contain a numbers of samples N_{ b } of 58183, 51083, 40894, 22691, 13157, 8302, 10869, and 6060, respectively, and for which the mean am values are 5.32, 13.96, 27.50, 37.71, 47.76, 63.56, 73.90, and 109.14 nT. 
In the iterative procedure we adopt, we make the initial assumption of uniform S_{am} (S_{am1}(UT, F) = 1). This gives initial estimates of s_{K}(UT, F) for each of the eight am ranges for the Canberra and Hartland aa stations studied (we here denote these initial values as s_{K1}). We then obtain initial A and B estimates (A_{1} and B_{1}) for each of the am bins, using equations (2)–(4) by fitting modelled sensitivity ratios s_{K} /S_{am} to the a_{K}/am ratio values in the relevant am subset, using the fitting procedure in the UTF parameter space described by Lockwood et al. (2018b). The modelled sensitivities were computed for 24 UT values (1 h apart) and 365 F values (daily) and their ratios then averaged into the same bins as for the observational data (namely, 3 h width in UT and 1/20 width in F). From these initial A_{1} and B_{1} values we can use equations (3) and (4) to compute the corresponding sensitivity value for each of the am stations, and then use equation (2) to recompute the sensitivity of the network of stations used in compiling the index, S_{am}, giving a new estimate S_{am2}(UT, F). Using these values in equation (5) gives revised (first iteration) values of s_{K}(UT, F), values (s_{K2}). This loop was repeated until the RMS deviation of the modelled s_{K} /S_{am} values for CNB and HAD from the observed a_{K}/am ratios converged on a constant, minimum value (to within an adopted uncertainty level of 0.001%). This iterative procedure yielded A values of 0.6116, 0.2727, 0.2083, 0.2010, 0.2001, 0.2001, 0.2003, and 0.2000 for the eight am bins (in order of increasing 〈am〉) and B values of 0.2890, 0.3293, 0.3631, 0.3786, 0.3711, 0.3163, 0.2800 and 0.2797. This iteration allows us to compute the am index sensitivity values, S_{am}(UT, F), selfconsistently rather than assuming it is constant at unity (the assumption that was employed by Lockwood et al., 2018b).
The righthand columns of Figures 7 and 8 show the UTF plots of the observed ratios 〈a_{K}〉/〈am〉 for the two current aa stations, respectively Hartland and Canberra. The rows are for the eight am activity level bins shown in Figure 6, with the highest activity at the top and the lowest at the bottom. The lefthand column of both figures gives the UTF plots of the modelled sensitivity of the am index, S_{am}, and the middle panels in both figures give the corresponding plots of the bestfit modelled sensitivity, s_{K} /S_{am}.
Fig. 7 Analysis of the sensitivity of the Hartland (HAD) station. Timeofday (UT)/timeofyear (F) plots of: (left column) the modelled sensitivity for the am index, S_{am}, for the current stations and sector weighting functions; (middle column) modelled values of the ratio s_{HAD}/S_{am} where s_{HAD} is the sensitivity of the Hartland magnetometer station for measuring its a_{K} values, a_{HAD}; and (right column) means of the observed values of the ratio 〈a_{HAD}〉/〈am〉 = s_{HAD}/S_{am}. All data are for eight UT bins 3 h wide and 20 F bins 18.25 days wide over the years 1959–2017 (inclusive). The panels are for am ranges (from top to bottom) of: am ≥ 70 nT; 60 ≤ am < 110 nT; 50 ≤ am < 90 nT; 40 ≤ am < 60 nT; 30 ≤ am < 50 nT; 20 ≤ am < 40 nT; 10 ≤ am < 20 nT; and 0 ≤ am < 10 nT shown in Figure 6. The modelled values are based on the mean am in each band which equals, respectively, 109.14, 75.94, 63.56, 47.76, 37.71 27.50, 13.96, and 5.32 nT. Modelled sensitivities are computed at points 1 h apart in UT and 1/365 apart in F and then averaged into the same sized UTF bins (3 h by 0.05) as used for the observations. Note that the lefthand plots are colourcontoured using the 0.8–1.2 scale given by the lower colour bar while the modelled and observed s_{HAD}/S_{am} sensitivity ratios both use the 0.5–1.6 scale given by the upper colour bar. In all plots unity values are coloured yellow. 
Fig. 8 The same as Figure 7 for the Canberra (CNB) station, giving UTF plots of: (left column) the modelled sensitivity for the am index, S_{am}, for the current stations and sector weighting functions; (middle column) modelled values of the ratio s_{CNB}/S_{am} where s_{CNB} is the sensitivity of the Canberra magnetometer station for measuring its a_{K} values, a_{CNB}; and (right column) observed values of the ratio a_{CNB}/am = s_{CNB}/S_{am}. 
Figures 7 and 8 both show that the local ionospheric conductivity term is more important at low geomagnetic activity with a strong peak around the minimum of the solar zenith angle χ (at F = 0.5 and 14 UT for Hartland and F = 0 and F = 1 and 4 UT for Canberra). This is reflected in the large values A for the lowest activity levels. This peak becomes increasingly less pronounced with increasing am activity level (i.e., A decreases) and at the largest am the pattern is determined mainly by the distance of the station from the midnight auroral oval.
3 Response functions of geomagnetic indices
From the derived values of A and B for each am activity level bin, we can use equations (2)–(4) to compute the (FUT) response pattern of any midlatitude range indices that is compiled using a mathematical algorithm. Rather than show results for all eight am activity bins in every case, we here show just two illustrative ones, representative of low and moderately high activity levels. We choose 10 ≤ am < 20 nT for low activity and 60 ≤ am < 110 nT for high activity. We avoid the lowest am bin because the lowest row of Figure 8 shows that the fit is not always as good as it is for other panels: this is a sensitivity effect associated with the lowest activity level that can be detected by a single station, compared to that for the am or ap index; that limit being lower for the indices, by virtue of the averaging of data from a number of stations. This means that there are times (as in the case of the bottom row of Fig. 8) when the station is not detecting any activity yet the am index is: in Figure 8 (for Canberra in the southern hemisphere) this is particularly true in midwinter (F = 0.5). We avoid the largest activity bin (am ≥ 70 nT) because it is based on the smallest number of samples. Instead we use the second largest and the second smallest activity bins as examples.
In all cases, the response functions are computed from the fitting procedure described in Section 2: the modelled sensitivities were computed for 24 different UTs (1 h apart) and 365 F values (daily) and for plotting of the UTF patterns these were then averaged into the same bins as for the observational data (namely, eight 3hour bins in UT and 20 18.25day bins of F). We also made 3hourly means of the sensitivity (for the same UT bins as the observations, i.e., 0–3 UT, 3–6 UT, etc.) and identified the maximum and minimum 3hourly value of each UTF pattern (S_{max} and S_{min}, respectively) as well as the mean (〈S〉 and standard deviation (σ_{S}) of the 160 values (eight UT values and 20 F values) in each pattern. As well as taking the maximum percentage deviations of from the mean (S_{max} and S_{min}), we quantify the percentage standard deviation V = (100σ_{S})⁄(〈S〉) for each am activity band. To evaluate the average behaviour, patterns were made for each of the eight am bands defined in Figure 6 and a weighted mean of the V values taken:
(6)where N_{ b } is the numbers of am observations in the band (given earlier).
We also computed the average response pattern for the index, S_{av}(UT, F) as the similarly weighted means of the eight patterns for the different activity levels
Table 3 summarises the results by comparing the largest positive and negative deviations of S_{av}(UT, F) for the tested indices as percentages of the mean, along with the metric V_{av} for various geomagnetic indices: results are given for both the modelling described above and from an empirical comparison with the am index.
Uniformity of average timeofday/timeofyear response, S_{av}(UT, F), of the various midlatitude geomagnetic range indices.
3.1 Modelled response functions of the an, as and am geomagnetic indices
Figure 9 compares the modelled response patterns (the UTF plots of index sensitivity) of the hemispheric an and as indices to that for am = (an + as)/2. These are all evaluated for the spatial distribution and weighting functions of stations for a selected example year which is 2014. At this time the IAGA codes of the stations in use are MGD, MMB, PET, POD, ARS, IRT, NVS, CLF, HAD, NGK, FRD, OTT, NEW, TUC, and VIC in the northern hemisphere and CNB, EYR, AMS, GNG, CZT, HER, PAF, AIA, PST, and TRW in the southern hemisphere (see Appendix A for the corresponding observatory names, locations and the intervals over which they were used to construct a given index). The computation procedure for deriving am and the longitudesector weighting functions is described at http://isgi.unistra.fr/Documents/am_LWFs_example.pdf.
Fig. 9 Timeofday (UT)/timeofyear (F) plots of the modelled sensitivity of (top row) the northernhemisphere an index, S_{an}; (middle row) the southernhemisphere as index, S_{as}; and (bottom row) the global index am = (an + as)/2, S_{am}. The left hand plots are for relatively high geomagnetic activity (defined as am = 74 nT, the mean of the 60 ≤ am < 110 nT band) and the right hand plots are for relatively low geomagnetic activity (defined as am = 14 nT, the mean of the 10 ≤ am < 20 nT band). All plots are for the stations and longitudinal sector weighting functions used in 2014. 
At both the moderately high and the low activity level examples shown (in the left and right and columns of Fig. 9, respectively) and in both hemispheres, there is a clear seasonal variation with enhanced index sensitivity in summer, S_{an} being largest around F = 0.5 and S_{as} being largest around F = 0 (which is the same as F = 1). If the longitudinal distributions of stations were ideal, the contours would all be vertical in these plots as there would be no UT variation. This is not quite the case, as S_{an} is slightly but consistently larger around 18 UT, and slightly lower around 04 UT at all times of year and all activity levels. S_{as} shows the converse behaviour, being similarly larger around 04UT and lower around 18 UT. Because many features in the S_{an} pattern are the converse of those in the S_{as} pattern, they are averaged out in am and the S_{am} patterns shown in the bottom panels of Figure 9 are much more uniform, especially for high geomagnetic activity. For 3hourly values, the largest value is S_{max} = 1.0284 for the lower activity range with a minimum of S_{min} = 0.9741 and hence the largest percentage deviations from 〈S_{am}〉 = 1, are 100 (S_{max} − 〈S_{am}〉)/〈S_{am}〉 = 2.8% and 100 (S_{min} − 〈S_{am}〉)/〈S_{am}〉 = −2.6%, respectively. Hence, in this case (for which 〈am〉 = 14 nT) the timeofyear/timeofday response pattern for am is uniform to within maximum deviations of ±2.8%. For the higher activity range (〈am〉 = 74 nT) the corresponding extrema are much smaller, being +0.21% and −0.25%. The patterns are complex and general features very weak but include very slightly stronger annual variation at 10–15 UT and a band of very slightly higher S_{am} around 5 UT. For weighted means of the average UTF response pattern over all activity levels, S_{av}(UT, F) as defined by equation (7), the extrema are +1.4 and −1.2%, as given by the top row in Table 3. The other metric that we use to quantify the flatness of the average UTF response pattern is V_{av} defined by equation (6). Table 3 shows that the modelled V_{av} value for the am index is 0.65%.
3.2 Modelled and observed response functions of the aa geomagnetic index
