We computed values of both H′ and R for 52 geomagnetic observatories from 1998 to 2020 to examine the spatial distribution of differences between these two definitions of horizontal change. The data consist of definitive 1998–2020 INTERMAGNET data , collected through the VirES server . From this dataset, we took the definitive magnetic field components of BN, BE and BH at 1 min cadence, allowing us to calculate 1 min resolution magnetic field derivative estimates. Mean monthly sunspot number was downloaded from SILSO . We also utilize the CHAOS-7 magnetic field model to model the effect of declination and intensity on the difference between the two methodologies. We focus on the largest events by only looking at the maximum difference between the methodologies in Fig. , and the largest events constitute the most important features on Figs.  and .

Figure  shows the absolute difference between R and |H′| at 50 observatories, with two observatories omitted due to anomalously high differences (more than 5000 nT min⁻¹, Narsarsuaq (NAQ, 61.17° N, 45.43° W) and Alibag (ABG, 18.64° N, 72.87° E)). We found a clear dependence on latitude for the relationship between R and H′. This is because there is a linear dependence on perturbation size, as from Eq. (), R-H′=R(1-cos⁡θ), and average perturbation magnitude increases proportionally to the background horizontal field strength, becoming higher closer to the poles. Three observatories were chosen for closer investigation as case studies at a variety of latitudes, namely Chambon-la-Forêt, France (CLF, 48.03° N 2.26° E), Tamanrasset, Algeria (TAM, 22.79° N 5.53° E), and Scott Base, Antarctica (SBA, 77.83° S 166.67° E). The statistical properties of |H′| and R for each of these observatories are displayed in Table . The largest differences between the methods occur during storm times, which are more common during solar maxima. This includes a maximum difference of 707 nT min⁻¹ for SBA on 29 October 2003.

We focus on CLF as a mid-latitude observatory that will have the broadest applicability for the United Kingdom and other countries at subauroral locations. A scatter plot between |H′| and R at CLF is shown in Fig. , with a Pearson Correlation Coefficient between the two quantities of 0.83. H′ is always less than R, and in particular, large R can occur frequently while H′ remains low, which is consistent with Eq. (). Figure  shows the absolute difference between |H′| and R at CLF between 1998 and 2020 along with the sunspot number over the same period. Solar maxima correspond to the highest difference between the methodologies, while solar minima (i.e. 2007–2011) consistently are contemporaneous with low values (less than 20 nT min⁻¹). In the following sections, we compare the two methods with available GIC data and we model the difference between the two quantities R and H′ for a given perturbation depending on the declination and inclination of the magnetic field model.

provides a small sample size of 25 space weather events recorded at Eyrewell, New Zealand (EYR, 43.47° S 172.39° E) from 2001 to 2015 and the associated GIC activity at the nearby Islington substation (43.54° S, 172.51° E). The data include the maximum value of |H′| recorded at that station on a given day, and consist of all dates where that maximum value was greater than 40 nT min⁻¹ and there are available GIC data at Islington. We verified that the Pearson correlation coefficient between |H′| and GIC was 0.845 and calculated R at those times along with the associated correlation coefficient between R and GIC activity at Islington. The full summary is available in Table .

We find that the correlation for this sample slightly favours H′ as a GIC activity indicator, as the correlation coefficient for R was 0.830. We found a high correlation between the two quantities |H′| and R with only one data point where the two differ significantly, for the 8 November 2004 event. This event consisted of a decrease in BN during an increase in BE, leading to a higher value of R relative to H′.

To demonstrate the difference between the two methods, for estimating the rate of magnetic field change we construct a model which estimates both H′ and R using a background magnetic field model. At a chosen epoch, we calculate BN and BE from the CHAOS-7 magnetic field model . We then add a spatially-constant perturbation of the geomagnetic field in the northward and eastward directions δBNδt and δBEδt and use equations () and () to calculate H′ and R on a 1°×1° resolution grid over all latitudes and longitudes. As R does not depend on BN and BE, but only on their time derivatives, R is the same everywhere; however, H′ displays significant variability.

Figure  shows the result of applying a perturbation of 1 nT min⁻¹ in an eastward direction on 1 July 2013 at 00:00 UTC, with no perturbation in the northward direction. Hence, R is 1 nT min⁻¹ at every location. In this scenario, as the perturbation is eastwards, a strong northward magnetic field will experience the perturbation as a rotation rather than a change in magnitude, and the eastward contribution to H′ will be vanishingly small as in Eq. (). As a result, close to the agonic line (where declination is equal to zero, so that the northward magnetic field component dominates), |H′| is significantly smaller than R (coloured dark blue). Conversely, close to the ±90° isogonic lines (where declination is ±90°), H′ and R become almost equal (coloured yellow). Figure  shows that there can be a significant difference between H′ and R, even for the same geomagnetic perturbation, which can result in significant differences between the predicted GIC activity levels, because of the varying angle between B^H and δBHδt.

