Estimation of geomagnetically induced current levels from different input data

Transcription

1 SPACE WEATHER, VOL. 4,, doi: /2006sw000229, 2006 Estimation of geomagnetically induced current levels from different input data Antti Pulkkinen, 1 Ari Viljanen, 2 and Risto Pirjola 3 Received 21 February 2006; revised 30 March 2006; accepted 23 April 2006; published 29 August [1] Pragmatic schemes for estimating geomagnetically induced current (GIC) values using different levels of knowledge about the physical quantities associated with the geomagnetic induction process are studied. The fundamental idea behind the proposed schemes is that as the knowledge about the detailed behavior of the relevant physical quantities decreases, the lack of knowledge is compensated by statistical characteristics of the geomagnetic induction process and GIC. It is shown that reasonable GIC estimates can be obtained if some information about the state of the geomagnetic environment is available. For example, single values of the time derivative of the magnetic field at the surface of the Earth or even a mere single value of the field deflection from the baseline level can sometimes be sufficient to give a reasonable statistical GIC estimate. It is also shown that the sampling rate of the magnetic field variations used to make the GIC estimates can be lowered down to about 60 s without crucially degrading the accuracy of the estimation. Citation: Pulkkinen, A., A. Viljanen, and R. Pirjola (2006), Estimation of geomagnetically induced current levels from different input data, Space Weather, 4,, doi: /2006sw Introduction [2] In an ideal case, geomagnetically induced current (GIC) (for a review on GIC, see, e.g., Kappenman [1996]; Boteler et al. [1998]; Pirjola [2002]; Molinski [2002], and references therein) flowing in the system of interest are measured directly and the impacts can be estimated from the measured GIC values. However, such ideal cases are rare and alternative ways to make estimates are needed. This may be the case, for example, when forecasting GIC. Because of the complexity of the phenomenon, it may not be feasible to forecast the exact behavior of GIC or GIC-related magnetic fluctuations but rather some overall properties like power spectrum, variance or mean of the fluctuations [e.g., Weigel et al., 2003; Wintoft, 2005; Pulkkinen et al., 2006]. In addition, in analyzing historical events often very limited information about, for example, magnetic field fluctuations is available. As an example, one may know only few or perhaps just one magnetic field value from which the GIC estimate needs to be made [e.g., Nevanlinna et al., 2001]. It is thus clear that some means to estimate the GIC levels from varying input data is needed. 1 NASA Goddard Space Flight Center, Greenbelt, Maryland, USA. 2 Space Research Unit, Finnish Meteorological Institute, Helsinki, Finland. 3 Geomagnetic Laboratory, Geological Survey of Canada, Natural Resources Canada, Ottawa, Ontario, Canada. [3] The aim of the work at hand is to explore a few pragmatic schemes that one can apply to estimate GIC using different levels of knowledge about the physical quantities associated with the geomagnetic induction process. More specifically, these quantities are the geoelectric field, the magnetic field at the surface of the Earth, the ground conductivity and the topology and the electrical parameters of the studied system. It is emphasized that the introduced schemes are the practical solution to a specific problem. It follows that various details of the geomagnetic induction and GIC phenomena are in many of the cases below neglected for the sake of simplicity and practicality; GIC levels are estimated. For example, the spatially three-dimensional nature of both the ground conductivity structure and the source field variations are almost completely ignored in this work and the reader is referred to other works if this type of details are of interest [e.g., Pulkkinen and Engels, 2005; Pulkkinen et al., 2006]. However, as we shall see, many of the details of the GIC phenomenon can be neglected if the requirements for the accuracy of the estimation are not exceedingly high. [4] The structure of the work is as follows: First, in section 2 we derive the essential theoretical relations and explain the schemes used to apply these relations. Some relatively strong assumptions are used in section 2 and the validity of these assumptions is studied in detail in section 3. One often encounters the problem of choosing an ap- Copyright 2006 by the American Geophysical Union 1of15