Figure 10 shows the F–UT response patterns for aa, S_{aa}(UT, F) for the same high and lowactivity ranges of am used in Figure 9. Note that the colour scale range is considerably expanded in Figure 10 compared to Figure 9 because S_{aa} shows much greater deviations from unity than S_{am}, as is to be expected because aa is derived from data from just two stations. The response pattern will depend on which pair of stations is employed to generate aa. In Figure 10 various years are chosen as examples of longlived aa station combinations: 2010 (contributing stations CNB and HAD), 1970 (TOO and HAD), 1930 (TOO and ABN) and 1890 (MEL and GRW). Table 4 shows that the minimum of the 3hourly aa sensitivity, S_{aa}, is reasonably constant for these years being −18.5% in 1890 and −16.7% in 2010 for low activity and −18.9% in 1890 and 16.1% in 2010 for high activity. The corresponding maximum values are almost constant at 12.8% for low activity and 12.0% for high activity. As shown in the next section, these extreme deviations from unity are actually smaller than those for Kp (ap) because, although aa is compiled using just two stations, those stations have been chosen to give as much cancellation of the stations’ diurnal and annual response sensitivity variations as possible (being in opposite hemispheres and about 10 h apart in local time).
Fig. 10 Timeofday (UT)/timeofyear (F) plots of the modelled sensitivity of the aa index, S_{aa}, for various years. The lefthand and righthand columns are for relatively high and low geomagnetic activity (defined as for Fig. 9), respectively. Plots are for: (a) and (b) 2010; (c) and (d) 1970; (e) and (f) 1930 and (g) and (h) 1890. 
Maximum and minimum percentage deviations of modelled 3hourly index sensitivities, S, from unity for selected years and middle and low geomagnetic activity levels.
The “checkerboard” response pattern for aa seen in Figure 10 is present for both low and high activity and is found in the aa data, as demonstrated by Figure 11. This plot shows the sensitivity of the aa index S_{aa} = (〈aa〉/〈am〉)S_{am}, where the ratio 〈aa〉/〈am〉 is taken from data for all available years (1959–2017). Note that the modelled sensitivity of the am index, S_{am}, is very close to unity at all F and UT and so the patterns for 〈aa〉/〈am〉 are almost identical to those shown. The lefthand and righthand panels are for high and low geomagnetic activity respectively. To get enough samples for this plot, high activity is defined in Figure 11 as am ≥ 40 nT both when averaging the observed 〈aa〉 and 〈am〉 values and when calculating the model sensitivity of am, S_{am}.
Fig. 11 Timeofday (UT)/timeofyear (F) plots of the observed sensitivity of the aa index, S_{aa} = (〈aa〉/〈am〉) × S_{am}, for the years 1959–2017. The lefthand and righthand columns are for high and low geomagnetic activity respectively. The low activity range is 10 ≤ aa < 20 nT as used in Figures 7 and 8, but to get sufficient samples, high activity is here defined as am ≥ 40 nT both when averaging the observed 〈aa〉 and 〈am〉 values and when calculating the model sensitivity of am, S_{am}. 
Table 4 shows that the peak deviations from unity are actually very similar in the two activitylevel cases. Note also that the response patterns for the hemispheric aa indices aa_{N} and aa_{S} are given, for the current pairing of aa stations, by Figures 7 and 8.
Figure 10 shows that there are no strong differences in the modelled aa sensitivity patterns for the years studied. This means that the effects of station changes and of secular drift in the magnetic field on the response pattern for aa have been small. There is an important point to clarify here about the effect of secular change in the intrinsic geomagnetic field. All the UTF response patterns in this paper have been normalised to unity. This means that any changes because of the drift in the average geographic latitude of the auroral oval will not be included. These effects have been studied by Lockwood et al. (2018a) and a new “homogeneous” aa index, aa_{H}, presented that makes allowance for this drift. The patterns presented in Figure 10 are the timeofyear/timeofday variations around the annual mean, and although the annual mean estimates will have varied because of secular drift, the patterns will not have changed much at all. However, there is a small secondary effect of the secular change in the field that is included in the patterns presented and this is in the UT at which midnight MLT occurs, which has an effect via the term T in equation (3). The fact that there is no detectable effect in Figure 10 shows that this effect on the response pattern is very small.
The fourth row of Table 3 gives the largest percentage deviations of the average pattern for the aa index and the V_{av} metric, both modelled and from data (by comparison with simultaneous am data).
3.3 Response function of the ap (kp) geomagnetic index
Our model cannot be employed to analyse the response pattern of the ap and Kp indices. This is because equation (2) requires the index to be compiled by linear mathematics so that the solar forcing term can be isolated and cancelled. In the compilation of Kp, the K_{S}/K factor used is a complex function of K and hence the process is achieved by a lookup table, rather than an analytic function and is also nonlinear as the lookup table used depends on the activity level. Hence we cannot use the same model analysis as applied above to am and aa. However, we can apply the databased approach (comparison with the am index) that was used in the previous section for aa. As for aa, we can estimate the response function for ap using the equation
Figure 12 compares the am and ap index response patterns over the full interval over which we have both, namely 1959–2017 (inclusive). The middle column of Figure 12 presents the UTF patterns of the ratio 〈ap〉/〈am〉 (again for eight 3hourly UT values, and twenty bins in F that are ΔF = 0.05 wide, and the eight am activity level bins shown in Fig. 6 and as employed in Figs. 7 and 8). The left hand column gives the modelled am sensitivity patterns, S_{am}(UT, F). Using these patterns, we can estimate the ap sensitivity S_{ap} using equation (8). The results are shown in the righthand column of Figure 12. Note that the patterns for S_{ap} are very similar indeed to those for (〈ap〉/〈am〉) in the middle column because S_{am} is so close to unity at all UT and F. Figure 12 shows a consistent pattern in S_{ap}(UT, F) with response at 0–8 UT being greater in northern hemisphere winter but that at all other UT being greater in northern hemisphere summer. At low activity levels, the 0–8 UT variation tends to dominate but the 8–24 UT variation increasingly dominates with increasing activity level.
Fig. 12 Timeofday (UT)/timeofyear (F) plots of: (lefthand column) the sensitivity of the am index, S_{am}, for 1988 (the midpoint of the interval 1959–2017 for which am data are available); (middle column) observed 〈ap〉/〈am〉 and (right column) S_{ap} = (〈ap〉/〈am〉)S_{am}. Data are for 1959–2017 (inclusive) and the am ranges defined in Figure 6. 
The values of S_{am} are very uniform and close to unity (note that the S_{am} scale is the lower one in Figure 12 and covers a smaller dynamic range than for the middle and right columns by a factor of six). This is particularly true for moderate and high activity. However, even at low activity, the differences between the pattern of 〈ap〉/〈am〉 and the corresponding pattern of (〈ap〉/〈am〉)S_{am} cannot be detected. The ap index is often treated as homogeneous but, in reality, the network of stations used has changed considerably even over the interval 1959–2017. Given the variability that this introduces, the use of the iterativelyderived factor S_{am} is probably not justified and in the remainder of this section we assume S_{am} is unity and just look at the ratio (〈ap〉/〈am〉). This has the advantage of making the analysis purely empirical (instead of the empiricalmodel mixture involved in S_{ap} estimates).
Figure 13 studies the relationship between the ap and am indices. The grey dots in Figure 13a are 3hourly values (ap and am) and the cyan points are 8point (1 day) running means (Ap* and Am*). The black dots are means (with error bars that are ±1σ) taken in bins of am that are Δam = 10 nT wide (only bins with six or more samples are shown). The mauve line shows am = ap and the plot shows that ap is consistently smaller than am. In addition, the difference grows with activity levels so that Ap* values are increasingly smaller than Am* values for larger activity. Lockwood et al. (2019b) point out that this is a significant factor when geomagnetic storms during the recent grand solar maximum, as defined using the Ap* data, are compared to those seen during 1868–1932 in the Aa* data or to the estimates for during the Carrington storm (that are often quantified by a proxy equivalent for Aa*). The overall mean of am is larger than that for ap by a factor η = 1.5339.
Fig. 13 (a) Grey points shows ap index values as a function of simultaneous am values for 1959–2017 (inclusive). Cyan points show the scatter plot of the corresponding 24hour (8point) running means, Am* and Ap*. The black points are means of ap (with error bars of plus and minus one standard deviation) as a function of means of am in am bins of width Δam = 10 nT (only means for bins containing six or more samples are shown). The mauve line is the ideal case for which ap = am. (b) and (c) Annual variations shown by means in fractionofyear (F) bins of width ΔF = 0.05. (b) shows (in black) the annual variations of 〈am〉 and (in mauve) 〈ap〉 × η, where η is the ratio 〈am〉_{all}/〈ap〉_{all} for means taken over the whole dataset. (c) shows the annual variation of the ratio of ap/am. (d) and (e) Diurnal variations shown by means for the eight UT values of both indices using the same color coding as in (b) and (c). 
Note that to a large extent taking 24hour running means reduces the effect of the UT variation in the sensitivity of any one 3hourly index; however, it does not remove it completely. This is because the 3hourly index value will generally have some variation within each 24hour interval and the phasing of this variation, relative to the UT variations in the index sensitivities, will influence the Ap* and Am* values and influence them differently.
Figure 13b and c compares the annual variations, by taking mean values in 20 bins of the fraction of the year, F, that are ΔF = 0.05 wide (18.25 days). The black dots and line show means of am, the mauve dots and line means of η × ap. It can be seen that the annual variations are very similar, but that the wellknown semiannual variation (Cortie, 1912, Chapman & Bartels, 1940; Russell & McPherron, 1973; Cliver et al., 2002; Le Mouël et al., 2004) is proportionally slightly larger, on average, in ap than in am. The average variation with F of the ratio of 〈ap/am〉 is shown in Figure 13c. Figure 13b and c show that ap also has an annual variation as it underestimates geomagnetic activity during northern hemisphere winter, which is perhaps not surprising given the dominance of northern hemisphere stations and the lower ionospheric conductivities in winter. Figure 13d and e presents the same analysis for diurnal variations. Figure 13d shows that there is a slight diurnal variation in am and a larger one in ap. The diurnal variation in the ratio of the two indices is shown in Figure 13e.
The average S_{ap}(UT, F) pattern gives percentage deviations of 11.2%, 25.4% and −21.6% for σ_{S}, S_{max} and S_{min}, respectively. Table 3 shows this is the least uniform of the indices tested in this paper. In Section 4 we develop and test an empirical correction for this.
3.4 Response function of the homogeneous aa index, aa_{H}