As observed in the previous section, the largest difference between the methodologies at SBA in the time period measured was 707 nT min⁻¹, and this difference is typical of high latitude stations. At CLF, the maximum difference was 107 nT min⁻¹. The effect of such large differences in the rate of change of B_H could be significant for modelling the induced geoelectric current depending on subsurface conductivity. For example, inferred an induced geoelectric field in South Island of New Zealand of around 1.5 V km⁻¹ with a perturbation of 100 nT over an inducing period of 30 s. This has the implication that a significantly larger geoelectric field could be predicted when using R compared to H′, having direct implications on GIC risk. The greatest difference between |H′| and R took place during the 2003 Halloween storms of 29–30 and 30–31 October 2003 for 17 of the total 52 stations we studied, and 47 of the 52 stations record the greatest difference between the methods when the Kp index is greater than 5. Hence the difference between the methods is largest during storm times, which is when the accuracy of a GIC proxy is the most critical, and indeed found that the largest GICs in the Mäntsälä gas pipeline from 1999 to 2019 were induced during the Halloween storms. This would therefore imply that studies into extreme geomagnetic activity e.g. may give different results depending on the methodology chosen. Of the five stations that demonstrate the greatest difference when Kp≤5, four record the greatest difference at midnight UTC possibly due to artifacts in the observatory time series, and the remaining station is Budkov, Czech Republic (BDV, 49.08° N, 14.02° E), recording the highest difference during a disturbance at 12:09 UTC on 1 March 2016. Differences in the two methods are compounded by rapid changes in the declination at storm times , which contribute to R but not H′.

We find that the publicly available data are not sufficient to come to a strong conclusion from direct comparison with GIC data, with only 25 data points. As GIC activity is heavily reliant on local conductivity structure and conductive network topology , we would also hesitate to come to a universal conclusion based on results for a single location and context. We would require a larger study with a sufficient location range within the dataset to make robust conclusions.

Most changes in the auroral electrojet induce northward geomagnetic disturbances , whereas sudden storm commencements are likely to also produce significant eastward geomagnetic disturbances .The ring current contributes to region 2 field aligned currents , and eastward geomagnetic disturbances have been shown to occur on the dawn side of the worldwide current system at the boundary between the region 1 and region 2 field aligned currents , suggesting a possible link between the ring current and eastward disturbances at auroral and subauroral latitudes. The equatorial electrojet can induce geomagnetic perturbations in both the northward and eastward directions . These results suggest that H′ and R may vary in their effectiveness as a GIC indicator depending on the regime within the nearby ionospheric current system, as an eastward geomagnetic disturbance is likely to produce a larger contribution to R compared to H′.

The resolution of magnetic field data in the northward and eastward directions at any INTERMAGNET observatory is 0.1 nT using a vector magnetometer . This is sufficient for measurements of geomagnetic field perturbation during storm times, where the perturbations are orders of magnitude larger than the resolution. This resolution also does not have a significant impact on the efficacy of either H′ or R as indicators of important GICs which would require mitigation or pose a threat to power networks. However, we find that a 0.1 nT measurement resolution in combination with the difference between the methods described in the previous section leads to times where R is many orders of magnitude larger than |H′|, in contrast to the typical ratio (1.4–1.8) derived in Sect. 3.1. We therefore comment on this artifact within the data and explain how this discrepancy arises.

During quiet times, measurement resolution is of a similar size to the average geomagnetic field perturbation at 1 min cadence. Analogue-to-digital conversion causes the data to be quantized in certain values e.g.. The resolution of magnetic field data has an effect on the directional distribution of the observed geomagnetic field perturbation after analogue to digital conversion, as there are only a finite number of options for a vector measurement on a magnetometer with a certain resolution. This has an effect on the difference between H′ and R, as the size of H′ is directly affected by the measured magnetic field direction (Eq. ).

We illustrate the effect of the resolution of a geomagnetic observation on the directional distribution of the measured vector magnetic field perturbation. We model the magnetic field as a background field B0 plus some horizontal perturbation vector P. We assume that the background field does not change with time, such that B0 is arbitrary and has no effect on the perturbation. We then assume that the perturbation has a fixed magnitude and a uniformly-distributed direction. The objective is then to find the measured direction of P following analogue-digital conversion. We define the perturbation magnitude as P and the resolution as ΔB.