2 propriate sampling rate of the ground magnetic field measurements to be used in GIC applications. The effect of different sampling rates is investigated in section 4. In section 5, we will give an explicit example on the usage of the schemes using GIC and geomagnetic data from the geomagnetic storm period of July Finally, in section 6 the findings of the work are briefly discussed and summarized. 2. Theoretical Relations [5] In the most general case, when the geoelectric field is known both as a function of space and time, GIC flowing through the grounding points of a discretely earthed system (such as a power grid) can be obtained from the matrix equation [Lehtinen and Pirjola, 1985; Pirjola and Lehtinen, 1985] I e ¼ ð1 þ YZ e Þ 1 J e ð1þ where Z e is the earthing impedance matrix and J e i ¼ X j6¼i V 0 ij R ij and Y is the network admittance matrix 8 >< 1 R Y ij ¼ ij X 1 >: k6¼i R ik i 6¼ j i ¼ j ð2þ ð3þ uniform in the area of the investigated system. Assuming a spatially uniform geoelectric field, we write in Cartesian geometry E 0 ¼ E x e x þ E y e y where e x and e y are the unit vectors in the horizontal plane. By substituting equation (5) to equation (4) we obtain Vij 0 ¼ E xl ij cos q ij þ Ey l ij sin q ij where l ij is the length of the straight line between nodes j and i and q ij is the angle between e x and the line from node j to i. Note now that as the electric field is expressible in terms of a potential field f (E 0 = rf), only the locations of the nodes affect the value of the voltage V 0 ij, not the path of the line between i and j as in the general case. By substituting the right hand side of equation (6) to equation (2) we obtain J e i ¼ E x X j6¼i l ij cos q ij R ij þ E y X j6¼i ð5þ ð6þ l ij sin q ij ¼ E x S e ix R þ E ys e iy ð7þ ij Using the right-hand side of equation (7), equation (1) can be written as where we have defined I e ¼ E x AS e x þ Ey AS e y ð8þ A ¼ ð1 þ YZ e Þ 1 ð9þ and Z i Vij 0 ¼ E 0 ds j ð4þ Finally, we see from equation (8) that if the geoelectric field driving GIC is spatially homogeneous, GIC flowing through a node of the system can be calculated simply by where E 0 is the surface geoelectric field, indices i and j indicate grounded nodes of the system and R ij is the line resistance between nodes i and j. The integration between nodes i and j is made along the conductor connecting the nodes. The earthing impedance matrix couples the earthing currents at the nodes of the system considered to the voltages between the earthing points and a remote Earth. Thus, if the earthing points are distant enough, making the influence of one earthing current on the voltage at another earthing point negligible, Z e is diagonal with the elements equalling the earthing resistances. The assumption about zero off-diagonal elements is usually made in practice [Pirjola, 2005]. [6] Often, only some approximate information about the geoelectric field is available. In many cases, information about the field is obtained only from one single point. Then, the geoelectric field is assumed to be spatially I e ¼ ae x þ be y ð10þ where constants a and b given by components of AS x e and AS y e, respectively, depend only on the electrical parameters and the geometry of the power system. Accordingly, a and b can be obtained also empirically if the geoelectric field and the corresponding I e are known. [7] Note that the equations above can be applied in both spectral and temporal domains. Also note that although the treatment above was given for discretely earthed systems, similar results, and especially relation (10) apply also for continuously earthed systems (e.g., buried pipelines) [Pulkkinen et al., 2001a, 2001b]. Thus the discussion below applies generally to any conductor system experiencing GIC. [8] The most direct way to apply equations (1) and (8) is, of course, to use the measured geoelectric field in the 2of15

3 area of the system of interest. However, first, the geoelectric field measurements tend to reflect the local geological structures (1 km) [e.g., Poll et al., 1989] thus smearing the field in spatial scales relevant for GIC calculations (100 km, a typical distance between the grounding points of high-voltage power transmission systems). Secondly, typically only magnetic field recordings are available for extended time periods and for sufficiently large spatial arrays. Thus the geoelectric field is usually determined in terms of model computations using the knowledge about the ground conductivity structure and the magnetic field variations. The simplest, and the most practical approach is to use the plane wave and locally layered conductivity structure approximation [Cagniard, 1953] and compute the geoelectric field using the relation ~E ðwþ xy ¼ ð Þ 1 Z ~ ðwþ B ~ w m 0 ð Þyx ð11þ where m 0 is the vacuum permeability, the surface impedance Z is a function of the ground conductivity structure, B(w) ~ yx indicate the horizontal components of the magnetic field spectrum and the tilde denotes quantities in