The K indices have long been a very important way to estimate geomagnetic activity. However, they have some basic and practical problems which restrict their reliability and applicability for long-term (centennial) studies. Here we discuss these problems, modify the K method and construct a new, straightforward, easily verifiable and homogeneous index, the so called Ah index, which is based on digital, hourly data and is dedicated for centennial studies. The local Ah indices correlate with the local K and ak indices very well and extend them typically by several decennia. Ah indices at all studied stations verify that geomagnetic activity has increased during the last century. However, the amount of centennial increase varies greatly with latitude, being largest at high latitudes, smaller at low latitudes and, unexpectedly, smallest at mid-latitudes. The centennial increase in the aa index is roughly twice larger than in the Ah index at similar mid-latitudes. This is due to the erroneous scaling of the aa index in the late 1950s, requiring aa to be revised. The global Ah index correlates uniquely well with the Ap index, better than aa or the not-K based IHV (Inter-Hourly Variability) index. Thus Ah yields the most accurate extension of the Ap index by roughly 30 years.
 Geomagnetic indices aim to estimate irregular variations, excluding regular variations like the solar quiet (Sq) daily variation. K index method defines irregular variations as the range (difference) between the upper and lower fitting quiet daily curves during each three-hour interval, associating this with an integer (0 to 9) [Bartels et al., 1939; Mayaud, 1980; Menvielle and Berthelier, 1991]. Since K scale is quasi-logarithmic, K indices are often linearized to equivalent amplitudes, the ak indices. Kp and Ap indices, based on 13 stations, are perhaps the most reliable long-term measure of global geomagnetic activity but only exist since 1932. The aa index runs since 1868 and was long the only centennial measure of geomagnetic activity.
 Despite their merit, K indices have some problems for long-term studies. K index is based on the observer's personal evaluation and, with changing observers, the long-term consistency is hard to guarantee. (Since 1990s K indices are mostly determined by computer methods; see Menvielle et al. .) The K method to “digitize” ranges to 10 values is problematic, forcing the observer to select, often arbitrarily, between two K values. This is particularly difficult for low K levels where the exact form of the Sq curve is important. Because of the great number of low K values, the selection has a large effect. Different selection principles may cause, e.g., the different distributions of low K values at different stations. This problem can be avoided by using continuous range values (“amplitudes”) instead of digitized K values. Also, the fixed K = 9 limit is problematic for long-term studies. During increasing activity such a limit may underweight the highest disturbance levels, leading to an erroneously small trend. (Changing the K = 9 limit would compromise homogeneity.) This problem can also be avoided using continuous range values with no artificial limits.
 A practical problem is that the early measurements are not in digital format at high sampling. Thus, examining the correctness and homogeneity of K and ak values is difficult. Recent analyses of aa suggest that the calibration was changed by roughly 2 nT in the 1950s [Jarvis, 2005; Lockwood et al., 2007]. Accordingly, the long-term consistency of aa is questionable and the index must be revised. Clearly other, more straightforward and easily verifiable measures of geomagnetic activity are needed for centennial studies. Data availability problem is corrected in the IHV index [Svalgaard et al., 2003, 2004], a recent alternative measure of geomagnetic activity. (IHV is calculated from absolute differences between successive hourly values of the H component during seven night hours in the aim to minimize the effect of the daily curve.) Using hourly digital values available at the World Data Centers (WDC), IHV can be examined in detail [Mursula et al., 2004; Mursula and Martini, 2006]. However, IHV is a measure of hourly variability while the K index is a three-hourly range measure. IHV is a daily index including only night-time activity, while the eight three-hourly K indices cover both day and night, which matters since geomagnetic activity has a strong diurnal variation. Thus, IHV and K indices measure somewhat different processes of the near-Earth space.
 Here we aim to combine the simplicity and easy verifiability of IHV with the basic principles of the K method by introducing a new index of geomagnetic activity, the Ah index (A for amplitude, analogue of the equivalent amplitude ak; h for hourly data), which is tailored for long-term studies.
2. Calculating the Ah Index
 We use six long-operating stations (Sodankylä SOD, Sitka SIT, Niemegk NGK, Cheltenham/Fredericksburg CLH/FRD, Honolulu HON, Tucson TUC) that have the longest and most uniform records of magnetic observations from early 1900s onwards (for more information, see Mursula and Martini [2006, Table 1]). They include two high-latitude (SOD, SIT), two mid-latitude (NGK, CLH/FRD) and two low-latitude (HON, TUC) observatories, allowing to study latitudinal differences in centennial evolution.