We provide an example, with P=2 and ΔB=1 in both the north and east directions. In this case, two perturbations at declinations of -10 and 12° from geographic north will produce measured perturbations with northward components of 2cos⁡(10°) and 2cos⁡(12°), with both perturbations producing a measurement of (2,0) in the (north, east) directions after analogue-digital conversion. An angle of 20°, however, will produce a perturbation of 2cos⁡(20°) and 2sin⁡(20°) in the northward and eastward directions, with the digital measurements being rounded to (2,1). This produces a measured perturbation at an angle of 26.6° and a magnitude of 5. Perturbation magnitudes can also change significantly during analogue to digital conversion, as illustrated in this example.

In Fig. , we model perturbations as normally distributed around zero in the northward and eastward directions to produce an illustration of the effect of analogue-digital conversion with as few assumptions as possible. Such a normal distribution is uniform about the mean and has rotational symmetry, which means that there is a uniform directional distribution of perturbation probabilities. This distribution is naïve, as the directional distribution of perturbations are not typically uniform, and the normal distribution will not be representative, especially in active times. However, the distribution allows us to illustrate the effect of digitisation in a controlled fashion.

We take the median perturbation magnitude as 0.45 nT min⁻¹, which was the value found for CLF. This was then used to find the corresponding covariance matrix for a bivariate normal distribution with rotational symmetry, which is the identity matrix multiplied by a factor of σ2. This was found using the Chi-squared distribution with two degrees of freedom (i.e. the Rayleigh distribution), for which the median m=σ2ln⁡(2), rearranging to give σ2=0.146 (nT min⁻¹)². We note that San Juan, Puerto Rico (SJG, 18.11° N, 66.15° W) had the lowest median perturbation of any station from 1998–2020 at 0.2 nT min⁻¹, producing a potentially even more significant effect in analogue to digital conversion. We then find the measured direction after analogue-digital conversion. In Fig.  we plot the probability of a given perturbation falling within each of 64 bins. The plot shows a significant change in the directional distribution favoring directions that are along the axes of measurement (either directly northward or eastward) due to the points (0,1), (0,2) and others. There is also an increase in frequency at 45° due to the point (1,1). The angular width of each bin is 5.625°. We would expect this quantization to also affect vertical magnetic field perturbations as well.

The 0.1 nT resolution of the data changes the directional distribution of digitised magnetic field perturbations compared to those that actually occurred when the resolution is similar to the median magnitude of the geomagnetic perturbation at a given observatory. Perturbations recorded directly in the eastward (for example) directions occur regularly (Fig. ), leading to the recorded magnetic field perturbation in the northward direction during quiet time being measured as 0 nT min⁻¹. Close to the agonic line, a perturbation in the eastward direction will have a vanishingly small contribution to H′ because the overall magnetic field vector is perpendicular to the magnetic field perturbation (as in Eq. ). This is also true for northward perturbations close to the ±90° isogonic lines. An eastwards perturbation close to the agonic line leads to a value of H′ which is orders of magnitude smaller than R, demonstrated in Fig.  with a modeled eastward magnetic field of 1 nT min⁻¹. As a result, stations near the agonic line may repeatedly have measurements with R≫H′. The agonic line passed through CLF around mid-2013 giving an opportunity to investigate the effect. The times when the recorded perturbation is directly eastward are highlighted by large values of the ratio Q between R and |H′|.

Q was calculated for each measurement (one per minute), and its rolling average was taken over one week (i.e., 10 080 points at 1 min cadence) for the full dataset. This allows us to focus on long-term variation. Following from Eq. (), Q=1|cos⁡(θ)| for the case where δBHδt≪BH, such that the rolling average QRA becomes: QRA(τ)=1n∑t=τ-nt=τ1|cos⁡(θ(t))|, where n is the number of minutes of data included in the rolling average. The peak in Q shown in Fig.  corresponds with BE approaching zero around the period when the agonic line moves westward through CLF in mid- to late-2013. The value of QRA greatly exceeds 10 from 2008–2020 and 100 in 2013. Similar results were found at other stations close to the agonic line, including Stennis Space Center, Bay St. Louis, United States (BSL, 30.35° N, 89.64° W) and Fort Churchill, Canada (FCC, 58.76° N, 94.09° W).