For comparison, it is useful to apply to the new homogenized aa index, aa_{H}, the same tests as used in the last section to study the constancy of the ap index. It is not instructive to test this index against the sensitivity model because the model is used to correct aa and give aa_{H}. Figure 14 compares with the am index and so corresponds to Figure 12. The lefthand column again gives the S_{am}(UT, F) patterns, the middle column (〈aa_{H}〉/〈am〉)_{(UT,F)} and the right hand column S_{aaH} = (〈aa_{H}〉/〈am〉)_{(UT,F)} × S_{am}(UT, F). The fluctuations around unity in the middle and right column are of smaller amplitude than for ap and in character are more of a regular UT variation (at all F), which means that averaging over a calendar day or making 8point running means (to give Aa_{H} and Aa_{H}*, respectively) will further reduce the effect of the nonuniformity of S_{aaH}(UT, F).
Fig. 14 The same as Figure 12 for the homogenized aa index, aa_{H}. Data are for 1959–2017 (inclusive) and the am ranges defined in Figure 6. 
Figure 15 corresponds to Figure 13 for the aa_{H} index. The agreement of aa_{H} and am (grey dots) and of Aa_{H}* and Am* (cyan dots) in part (a) are both good and linear. In parts (b) and (c), the annual variations of 〈aa_{H}〉 and 〈am〉 with F are very similar (with aa_{H} being just very slightly lower values round F = 0.5 and very slightly higher around F = 0 and F = 1). Parts (d) and (e), show there is a UT variation in average aa_{H} that is somewhat greater than that in am. The mean S_{aaH}(UT, F) pattern gives percentage deviations of 11.20%, 25.4% and −21.6% for σ_{S}, S_{max} and S_{min}, respectively. Table 3 shows that the average pattern gives percent deviations of 5.54%, +12.6% and −12.3% for σ_{S}, S_{max} and S_{min}, respectively for aa_{H} which is considerably better than for the uncorrected aa data and therefore also better than for ap.
4 A corrected ap index, ap_{C}
In this section, we present a method for empirically correcting the ap index to allow for its nonuniform response function. We do this using the same basic principle as introduced by Bartels and still used today to compile ap, namely activitydependent lookup tables to correct the timeofday/timeofyear response. A difference is that we are applying it to an index and not the data from a single station. The tables are provided in the form of arrays of values that can be interpolated to give the multiplicative correction factor required for a given ap, UT and F. We recommend use of Piecewise Cubic Hermite Interpolating Polynomial, (PCHIP) interpolation because it maintains smooth changes in gradient through the data points. Tests show that, although in general that it can sometimes give excessive oscillation between points, in this case it gives more accurate value than a linear interpolation).
Activity level in previous sections was quantified using am index bands, but that cannot be applied here; instead, ap must be used as the aim is to correct ap even at times when am was not measured. Figure 16 studies the distribution of ap values. The blue line gives the c.d.f of the (quantised) 3hourly ap values and the mauve line the (almost continuous) Ap* values. The vertical grey and white bands divide the Ap* distribution into 20 percentiles, each containing N_{20} = 8612 samples. The separation of these percentiles is smaller than the separation of the ap levels below 22 and so all these levels contain considerably more than N_{20} samples. The ap = 22 level contains very close to N_{20} samples level and the ap = 27 and ap = 32 levels combined give slightly more than N_{20} samples, but we have to combine all ap levels greater than 39 to get more than N_{20} samples in the tail of the distribution. Hence we derive corrections for ap levels of 2, 3, 4, 5, 6, 7, 9, 12, 15, 18, 22, 27 combined with 32, and 39 and greater. We also deal with each of the eight UTs of the ap index separately. For each of these UTap combinations we then compute the empirical correction factor f_{c}(UT, F, ap) = 〈ap〉/〈am〉 in 20 equalwidth bins in F. These can be used to convert an ap value for a general F and UT using PCHIP interpolation between the relevant 20 f_{c}(F) values. Figure 17 presents contour plots of daily f_{c}(ap, F) values for the eight UT ranges.
Fig. 16 The distributions of ap and Ap* values. The mauve and blue lines are the c.d.f.s of, respectively, Ap* and ap values for 1959–2017, inclusive. The black line is the histogram of N/N_{max}, where N is the number of Ap* samples in bins 0.5 wide and N_{max} is the maximum value of N. The vertical white and grey bands divide the distribution of Ap* into the 20percentiles, each containing ΣN/20 = 8612 samples. 
Fig. 17 Plots of fits to the ratio of am/ap as a function of time of year F and ap value for the 8 UT’s of the ap and am index samples, derived as described in the text so that they can be used to generate the corrected ap index, ap_{C} from ap. 
This allows us to turn every ap value into a corrected value ap_{C} = ap × f_{c}(UT, F, ap). Figure 18 compares the ap_{C} and am values, in the same format as Figures 13 and 15. Part (a) shows that there is very good agreement between ap_{C} and am and between Ap_{C}* and Am*. Parts (b) and (c) shows that the average annual variations of ap_{C} and am are very similar and parts (d) and (e) even their UT variations are well matched. The mean S_{apC}(UT,F) pattern gives percentage deviations of 1.78%, 5.0%, and −4.6% for σ_{S}, S_{max} and S_{min}, respectively. Table 3 shows this is a major improvement compared to ap.
Fig. 18 Same as Figures 13 and 15 for the corrected ap index, ap_{C}. 
The coefficients needed to implement the correction to ap are given in the Supporting Information file attached to this paper. The metadata given in the header to that file also gives some MATLAB program code that converts ap into ap_{C}.
5 Conclusions and recommendations
5.1 Conclusions
We have presented analysis of the timeofday (UT)/timeofyear (F) response patterns of various planetary geomagnetic indices. In general, responses depend on the level of geomagnetic activity because the effects of EUV/Xray generated solar conductivity local to the observatory dominate the response at low geomagnetic activity, but the effects of the great circle distance between the observatory and the midnight sector auroral oval dominate at high geomagnetic activity.
The diurnal variations in the hemispheric an and as indices are small because of the use of station groupings that are as even in longitudinal separation as possible. On averaging to give the am index, the seasonal variations cancel to a very large extent and so the index sensitivity, S_{am}, is close to unity at all F and UT. This is valid for threehourly values to within extrema of ±2.9% at low geomagnetic activity, falling to just ±0.5% at high geomagnetic activity. As an overall average, the modelled (UTF) response pattern of am is constant to within a standard deviation of 0.65% with extreme deviations of +1.4% and −1.2%.
The ap (Kp) index station distribution is not uniform but the use of the K to Ks conversion tables makes allowance for this. These tables were constructed using data from a limited number of years when solar and geomagnetic activity were at moderate levels only. Hence it not that surprising that they may not be ideal over all the 1957–2017 interval tested here. The general similarity of the average annual and diurnal variations of ap and am in Figure 13 indicates that the tables are performing well in that they do not introduce major errors. The right hand column of Figure 11b, however, shows that there are spurious timeofyear variations in the response of ap that depend strongly on UT. For 0 < UT < 10 h there are peaks at almost all activity levels around F = 1 (northern hemisphere winter), whereas at all other UT the peak is around F = 0.5 (northern hemisphere summer). Averaging over all UT gives peaks in the sensitivity at the equinox at that can be seen in Figure 13c, which reveals that that ap exaggerates slightly the semiannual variation in geomagnetic activity. There is also a spurious net annual variation with values lower around F = 1 than F = 0.5. Note that Figure 4a is also consistent with this, showing the greatest average response in ap to solar wind forcing at equinox and a greater response around the June solstice than around the December solstice. Figure 13d shows there is also a persistent net diurnal variation in the ap response with a minimum at 9–12 UT. These average values however hide the fact that the ap response function is a complex and variable function of UT, F and activity level, as shown by Figure 11. We note that these spurious diurnal and annual variations in ap arise because this index employs concentrations of stations in certain regions (particularly Europe). But this, and making the data from all stations mimic the Niemegk reference station by converting to K_{S,} does also have advantages in noise suppression because one is averaging different estimates of the same thing and, when averaged over a whole year of continuous data (for which 〈S_{ap}〉_{ τ=1yr} = 1), the spurious variations with UT and F are largely averaged out – hence the ap index almost certainly provides the most reliable and accurate estimate on annual timescales. This is reflected in the response to solar wind forcing shown in the right hand panels of Figure 4 which reveals that ap performs slightly, but significantly, best on annual averaging timescales. However, we also note that the uneven pattern of S_{ap} will introduce some random sampling noise, depending on the time of year and UT at which the quasirandomlyoccurring largest interplanetary disturbances happen to hit Earth’s magnetosphere.
The sensitivity modelling indicates that the aa index response pattern has remained very constant over time and is comparable, and actually slightly better than, that derived empirically for ap. This is because the southern hemisphere data is given equal weight in aa, whereas it has much lower weight in ap.
Our analysis shows that the new “homogeneous” aa index aa_{H} (Lockwood et al., 2018a, b) performs considerably better than aa, having a flatter UTF response at all activity levels. Naturally, being based on just one station in each hemisphere it cannot match the performance of am, but it has the advantage (for studies of longterm change or requiring large sample numbers) of extending back to 1868 whereas am extends back to only 1959.
We have also presented a method for correcting the ap index for its uneven response pattern. The correction presented here uses all ap and am data since 1959, despite the fact that the network of stations used to generate the am index has undergone (relatively minor) changes in that interval and that used to generate the ap index has undergone considerable changes. This means that, in effect, we are accepting the tabular KtoK_{S} conversions inherent in ap for the purposes in deriving a correction factor to ap, f_{c}(ap, F, UT), but this is one step removed from accepting them for the purposes of compiling ap itself. Hence this is a first order correction. It would be possible to make a more detailed correction and analyse the response and associated f_{c}(ap, F, UT) factors for each of the various combinations of stations used to construct ap after 1959, as given by Appendix A. However, this would introduce other uncertainties caused by the reduction in the number of samples in each case. In addition, this would still not account for the several changes to the ap network made before 1959. The optimum correction to ap would be to correct the K indices for each station individually using the station sensitivity model and then average them together as this could be done in a consistent way for all the data (since 1932). However, this is a large task and well beyond the scope of this paper.