the spectral domain. The minus sign in equation (11) applies for the ( E ~ y, Bx ~ ) pair. We emphasize that the simple local plane wave method has been shown to be a very reasonable approach for GIC computations in cases where the horizontal gradients of the ground conductivity can be locally neglected and provided that the measurements of the magnetic field variations are obtained sufficiently close to the location where GIC is computed [e.g., Viljanen et al., 2004]. In general, sufficiently close depends on the characteristics of the source currents (in the auroral region mainly in the ionosphere), in Finnish GIC studies sufficiently close has been found to be of the order of 200 km. [9] Before proceeding further, let us underscore the equivalence of the relations using the magnetic field or its time derivative. By using the basic properties of the Fourier transformation, it can be shown that ~B ðwþ ¼ 1 iw d ~ B dt ðwþ ð12þ from which it follows that in any relation we may replace the magnetic field spectra B(w) ~ with the spectra of the time derivative of the magnetic field just by compensating with the factor iw. [10] If the only available data is j B(w)j ~ (or equivalently j d~ B (w)j), the loss of the phase information causes that dt we are not able to reconstruct the temporal behavior (by inverse Fourier transform) of the geoelectric field. Instead, we are left with the spectral domain estimates via relation j E ~ ðwþ xy j¼ 1 j Z m ~ ðwþjj B ~ ðwþ yx j 0 ð13þ from which, in the case of a spatially uniform electric field the spectral domain GIC levels can be estimated again using the linear relation j ~ I e ðwþj jajj E ~ ðwþ x jþjbjj E ~ ðwþ y j ð14þ where follows from the triangle inequality. Although the exact temporal behavior cannot be reconstructed from the given data, an integrated estimate can be obtained for finite signals via Parseval s theorem, that is, Z 1 1 ji e ðþj t 2 dt ¼ Z 1 1 j ~ I e ðwþj 2 dw ð15þ where the left-hand side is related to the total ohmic work done by GIC. [11] A special case of the formulation above is obtained for discrete set of values if instead of j ~ B(w i )j, we know only the variance s b 2 of jb(t i )j (or equivalently of jdb(t i )/ dtj) and the basis spectrum, denoted by j ~ f (w i )j, of the variations. In other words, we assume that the shape of the spectrum is static on a bilogarithmic scale (see Figure 1) and that the spectrum j ~ B(w i )j can be expressed as aj ~ f (w i )j. In this case, we can write for finite Fourier transform pairs by applying the discrete form of the Parseval s theorem s 2 b ¼ 1 X jbt ð i Þj 2 ¼ 1 X N N 2 j B ~ ðw i Þj 2 ¼ a2 X N 2 j ~ f ðw i Þj 2 ð16þ t i w i where the mean of the signal is assumed to be zero and t i indicates the ith time step, w i the ith angular frequency and N is the number of samples. The assumption about the zero mean is made for all signals from here on. By solving a 2 from equation (16) we can estimate the variance of the electric field as s 2 e ¼ 1 X jet ð i Þj 2 ¼ 1 X j Z ~ ðw i Þj 2 j B ~ ðw i Þj 2 N t i N 2 w i m 2 0 X j Z ~ ðw ¼ s 2 w i Þj 2 j ~ f ðw i Þj 2 i X b m 2 0 j ~ f ðw w i Þj 2 ð17þ i from which a quantity related to the work done by GIC can be obtained by multiplying the result with N. Finally, using relations (14) and (17) and Schwarz s inequality we obtain an upper estimate for the variance of GIC s 2 I e qffiffiffiffiffiffiffiffiffiffiffiffi jaj2 s 2 e x þjbj 2 s 2 e y þ 2jabj s 2 e x s 2 e y w i ð18þ 3of15

4 magnetic field behavior need to be made. Also, it is seen that the strength of the electric field depends on how long db x /dt holds its sign; if, after prolonged positive db x /dt, the sign changes, the electric field magnitude starts to decrease. [14] To account for the features mentioned above, it is assumed that the time derivative of the magnetic field is composed of single pulses defined by zero crossings of the field (see Figure 2). We assume that each pulse has a sinusoidal temporal form; that is, each pulse is defined in terms of the height of the pulse (db x /dt) max and the temporal length l p between the changes of the sign. Then, the peak electric field produced by the waveform is estimated by applying the plane wave formula (see equations (12) and (13)) j E ~ ðwþ y j¼ 1 m 0 w j~ ZðwÞjj d~ B x dt ðwþj ð20þ Figure 1. Basis spectrum j ~ f (w i )j and the rescaled spectrum aj ~ f (w i )j on a bilogarithmic scale with arbitrary units. where s ex 2 and s ey 2 denote variances of the x and y components of the electric field, respectively. [12] The formulations above are strictly valid only for infinite series and any finiteness naturally introduces inaccuracies to the formulation. However, as our notion about static basis spectrum (i.e., the spectrum does not change as a function of time) already