 We use hourly data available at WDC. Note that the very early hourly values (at most stations until 1915) were hourly spot values, not hourly means [Mursula and Martini, 2006]. Since spot values include more variability than means, the early Ah indices would, without due correction, remain too large and their centennial increases too small. We have corrected the Ah indices for this effect in a similar way as the IHV indices [Mursula and Martini, 2006] using high-sampling data for the more recent years. While the sampling correction of the Ah indices is presented in detail elsewhere (see D. Martini and K. Mursula, Centennial geomagnetic activity studied by a new, reliable long-term index, submitted to Journal of Atmospheric and Solar-Terrestrial Physics, 2007), note that the reduction needed for Ah, roughly 20%, is smaller than the 30% reduction typically required for IHV [Mursula and Martini, 2006].
 To derive the Ah index we first calculate the quiet daily variation. We use local IHV indices [Mursula et al., 2004; Mursula and Martini, 2006] to find the five quietest days in each month at each observatory. Using local rather than global quiet days gives a better account of local conditions. Also, the official global quiet days exist only since 1932. The Sq curve is the average daily curve of the H component in the five quietest days of each month. This takes into account the seasonal variation of the quiet daily curve even more accurately than the seasonal model curves typically used by observers. Also, the quiet daily variation is no longer subjective and can be easily reproduced and examined.
 As for the K index, the quiet daily curve is fit to the data in each three-hour interval as an upper and lower limiting envelope curve. The difference (range) between the two envelope curves is the Ah index of the respective three-hourly interval. No digitization of the range value is made, contrary to the K method. Accordingly, the continuous range (or amplitude; analogue to ak) is the fundamental, linear parameter. Thus, the Ah index solves the above mentioned problems of the K method in long-term studies.
3. Centennial Increase and Its Latitudinal Variation
 We have depicted the yearly Ah indices for the six stations in Figure 1. Despite large differences in absolute level, all six Ah series depict the same qualitative long-term pattern [Mursula et al., 2004; Mursula and Martini, 2006]: on top of solar cycle variation, there is an increase from the beginning of the century until about 1960, then a dramatic dropout in early 1960s, and a weaker increase thereafter. (A two-line fit emphasizes this behavior.) A qualitatively similar behavior is also found, e.g., for Ap, aa, and IHV indices [Mursula et al., 2004; Mursula and Martini, 2006]. However, the various indices differ significantly in quantitative details, e.g., the centennial trend.
 We have quantified the centennial increase in Ah in two ways. First, we calculated the average values of Ah during the last (1979–2000) and first (1901–1922) 22 years of the last century. (Note that the stations cover slightly different fractions of the first 22 years.) Thereby one can quantify the centennial increase between the beginning and end of the last century, neglecting everything (e.g., the local peak around 1960) in between these time intervals. This method is independent of the normalization of indices. We have depicted the average levels and the percentage changes (relative centennial increases) in Table 1. All six Ah series depict clearly larger values at the end of the century. The centennial increases at the six stations depict the same latitudinal ordering as the IHV indices [Mursula and Martini, 2006]. Although the increases can not be simply compared because of different start years, it is clear that the largest increases are found at high latitudes (SOD, SIT), smaller increases at low latitudes (TUC, HON) and, surprisingly, the smallest increases at mid-latitudes (NGK, CLH/FRD). This latitudinal ordering of the centennial increases is systematic and even more clear than in IHV.
Table 1. Mean Values of Ah Indices for Six Stations at the Beginning and End of the Last Century, Their Relative Increase, and the Slope of the Best-Fit Line for the Mean Normalized Values in 1914–2000a
Station or Index
1000 * Slope
Beginning is until 1922 and end is 1979–2000. Similarly for the mean normalized 6-station (1914–2000) and 3-station (1902–2000) global Ah and for the mean normalized aa-1914 and aa-1902 indices.
 Second, we calculated the slopes of the best fit lines in 1914–2000 (see Table 1). For that, we normalized Ah indices by their means in 1914–2000. One can see that the latitudinal ordering is clearly valid also for slopes. Mid-latitude stations depict quite a small slope while slopes at low (resp., high) latitudes are more than twice (five times) larger.
4. Global Ah Indices
 We have taken the latitude variation of trends into account by including stations from different latitudes in our estimates of global Ah indices. We have constructed two versions of global Ah indices. First, each local Ah index was normalized by its mean in 1914–2000 and then averaged, yielding a six-station global index Ah6. In order to extend to earlier years, we constructed similarly the Ah3 index for 1902–2000 using one station from high (SIT), mid (NGK) and low (HON) latitudes.
Table 1 shows the relative increases and slopes for Ah6 and Ah3. Ah6 depicts an increase of 13.6% and a slope of 1.64 · 10−3 in 1914–2000, to be compared with 37.6% increase and 4.48 · 10−3 slope for aa (aa-1914 in Table 1). Similarly, Ah3 shows a 36.8% increase and a slope of 3.57 · 10−3 in 1902–2000 to be compared with 62.0% and 5.88 · 10−3 for aa (aa-1902 in Table 1). Thus, the centennial increase in global Ah indices is clearly lower than in aa. Also, the centennial increase in Ah at mid-latitudes is about 21%, i.e., much smaller than the 62% increase in aa, also based on mid-latitude stations. Clearly, aa greatly exaggerates the centennial increase. However, because the qualitative similarity of centennial change, Ah3 and aa are fairly well correlated (cc = 0.946).