Table 5 quantifies the improvements to the aa and ap indices made by aa_{H} and ap_{C}, respectively, using am as the calibration standard. Note that in the case of aa_{H}, the correction is made through application of the station sensitivity model, whereas for ap_{C} it is purely empirical. The most revealing division of activity levels is to consider the lowest 20% of samples (as determined by am), the highest 20% and the remainder in the inter 20percentile range. The bottom two rows of Table 5 give the factor improvement brought about in the corrected indices: ideally these values should be infinite as am is used to correct the data and then used to test the corrected index. However, this is not going to be the case for a variety of reasons: in the case of aa_{H}, the main reason is that by averaging many stations am will always suppress geophysical and instrumental noise more effectively than aa which is based on just two stations; in the case of ap_{C}, the main reason is the effect of changes in the ap observation network. It can be seen that improvements, in terms of lowering σ_{S} (flattening the response function) are always in excess of three and that for extrema the improvement in S_{max} is always better than that in S_{min}. In both cases the improvement for the bulk of the distribution is always better than for the lower 20% and the upper 20%.
Standard deviations and maximum and minimum percentage deviations from unity of observed 3hourly index sensitivities, S, estimated assuming the am index is ideal for 1959–2017, for low, middle and high geomagnetic activity levels. The bottom two rows give the factor by which the corrected index is improved in terms of the uniformity of its response.
Because taking running means over 24 h averages out the UT dependence, the improvements in Aa_{H}* and in Ap_{C}* are not as great as for the 3hourly values aa_{H} and ap_{C.} Nevertheless, the ranking order of storm days in these indices can be considerably different from those for Aa* and in Ap*. The occurrence of storm days, as quantified by Aa_{H}* since 1868 will be the subject of another paper. Lockwood et al. (2019b) have studied the behaviour and ranking order of extreme events since 1932, as quantified using the Ap_{C}* index.
5.2 Recommendations
The major differences between the indices studied in this paper arise from the geographic distribution of stations used to compile them. The aa index uses just two stations in order to give a long data sequence. The am index has been designed to make the stations’ distribution as even as possible and so give the most even timeof day/timeofyear response that is allowed by the availability of land that is suitable for housing a magnetometer. On the other hand, the ap index is, and has always been, dominated by European stations and is constructed in a way that recalibrates all data to a European station (Niemegk): we have shown that because it is averaging more data that is similar (or is made similar), this gives it better sensitivity and more accurate annual means but an uneven timeofday/timeofyear response. Lockwood et al. (2018a, b) have presented a modelbased means for correcting the aa index, to give the “homogeneous” aa index aa_{H}, that allows for both longterm secular changes and the unevenness of the timeofday/timeofyear response pattern caused by the use of just two stations. In this paper, we have used empirical comparisons with the am index to generate a corrected ap index, ap_{C}, that allows for the unevenness of the timeofday/timeofyear response pattern of ap identified in this paper. Because of the differences between these corrections, their potential applications are different. The corrections to aa are based on physical model and employ parameters that can be projected into and measured in the future (the stations’ geographic and geomagnetic coordinates and the solar declination angle): hence we recommend that it is used for both reevaluation of past studies and for studies that extend into future. On the other hand, the ap corrections (to give ap_{C}) are empirical and based on comparison with past am index data and so will become increasingly unreliable with time for future data. Hence ap_{C} will be useful in reevaluation of some past work for which evenness of the timeofyear/timeofday response to solar wind forcing is important (for example studies involving the ranking order of the largest geomagnetic storms, Lockwood et al., 2019b, c) but it would not make sense to recommend its use for future work with that requirement, because as long as the am index is available it is the better option in this context. That is not to say, that the ap index is not of value in other contexts. For example, in annual means of the data, the timeofyear/timeofday response is averaged out and so long term trends are well monitored and for such applications keeping the same index (and keeping its compilation as homogeneous as possible) is important. We note in that ap extends back 27 years further into the past than am and so for studies where the subannual response is important, ap_{C} can be used to extend the am data sequence back a further 27 years to 1932.
Looking to any potential development of the networks of stations, it is important that any changes made enhance, rather than undermine, the strengths of the index in question. From the above, we would argue that in the case of aa the strength is longevity. In the case of ap it is part longevity and part sensitivity brought about by averaging many nearby stations and calibrating all station to Niemegk. For longevity, homogeneity of compilation is by far the most important consideration, meaning that both past and future station changes are undesirable. In the case of aa, the station sensitivity model can be used to minimise the effects of station changes, as does the scaling to Niemegk in the case of ap. Hence moving ap stations to make the geographic distribution more even (and make ap more like am) would not be desirable change as it would undermine the advantages that ap has. For am, the choice of stations and the method of their combination have been shown to be good in that it gives a very flat timeofday/timeofyear response pattern, which is the index’s most important strength. This is true at all levels of geomagnetic activity, except the very quietest which suggests a sensitivity issue. In terms of improving the network distribution for am, the major limitation is the availability of accessible land on which a magnetometer can be placed and operated. However, there may be advantages for some applications in increasing the numbers of stations in all the longitude sectors to give increased averaging out of local site and instrumentation effects and so increase sensitivity. However, so as not to disrupt the evenness of the response, one would want to make similar, matching, improvements in all sectors.
Acknowledgments
The authors are grateful to the staff of The ISGI, France and collaborating institutes for the compilation and databasing of the am index which were downloaded from http://isgi.unistra.fr/data_download.php. We also thank the staff of Geoscience Australia, Canberra for the southern hemisphere aastation Kindex data, and at British Geological Survey (BGS), Edinburgh for the northern hemisphere aastation Kindex data. For the SuperMAG indices data we gratefully acknowledge: Intermagnet; USGS, Jeffrey J. Love; CARISMA, PI Ian Mann; CANMOS; The SRAMP Database, PI K. Yumoto and Dr. K. Shiokawa; The SPIDR database; AARI, PI Oleg Troshichev; The MACCS program, PI M. Engebretson, Geomagnetism Unit of the Geological Survey of Canada; GIMA; MEASURE, UCLA IGPP and Florida Institute of Technology; SAMBA, PI Eftyhia Zesta; 210 Chain, PI K. Yumoto; SAMNET, PI Farideh Honary; The institutes who maintain the IMAGE magnetometer array, PI Eija Tanskanen; PENGUIN; AUTUMN, PI Martin Connors; DTU Space, PI Dr. Rico Behlke; South Pole and McMurdo Magnetometer, PI’s Louis J. Lanzarotti and Alan T. Weatherwax; ICESTAR; RAPIDMAG; PENGUIn; British Artarctic Survey; McMac, PI Dr. Peter Chi; BGS, PI Dr. Susan Macmillan; Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (IZMIRAN); GFZ, PI Dr. Juergen Matzka; MFGI, PI B. Heilig; IGFPAS, PI J. Reda; University of L’Aquila, PI M. Vellante; BCMT, V. Lesur and A. Chambodut; Data obtained in cooperation with Geoscience Australia, PI Marina Costelloe; SuperMAG, PI Jesper W. Gjerloev. SuperMAG data are available from http://supermag.jhuapl.edu/indices/?layers=SME.UL. The work at University of Reading (UoR) is supported by the SWIGS NERC Directed Highlight Topic Grant number NE/P016928/1 with some additional support from STFC consolidated grant number ST/M000885/1. The work at École et Observatoire des Sciences de la Terre (EOST) is supported by CNES, France. Initial work for this paper was carried out by IDF as part of his PhD studies at Southampton University, where he was a parttime student: we are grateful to Rutherford Appleton Laboratory and to Southampton University for supporting that PhD work. CH is supported on a NERC PhD studentsship as part of the SCENARIO Doctoral Training Partnership. The editor thanks Lauri Holappa and an anonymous referee for their assistance in evaluating this paper.
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Appendix A
Stations used to compile rangebased global indices.
am index magnetometer stations.
ap (kp) index magnetometer stations.
aa index magnetometer stations.
Supplementary material
Supporting information file Access here
All Tables
Bands of range values used to generate quantized Kindices for a station with a lower limit of the K = 9 band of L. ΔH_{ X or Y } is the range between extreme values in the 3hour intervals of the northward or westward horizontal component, whichever is the larger. The right hand column gives the quantized a_{K} values ascribed to the Klevels using the “K2aK” or “midclass amplitudes” scale.
Linear correlation coefficients between midlatitude range indices and the SuperMAG and auroral electrojet indices: r and r* are for 3hourly values and the 8point running means, respectively. The data used are for 1996–2017 (inclusive). This yields 61368 3hourly samples (am, ap and aa) and 6136124 runningmean samples (Am*, Ap* and Aa*). For all correlations, the large number of samples ensures that the correlation significance level, derived by comparison with the AR1 red noise model, is 100% to within at least three decimal places for all cases. The maximum SME and minimum SML in each 3hour intervals is SME_{max} and SML_{min} respectively.
Uniformity of average timeofday/timeofyear response, S_{av}(UT, F), of the various midlatitude geomagnetic range indices.
Maximum and minimum percentage deviations of modelled 3hourly index sensitivities, S, from unity for selected years and middle and low geomagnetic activity levels.
Standard deviations and maximum and minimum percentage deviations from unity of observed 3hourly index sensitivities, S, estimated assuming the am index is ideal for 1959–2017, for low, middle and high geomagnetic activity levels. The bottom two rows give the factor by which the corrected index is improved in terms of the uniformity of its response.
All Figures
Fig. 1 Maps of networks of stations currently contributing to (a) the Kp (and hence ap) index, (b) the am index and (c) the aa index. In each map, the light grey bands are typical locations of the auroral oval and dark grey bands are ideal middle geomagnetic latitudes for stations to give a Kindex value, being close enough to give a large signal, but far enough away that the response is monotonic because, for all but the very largest disturbances, the auroral oval approaches the station as the activity level increases. Details of these stations, and others used in the past, are given in Appendix A. Images courtesy of the International Service of Geomagnetic Indices (ISGI). 