implicitly implied, one wants to apply these ideas in a piecewise fashion for limited time windows. This is obvious for practical applications of the method as time varying information about GIC is likely needed. In the case of a piecewise application of the method, the assumption about the invariance of the basis spectrum j ~ f (w i )j can be relaxed by updating the spectrum as the new data is accumulated. It should also be noted that if such an updating is not done, the right-hand side of equation (17) is just a multiplication of s b 2 by some constant which can be, in principle, also obtained empirically. Such an empirical connection was used recently quite successfully by Wintoft [2005]. [13] Let us then assume that the information about the electric field or the magnetic field variations is even more limited. Suppose that just the maximum value of the time derivative of the horizontal magnetic field is known. By assuming a homogeneous ground conductivity structure and by inverse Fourier transforming equation (11), we obtain the well-known time domain relation where now w = p/l p and j d~ B x dt (w)j =(db x/dt) max. The basic reason why some success can be anticipated using this crude method is that rather than capturing all the finescale variations, as seen from equation (19), only a rough effective estimate on the area covered by db x /dt needs to be obtained (see also Figure 2). The validity of this approach is investigated in detail in section 3. [15] Moving to even more limited information about the field variations, we assume that only one single (time domain) value of the magnetic field deflection B from the baseline value is known. It is obvious that some very strong assumptions about the field variations are required to relate a single B value to GIC. We attack the problem in terms of probabilistic linking between B and its time derivative db/dt. Instead of the field value at one single station, we use the local variant of the AE index [Davis and Sugiura, 1966], the IE index [Kauristie et al., 1996] derived from the IMAGE magnetometer array [Lühr et al., 1998] (see Figure 3) recordings and the die index which is E y ðþ¼ t pffiffiffiffiffiffiffiffiffiffi 1 pm 0 s Z 1 0 db x ðt t 0 Þ 1 p ffiffiffi dt 0 dt 0 t 0 ð19þ where s is the ground conductivity. As is seen from equation (19), also the history of db x /dt affects the electric field value at time t so some assumptions about the Figure 2. Sinusoidal pulse having wavelength of 2l p and amplitude (db/dt) max plotted on top of actual db/dt data. 4of15

5 Figure 3. Locations of the IMAGE magnetometer stations. Geographic coordinates are used. Location of the station at Pello is indicated by a circle. defined as max(jdh/dtj) (H= X + Y) where the maximum is taken over all IMAGE stations. IE and die thus provide us regional upper estimates of the amplitude of the ionospheric current and db/dt. [16] The statistical link between IE and die is established in terms of conditional probability distributions P(dIEjIE) = P(dIE \ IE)/P(IE), which will be calculated for a set of different IE values. As the dynamics of the system most probably varies between different geomagnetic regions, we calculate the distributions for four different local time sectors. 10-s time resolution IMAGE data between 1 January 1995 and 1 March 2005 are used to compute the distributions and visually detected daily baselines are removed from the IMAGE data prior to the analysis. [17] Figure 4 shows the computed conditional probability distributions. It is seen that, as anticipated, the distributions are slightly different for different local time sectors. Most importantly, it seen that the distributions vary for different values of the IE index; there is a clear statistical coupling between IE and die. Interestingly, it is also seen that when moving to larger values of the IE index, a peak in the probability for obtaining certain level of die index is obtained. This peak value provides a natural rule for the selection of a die index value corresponding to a certain IE index value. For example, say, we observe a field magnitude of about 1300 nt (from the baseline level) at the local post midnight region. The corresponding db/dt value is found from the bottom right corner of Figure 4 from which we see that the corresponding maxima of probability of die is obtained roughly at about 3 nt/s. This value can then be used as a peak value for the sinusoidal pulse model discussed above which finally enables the estimation of the electric field and GIC magnitude. Note that to analyze extreme events with extreme field amplitudes, it is obvious that the statistics need to be extended to cover values greater than IE = 1350 nt. Also, it should be noted that similar statistics can be computed to couple the magnetic field