5. Comparison With Ap
 Annual Ap correlates well with aa (cc = 0.95), but even better with Ah3 (0.97) and Ah6 (0.98). We have depicted in Figure 2 the yearly Ap and the (correlated) Ah3 and aa together with the best fit lines in 1932–2000 and in 1902–2000. The lines for Ap and Ah3 in 1932–2000 are so close that they can hardly be distinguished (slopes 0.0116 and 0.0065) but the aa line shows a larger increase (slope 0.0536). The difference between Ah3 and aa becomes even more clear in the early years (Ah3 slope is 0.0606, aa slope is 0.0965).
 We have depicted Ap-Ah3 and Ap-aa differences in 1932–2000 in Figure 3. Despite fluctuations, Ap-aa is systematically above zero until about 1960 and below zero thereafter. This is in a good agreement with recent studies [Lockwood et al., 2007; Jarvis, 2005; Clilverd et al., 1998], concluding that the calibration of aa fails in late 1950s because of the change of the northern aa station from Abinger to Hartland. The average value of Ap-aa difference in 1932–1960 is 1.1 nT and −0.80 nT in 1965–2000, yielding a step of about 2 nT, in a good agreement with earlier estimates. On the other hand, Ap-Ah3 shows no systematic trend off zero, in agreement with the result that Ap and Ah indices are more consistent and better correlated than Ap and aa.
 We have also calculated the correlation between Ap and local Ah′s (see Table 2). The highest correlations are found for the two mid-latitude stations. This is understandable since most Kp stations are at mid-latitudes. Note that most correlations between Ap and local Ah indices are better than between Ap and aa. (For low-latitude stations correlations are smallest: for HON it is slightly smaller and for TUC equal to that with aa.) The agreement between Ap and all Ah indices is due to the fairly similar basic definitions and the fact that both indices include geomagnetic activity from all local time sectors. Since Ap correlates with the mid-latitude Ah indices better than with aa, these similarities must be more important than the differences (e.g., different sampling frequency).
Table 2. Correlation Coefficients Between Ap and the Local Ah Indices and Ap and the Local IHV Indices at Six Stations
 We have included in Table 2 the correlation of Ap with local IHV indices. These correlations are also fair but remain clearly below those between Ap and local Ah. (For HON they are equal.) Contrary to correlations between Ap and local Ah, the correlations between Ap and local IHV are, in all cases, below the correlation between Ap and aa. This is because IHV differs from the main principles of the K method. Thus, Ah rather than IHV, should be used to extend the Kp and Ap to earlier times.
 We have calculated the average values of SOD ak indices in 1914–2000 in the eight three-hourly UT (or LT; SOD LT = UT + 2.5 h) sectors separately, depicting a strong diurnal variation with activity maximum in the night (Figure 4). The same three-hourly averages calculated for the SOD Ah index depict a very similar diurnal distribution as SOD ak (see Figure 4). The correlation coefficient between the curves is 0.981, implying a probability better than 99.992%. (Note the different absolute scales because of different normalization.) This extremely good correlation yields compelling evidence for the detailed success of the Ah index. Note also that the strong diurnal variation of activity denies a detailed comparison between the K based indices (which include local activity from all LT sectors) and IHV (which includes night sector only). This is a fundamental difference between the K (including Ah) indices, and the IHV index, and suggests Ah rather than IHV to be used as a long-term proxy or extension of local and global K indices.
 We have discussed here the K index method of geomagnetic activity and its problems for long-term studies. We have modified the K method so that these problems can be avoided and hourly magnetic data can be used to calculate a new, verifiable and homogeneous index, the Ah index, which is dedicated for centennial studies. The Ah indices at six stations studied verify that geomagnetic activity has increased during the last century. However, the amount of centennial increase varies greatly with latitude so that the increase is largest at high latitudes, smaller at low latitudes and, unexpectedly, smallest at mid-latitudes [Mursula and Martini, 2006].
 While the centennial increase in the aa index is roughly twice larger than in mid-latitude Ah index, comparison with the Ap index verifies that the scaling of the aa index was changed by a few nT in late 1950s, and that the index must be revised [Lockwood et al., 2007; Jarvis, 2005]. Local and global Ah indices correlate extremely well with the Ap index, better than aa or the IHV index. Ah indices depict a closely similar trend with Ap since 1932, while aa shows a clearly larger trend. Local Ah indices depict a very similar diurnal variation as local K indices, yielding compelling evidence that the new method preserves the essential properties of the original K indices. Local Ah indices can be used to extend the local K and ak indices by several decennia, and the global Ah index gives the most reliable extension of the Ap index by roughly 30 years.
 We acknowledge the support from the Thule Institute of the University of Oulu and the Hungarian Scientific Research Fund (NI 61013).