In the text 
Fig. 2 Variations in geomagnetic range indices for 27 October 2003 to 2 November, showing the “Halloween storms”: (a) 3hourly values am, f × ap, aa and aa_{H} (b) their 24hour (8point) running (“boxcar”) means Am*, f × Ap*, Aa* and Aa_{H}*. The ap and Ap* values have been multiplied by f = 〈am〉_{all}/〈ap〉_{all}, the ratio of overall means of am and ap for 1995–2017, to allow for the difference between the scaling of ap and that for other indices. Circles, triangles, squares and diamonds are for am (Am*), f × ap (f × Ap*), aa (Aa*), and aa_{H} (Aa_{H}*), respectively. Points are colourcoded by the UT of observation in (a) and vertical grey lines are at UT = 0. 

In the text 
Fig. 3 Scatter plots for 1996–2017 (inclusive) of the midlatitude range indices with the maximum of the SME index, SME_{max}, seen in the same 3hour intervals by the SuperMAG global magnetometer network. (a) (grey) 3hourly SME_{max} values as a function of 3hourly am and (orange) 24hour running means of SME_{max} as a function of corresponding running means of am, Am*. Black dots are means in 1percentile ranges of Am* (giving 614 samples in each bin) and the horizontal and vertical error bars are ±1 standard deviation. The mauve line is a fourthorder polynomial fit to the 3hourly values. The RMS deviation of the observed 3hourly SME_{max} values from the fitted polynomial value for the corresponding am, Δ_{RMS} is given, as is the correlation coefficient r between 3hourly SME_{max} and am values. (b) The same as (a) for 3hourly ap and its 24hour running mean, Ap*. (c) The same as (a) for 3hourly aa and its 24hour running mean, Aa*. In each panel the cyan line is a linear fit to the 3hourly values for SME_{max} < 750 nT, and is plotted to gauge the deviation from linearity of the data for larger SME_{max}. 