deflections, for example, to the local geoelectric field or directly to GIC if required information about the ground conductivity structure and/or network parameters is available. [18] The ultimate case regarding the lack of information about the state (at the time instant of interest) of the geomagnetic environment is that we have no data at all. In a context of GIC, such a case refers to a sole statistical estimation of GIC occurrences [e.g., Pulkkinen et al., 2001b]. Such statistics can be used, for example, to estimate the largest possible GIC or geoelectric field amplitudes [see, e.g., Langlois et al., 1996; Boteler, 1990]. Although statistical occurrences and extreme value estimates are of great interest, we focus here only on situations where at least some input is available for the time instant of interest and thus the usage of such statistics is not discussed any further. 3. About the Validity of the Assumptions Made in Section 2 [19] The formulations introduced in the previous section assumed essentially two different and relatively critical characteristics about the behavior of the time derivative of the ground magnetic field: (1) some sort of stability of the basis spectrum j ~ f (w i )j and (2) pulsed sinusoidal shape of the temporal behavior. In the following, we will demonstrate that these assumptions are reasonable for magnetic field variations in the auroral zone. [20] Let us first investigate the properties of the db/dt spectrum. We will constrain our analysis to 1-s time resolution magnetic field data from station Pello (see Figure 3) of the IMAGE array for the geomagnetic storm period of October 2003 [see Pulkkinen et al., 2005]. First, we compute the time-frequency transformation (TFT) of db x /dt using the short-time Fourier transform. Fourier transforms were computed using the Hanning window of 120 min length with 60 min overlapping between segments thus giving separate spectra for each hour. [21] TFT of db x /dt and the IE index are shown in the middle and bottom plots of Figure 5. It can be seen that, despite the statistical uncertainty due to a limited number of samples, during the most active times (IE > 500 nt, time periods indicated by boxes in Figure 5) the spectrum of db x /dt tends to be a relatively flat curve at frequencies below of about 10 2 Hz; for higher frequencies the spectral power of the fluctuations decreases sharply. Though for frequencies higher than 10 2 Hz the temporal change of 5of15

6 Figure 4. Conditional probability distributions P(dIEjIE) computed for four different local time sectors. The five lines moving from left to right correspond to the IE index values 150, 450, 750, 1050, and 1350 nt, respectively. the spectra is more complicated, the behavior of the spectra at lower frequencies is the type of stability that we were looking for. We note that the flatness of the db x /dt spectrum below periods of about 10 2 Hz is, in an average sense, actually very typical feature for auroral magnetic field fluctuations, as shown in terms of structure function analysis of the field variations by Pulkkinen et al. [2006]. 2 [22] Next we calculate the ratio s dbx /dt/s 2 ey for 1-hour segments of the investigated period. According to equations (12) and (17) the ratio should remain constant if the basis spectra of the field variations does not change. The electric field is calculated by applying the plane wave method in the temporal domain: a uniform Earth with conductivity s =10 3 S/m is used and we integrate the magnetic field variations 600 s backward in time (extending the integration time did not change the results). The results of the analysis are shown in the top plot of Figure 5. 2 As is seen, the ratio s dbx /dt/s 2 ey is relatively stable for the two rightmost time periods with flat spectra below frequencies of about 10 2 Hz (indicated by the boxes), value of the ratio being around (nt/s) 2 /(mv/km) 2. The different value of the ratio for the leftmost period may result from the fact that the spectrum for the period slightly deviates from those of the other two periods. Namely, the power above frequencies of 10 2 Hz is slightly higher for this period. In any case, it is seen that 2 the ratio s dbx /dt/s 2 ey does not vary wildly: the range of the variation is roughly one order of magnitude. [23] Let us then investigate if the temporal behavior of the time derivative of the magnetic field can be approximated, to any reasonable accuracy, by sinusoidal pulses. We will again use 1-s magnetic field data from Pello for the period October The electric field calculated above (homogeneous Earth with s =10 3 S/m) is used as the true electric field to which the pulsed results are compared. [24] First, we identify zero crossings in the db x /dt data. Two successive zero crossings define a pulse having height (db/dt) max and length l p (see Figure 2). These two values of each pulse are recorded. Also the maximum absolute value of the true geoelectric field corresponding to each pulse is recorded. An example of identified pulses is shown in Figure 6 where only pulses having an amplitude over 5 nt/s and a length over 100 s are shown. It is seen that the db x /dt pulses often coincide fairly well with the pulses of the geoelectric field. However, as seen for example from the pulse just before 2000 UT, sometimes negative pulse in the db x /dt corresponds rather to the decrease of the geoelectric field amplitude than to a new pulse originating from the zero level. 6of15