Using the directional distribution of magnetic field perturbations described in Fig. , we can make a prediction of Q as a function of declination for an observatory with a similar total horizontal field intensity as CLF (40 000 nT). We use a bivariate normal distribution to simulate the magnetic field perturbation distribution, with covariance matrix 0.146I as previously noted, where I is the 2×2 identity matrix. Predicted mean Q over the full digitised distribution for a range of declinations is shown in Fig. , and illustrates that an increase in Q, due to the limited resolution of the file format, is expected around the agonic line, and that other smaller local maxima in Q also exist at other declinations. This corroborates well with Fig. . The larger range of Q variations in Fig.  compared with Fig.  is because a bivariate normal distribution is unlikely to be an accurate model of the geomagnetic field perturbation.

The dominant contributions to large values of Q when the declination is small are measurements where δBNδt vanishes (at the recorded resolution of the observations) and when δBEδt is nonzero. An example of this behaviour is shown in Fig. , which is consistent with the hypothesis that high values of Q result from fixed resolution in the file format and proximity to the agonic line. Measurements with the greatest value of Q most commonly coincide with δBEδt also being small (but nonzero), due to smaller perturbations being both more frequent and more influenced by the measurement resolution. Consequently, these intervals do not have a significant contribution to the average absolute difference between |H′| and R. Again, this is the case for other observatories we investigated close to the agonic line.

To indicate the potential benefits of improving measurement resolution, we calculated the mean deviation between the observed and digitised declination of the magnetic field perturbation as a function of analogue to digital resolution for the bivariate normal distribution with again a covariance matrix of 0.146I as used previously. We found that a resolution of 0.1 nT (shown in Fig. ) leads to a mean deviation in the measurement of declination of 4.12°. These mean deviations would be reduced to 0.45 and 0.01° for resolutions of 0.01 and 0.001 nT respectively, both of which would greatly improve the representation of the observed magnetic field data and reduce the difference between |H′| and R near the agonic line during geomagnetically quiet times.

This investigation implies that the current resolution is having an observable impact on the recorded directional distribution of geomagnetic field perturbations during quiet times, and there are a significant number of instances where no geomagnetic perturbation is recorded in either the northward or eastward directions. We recommend improving the resolution of measurements of the geomagnetic field to 0.01nT to mitigate this artifact. This is feasible with current fluxgate magnetometer systems able to reach a precision of around 10 pT .

The baseline-subtracted horizontal variation field vector ΔBH is a quantity occasionally used to isolate the externally-driven magnetic field and therefore highlight rapid changes, for example in machine learning research e.g.. In these studies, the quiet time background magnetic field is removed so that a regression model can predict solely the externally driven part of the measured variation, simplifying the relationship between inputs (i.e. solar wind parameters) and outputs (i.e. rapid variation in ground level magnetic field) to produce a more explainable and more easily trained statistical model. It might be expected that removing the baseline magnetic field before calculating H′ has the potential to remove the effect of the relative magnitudes of different magnetic field directions in Eq. (). However, we find that subtraction of the baseline geomagnetic field does not qualitatively change the difference between R and H′.

Baseline field subtraction does not change the value of R because it does not depend on the background field. For H′ however, the situation is different, and in fact the removal of the baseline can occur either before or after BH is calculated from BE and BN with different effects in each case. Removal of the baseline magnetic field after H′ has been calculated introduces an additional term to Eq. () for the rate of change of the magnetic field model. However, this term is vanishingly small compared to the rate of change of the observations, leading to an equation that follows Eq. (). This method therefore does not eliminate the difference between H′ and R. Alternatively, the baseline magnetic field can be removed before combining BE and BN to produce BH, producing the scalar magnitude of the baseline-subtracted horizontal variation field vector ΔBH. Following the subtraction of the previous quantity ΔBH(t-δt) from this quantity to estimate δ(ΔBH)δt, the relative contribution of the horizontal magnetic field perturbation in a given direction is now proportional to the magnitude of the baseline subtracted magnetic field vector ΔBH in that direction since 5δΔBHδt=1ΔBH⋅ΔBNδΔBNδt+ΔBEδΔBEδt=ΔB^H⋅δδtΔBH where ΔBN and ΔBE are the northward and eastward components respectively of ΔBH. As ΔBH does not have a symmetric directional distribution and has a significantly different distribution from any rapid geomagnetic field perturbation depending on location , removal of the baseline magnetic field does not remove the difference between H′ and R. A large geomagnetic perturbation nearly perpendicular to ΔBH will still cause relatively little change in the overall magnitude of ΔBH, and will therefore lead to a large discrepancy between H′ and R. This means that a machine learning model that uses a baseline subtracted magnetic field variation vector as the target for its training and testing, for example as its GIC indicator, will still give different results depending on which of the two different methods was used to calculate the horizontal magnetic field perturbation.