In the text 
Fig. 4 Scatter plots of geomagnetic indices as a function of normalized power input into the magnetosphere computed from nearEarth solar wind observation, P_{ α }/〈P_{ α }〉_{all}, where the average is over the full period considered (1995–2017, inclusive). The left hand panels are for daily means, the right hand panels for annual means. The top panels are for the ap index, the middle for aa index, the bottom for am index. For the daily data, linear regression fits are shown for: (red line) 91 days around the June solstice; (blue line) 91 days around the December solstice; and (orange line) 91 days around either equinox). For annual means the cyan lines are linear regression fits for all data. The number of valid daily P_{ α } data points is N = 8375 (an availability of 99.7%) and for annual means is N = 23. The bestfit coupling exponent used to generate P_{ α } is α = 0.44 for am and aa and α = 0.48 for ap. The linear correlation coefficients, r, and the Root Mean Square (RMS) linear fit residual ε (as a ratio of the overall mean value of the index) are given in each panel. 

In the text 
Fig. 5 The annual and semiannual variations in geomagnetic indices and estimated power input into the magnetosphere, P_{ α }, for coincident data from 1995 to 2017, inclusive. In each panel the coloured line shows mean values of daily means of the geomagnetic index in 30 equalwidth bins of timeofyear, F, smoothed with a 3point running mean. The black line is the bestfit variation of the nearcontinuous P_{ α } data for the same interval processed the same way. (a) is for the Ap index; (b) is for the Aa_{H} index, and (c) is for the Am index. In each panel, two goodness of fit metrics are given: the correlation coefficient r and the Root Mean Square (RMS) fit residual, ε, as a ratio of the overall mean value of the index. 