7 Figure 5. (top) Ratio between the variance of db x /dt (taken over 1-hour segments) and the variance of the modeled y component of the electric field for October (middle) The 60 min moving average of the IE index. (bottom) Time-frequency representation of db x /dt at the station Pello (see Figure 3). The boxes indicate the time periods during which spectra below frequencies of about 10 2 Hz are a relatively flat curve. [25] In Figure 7 we show the probability distribution of the pulse lengths. Since we are not interested in extremely small geoelectric field values here, only pulses having an amplitude over 1 nt/s and a length over 20 s were used in the final analysis. Altogether 2398 pulses were identified. As is seen from Figure 7, db x /dt pulses do not have any statistically preferred characteristic length. Further, there is no correlation between (db x /dt) max and l p (linear correlation coefficient for the identified pulses was 0.01). It follows, that because of the absence of the characteristic length and correlation between (db x /dt) max and l p, the selection of the pulse length is very ambiguous if only the peak (db x /dt) max is known. The best we can do here is to use the mean value, 60 s, of the computed distribution. [26] Finally, in Figure 8 we show the comparison between the true geoelectric field pulse heights and the pulse heights estimated using equation (20). Both actual measured pulse lengths and the mean l p = 60 s were used in calculating the estimate. Interestingly, it is seen that the pulsed method seems to give quite reasonable estimates for the peak geoelectric field values, the linear correlation coefficient (computed over all pulses) being in both cases around 0.8. Furthermore, the average l p gives the peak geoelectric field values with almost equal accuracy to that of varying measured l p. This indicates that accurate estimation of l p is not as crucial as could perhaps at first be anticipated; reasonable estimates are obtainable with quite limited information about the magnetic field variations. These considerations also give a more quantitative justification for the usage of empirical linear coefficients linking single db/dt values to the geoelectric field and GIC [see also Nevanlinna et al., 2001; Viljanen et al., 2001]. 4. On the Required Sampling Rate of the Ground Magnetic Field [27] Large GIC, especially in the auroral region, are often related to rapid variations of the ground magnetic field and associated temporal scales vary within wide range as was demonstrated by the distribution of the db x /dt pulse lengths in Figure 7 [see also Pulkkinen et al., 2005, 2006]. This rises a question about the lowest possible sampling rate of the ground magnetic field required for GIC applications; too low sampling rate obviously smoothes some of the most rapid variations and corresponding GIC estimates my be significantly degraded. [28] We investigate the degradation of the GIC estimates as a function of decreasing sampling rate as follows: (1) 1-s magnetic field data (x component) from Pello for the geomagnetic storm period of October 2003 is used as a reference data to which the averaged data is compared. (2) Using the reference field measurements, we construct average data sets (simple nonweighted moving 7of15

8 Figure 6. (top) The y component of the geoelectric field computed from the magnetic field measurements at Pello. (bottom panel) Time derivative of the x component of the magnetic field db x /dt measured at Pello. Shaded regions indicate the pulses (peak amplitude over 5 nt/s and length longer than 100 s) identified from db x /dt data. The data are shown for 30 October As is seen also from the sign in front of the right-hand side of equation (19), it is rather db x /dt than db x /dt that follows the geoelectric field variations. Figure 7. Probability distribution of the pulse lengths determined from the db x /dt data. See text for details. 8of15