In the text 
Fig. 6 Cumulative probability distribution (c.d.f, mauve line) and histogram of number of am samples in bins Δam = 1 nT wide (N, shown by the black line as N/N_{max}, where N_{max} is the maximum value of N) for all am data in the years 1959–2017 (inclusive). The grey bars give the eight overlapping am bands employed in this paper: 0 ≤ am < 10 nT, 10 ≤ am < 20 nT, 20 ≤ am < 40 nT, 30 ≤ am < 50 nT, 40 ≤ am < 60 nT, 50 ≤ am < 90 nT, 60 ≤ am < 110 nT, and am ≥ 70 nT which contain a numbers of samples N_{ b } of 58183, 51083, 40894, 22691, 13157, 8302, 10869, and 6060, respectively, and for which the mean am values are 5.32, 13.96, 27.50, 37.71, 47.76, 63.56, 73.90, and 109.14 nT. 

In the text 
Fig. 7 Analysis of the sensitivity of the Hartland (HAD) station. Timeofday (UT)/timeofyear (F) plots of: (left column) the modelled sensitivity for the am index, S_{am}, for the current stations and sector weighting functions; (middle column) modelled values of the ratio s_{HAD}/S_{am} where s_{HAD} is the sensitivity of the Hartland magnetometer station for measuring its a_{K} values, a_{HAD}; and (right column) means of the observed values of the ratio 〈a_{HAD}〉/〈am〉 = s_{HAD}/S_{am}. All data are for eight UT bins 3 h wide and 20 F bins 18.25 days wide over the years 1959–2017 (inclusive). The panels are for am ranges (from top to bottom) of: am ≥ 70 nT; 60 ≤ am < 110 nT; 50 ≤ am < 90 nT; 40 ≤ am < 60 nT; 30 ≤ am < 50 nT; 20 ≤ am < 40 nT; 10 ≤ am < 20 nT; and 0 ≤ am < 10 nT shown in Figure 6. The modelled values are based on the mean am in each band which equals, respectively, 109.14, 75.94, 63.56, 47.76, 37.71 27.50, 13.96, and 5.32 nT. Modelled sensitivities are computed at points 1 h apart in UT and 1/365 apart in F and then averaged into the same sized UTF bins (3 h by 0.05) as used for the observations. Note that the lefthand plots are colourcontoured using the 0.8–1.2 scale given by the lower colour bar while the modelled and observed s_{HAD}/S_{am} sensitivity ratios both use the 0.5–1.6 scale given by the upper colour bar. In all plots unity values are coloured yellow. 