9 Figure 8. Scatterplot of the true geoelectric field pulse heights and the heights estimated from the db x /dt pulses. (left) Measured pulse lengths l p were used to estimate the geoelectric field. (right) Average value of l p = 60 s was used for all pulses. The solid line shows the linear fit to the data. The linear correlation coefficient for the left plot is 0.81 and for the right plot Linear coefficient (b in the equation of the form y = a + bx) for the left plot is 0.90 ± 0.01 and for the right plot 0.88 ± average) representing different sampling rates of the data. (3) The time derivatives of the reference and averaged data sets are computed. (4) The reference and averaged data sets are used to compute the geoelectric field using the plane wave method of equation (19). Again, a uniform Earth having conductivity s = 10 3 S/m and the 600-s backward integration are used. (5) The averaged data sets are resampled using a linear interpolation to the 1-s sampling rate of the reference data. (6) The reference and averaged data sets are compared by calculating the linear correlation between the time series and by taking the ratio between the maximum absolute values in the time series. [29] The results of the analysis are shown in Figure 9. The fluctuation of the curves can be attributed to the finite length of the investigated time series. We can observe that although the general behavior of the field amplitude is well represented also by the averaged data, the time derivative is extremely susceptible for temporal averaging; the peak db x /dt is underestimated by more that 60% using sample rates above 10 s and linear correlation to reference data practically disappears for sample rates above 60 s. However, since GIC is driven by the geoelectric field and not by db x /dt, the main focus should be on the behavior of the geoelectric field. Quite surprisingly, it is seen that the temporal averaging has significantly smaller effect on the geoelectric field; the 60-s sampling rate is still able to capture the most of the variations of the field (correlation coefficient about 0.9) and also the peak field value is still more than 80% from that of the reference data. However, above the 60-s sampling rate the capability to reproduce the geoelectric field steadily degrades and, for example, with 180-s sampling rate the peak geoelectric field is underestimated already by 40%. [30] The reason for the lower susceptibility of the geoelectric field to the averaging of the magnetic field data can be understood by looking at the basic form of equation (19). As was already mentioned above, the geoelectric field at a time instant t is determined by not only the time derivative of the magnetic field at t but also by the history of the time derivative, the p ffiffiweight of the history being inversely proportional to t. Accordingly, the high-frequency variations of the time derivative of the magnetic field are smoothed, or in other words, the db x /dt variations are low-pass filtered when computing the geoelectric field. [31] On the basis of the analysis above, we may conclude that for the used setting, the 60-s sampling rate of the ground magnetic field is able to capture essentially the same features of the surface geoelectric field variations as that of the 1-s sampling rate. In addition, as we can see from Figure 9, when moving from the 20-s sampling rate to higher rates, both the correlation and the ratio of the peak values of the geoelectric fields tend to saturate indicating that higher resolution data may not give any additional information from the viewpoint of the geo- 9of15

10 Figure 9. (top) Linear correlation between the 1-s time resolution reference data from station Pello and the averaged data. (bottom) Ratio (percent) between the maximum absolute values of the reference and averaged data. Thick solid line, magnetic field; thin solid line, modeled surface geoelectric field; dashed line, time derivative of the magnetic field. See text for details. electric field behavior. This is in line with the observation made from Figure 5 that the power of db x /dt fluctuations decreases sharply above frequencies of about 10 2 Hz. [32] There are two important aspects in the geoelectric field response to the magnetic field fluctuations that may alter the results above. Let us discuss these aspects briefly. (1) As is seen from equation (11), the geoelectric response is dependent on the driving magnetic field fluctuations. The analysis was performed here for auroral region magnetic field variations and for an extreme storm period. Thus, for example, for lower-latitude GIC estimates where other geophysical processes, like those related to ring current dynamics dominate, the sampling rate requirements can possibly be relaxed from those above. This matter is, however, a subject of another investigation. (2) Via the surface impedance, the geoelectric response is also a function of the ground conductivity structure. It