In the text 
Fig. 8 The same as Figure 7 for the Canberra (CNB) station, giving UTF plots of: (left column) the modelled sensitivity for the am index, S_{am}, for the current stations and sector weighting functions; (middle column) modelled values of the ratio s_{CNB}/S_{am} where s_{CNB} is the sensitivity of the Canberra magnetometer station for measuring its a_{K} values, a_{CNB}; and (right column) observed values of the ratio a_{CNB}/am = s_{CNB}/S_{am}. 

In the text 
Fig. 9 Timeofday (UT)/timeofyear (F) plots of the modelled sensitivity of (top row) the northernhemisphere an index, S_{an}; (middle row) the southernhemisphere as index, S_{as}; and (bottom row) the global index am = (an + as)/2, S_{am}. The left hand plots are for relatively high geomagnetic activity (defined as am = 74 nT, the mean of the 60 ≤ am < 110 nT band) and the right hand plots are for relatively low geomagnetic activity (defined as am = 14 nT, the mean of the 10 ≤ am < 20 nT band). All plots are for the stations and longitudinal sector weighting functions used in 2014. 

In the text 
Fig. 10 Timeofday (UT)/timeofyear (F) plots of the modelled sensitivity of the aa index, S_{aa}, for various years. The lefthand and righthand columns are for relatively high and low geomagnetic activity (defined as for Fig. 9), respectively. Plots are for: (a) and (b) 2010; (c) and (d) 1970; (e) and (f) 1930 and (g) and (h) 1890. 

In the text 
Fig. 11 Timeofday (UT)/timeofyear (F) plots of the observed sensitivity of the aa index, S_{aa} = (〈aa〉/〈am〉) × S_{am}, for the years 1959–2017. The lefthand and righthand columns are for high and low geomagnetic activity respectively. The low activity range is 10 ≤ aa < 20 nT as used in Figures 7 and 8, but to get sufficient samples, high activity is here defined as am ≥ 40 nT both when averaging the observed 〈aa〉 and 〈am〉 values and when calculating the model sensitivity of am, S_{am}. 

In the text 
Fig. 12 Timeofday (UT)/timeofyear (F) plots of: (lefthand column) the sensitivity of the am index, S_{am}, for 1988 (the midpoint of the interval 1959–2017 for which am data are available); (middle column) observed 〈ap〉/〈am〉 and (right column) S_{ap} = (〈ap〉/〈am〉)S_{am}. Data are for 1959–2017 (inclusive) and the am ranges defined in Figure 6. 

In the text 
Fig. 13 (a) Grey points shows ap index values as a function of simultaneous am values for 1959–2017 (inclusive). Cyan points show the scatter plot of the corresponding 24hour (8point) running means, Am* and Ap*. The black points are means of ap (with error bars of plus and minus one standard deviation) as a function of means of am in am bins of width Δam = 10 nT (only means for bins containing six or more samples are shown). The mauve line is the ideal case for which ap = am. (b) and (c) Annual variations shown by means in fractionofyear (F) bins of width ΔF = 0.05. (b) shows (in black) the annual variations of 〈am〉 and (in mauve) 〈ap〉 × η, where η is the ratio 〈am〉_{all}/〈ap〉_{all} for means taken over the whole dataset. (c) shows the annual variation of the ratio of ap/am. (d) and (e) Diurnal variations shown by means for the eight UT values of both indices using the same color coding as in (b) and (c). 

In the text 
Fig. 14 The same as Figure 12 for the homogenized aa index, aa_{H}. Data are for 1959–2017 (inclusive) and the am ranges defined in Figure 6. 

In the text 
Fig. 15 Same as Figure 13 for the homogenized aa index, aa_{H}. Data are for 1959–2017 (inclusive). 

In the text 
Fig. 16 The distributions of ap and Ap* values. The mauve and blue lines are the c.d.f.s of, respectively, Ap* and ap values for 1959–2017, inclusive. The black line is the histogram of N/N_{max}, where N is the number of Ap* samples in bins 0.5 wide and N_{max} is the maximum value of N. The vertical white and grey bands divide the distribution of Ap* into the 20percentiles, each containing ΣN/20 = 8612 samples. 

In the text 
Fig. 17 Plots of fits to the ratio of am/ap as a function of time of year F and ap value for the 8 UT’s of the ap and am index samples, derived as described in the text so that they can be used to generate the corrected ap index, ap_{C} from ap. 

In the text 
Fig. 18 Same as Figures 13 and 15 for the corrected ap index, ap_{C}. 

In the text 
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