follows, that for different regions having different conductivity structures, the sampling requirements can be either relax or tightened from those above. We made some tests with realistic layered Earth structures to investigate this. The used Earth models were the models of southern Finland, central Finland [e.g., Viljanen et al., 1999] and Canadian Québec and British Columbia [e.g., Boteler and Pirjola, 1998]. The results show (not shown here) that the effect of the varying sampling rate is also for these more complex Earth models very similar to those above. Especially, the degradation of the correlation is almost the same for all models; with 60-s sampling rate the correlation between the modeled geoelectric fields is about 0.9. The peak value of the 60-s data, however, is for the Finnish Earth models only about 60% from that of the reference data. Thus, although the results obtained using the homogeneous Earth give is us some rule of a thumb, more detailed estimates about the effects of varying sampling rates on the modeled geoelectric field require information about the local ground conductivity structure. 5. Example on the Usage of the Methods [33] Here we give an explicit example on the usage of the methods discussed above. We will focus this time on the analysis of the geomagnetic storm period of July 2000 from which we have both actual measured GIC and ground magnetic field data. We will use GIC recordings from the Finnish natural gas pipeline at Mäntsälä pipeline section [Pulkkinen et al., 2001b] and the geomagnetic recordings from the Nurmijärvi Geophysical Observatory located about 30 km southwest from the GIC measurement site. Both data are obtained with a 10-s temporal resolution. We will approximate the ground conductivity structure with a homogeneous Earth having effective conductivity of s eff = S/m and the distributed source transmission line theory is used to obtain the theoretical estimates a = 70 Akm/V and b = 88 Akm/V for the measurement site at Mäntsälä [Pulkkinen et al., 2001a, 2001b]. The overall characteristics of the event are shown in Figure 10 where the measured GIC, x component of the magnetic field and its time derivative measured at Nurmijärvi are shown. 10 of 15

11 Figure 10. (top) GIC measured at Mäntsälä pipeline section during July (middle) The x component of the magnetic field (B x ) measured at Nurmijärvi Geophysical Observatory. (bottom) Time derivative of the x component of the magnetic field ( db x /dt) measured at Nurmijärvi. Dashed lines show the times of maximum GIC, (left) db x /dt (practically overlapping that of GIC) and (right) B x. The visually detected baseline was removed from B x prior to plotting. [34] First, we will use the full knowledge about the temporal variations of the ground magnetic field and apply equations (11) and (10) to obtain GIC. The result is shown in Figure 11, where both the measured and the modeled GIC are shown. As is seen, the modeled GIC reproduces the measured GIC very well: the linear correlation of 0.83 between the two time series can be considered in the context of complex auroral processes very high. [35] Then, we abandon the information about the phase of the magnetic field variations and use equations (13) and (14) to estimate the spectrum of the GIC fluctuations. The result is shown in Figure 12. Again, the basic features of the measured time series are reproduced very well and as can be easily verified, as stated also by equation (14), the modeled spectrum gives an upper estimate for the spectral power of the GIC fluctuations. The reason for greater overestimation at lower frequencies is likely due to the usage of the homogeneous Earth model. Low-frequency fluctuations penetrate deeper to the Earth where the actual conductivity is probably larger than the used s eff = Then, as a special case of the previous procedure, we assume that the spectrum of db x /dt and db y /dt measured at Nurmijärvi can be expressed as aj ~ f (w i )j where only a varies as a function of time. The analysis is carried out in a piecewise fashion for 30-min time windows and the basis spectrum j ~ f (w i )j is constructed from the average of these time windows for the entire period of July Note that as we are using the time derivatives of the field, equation (17) takes the form s 2 e ¼ s 2 X db=dt j Z X ~ ðw i Þj 2 j ~ f ðw i Þj 2 j ~ f ðw w i Þj 2 w i w 2 i i m 2 0 ð21þ Equation (21) is used with equation (18) to compute the variance of GIC for the analyzed 30-min windows. The results of the computations are shown in Figure 13 from which it is seen that although the accuracy of the estimates clearly degrades as statistical information about the field fluctuations (the form of the basis spectrum) was used, we still obtain estimates well within order of magnitude accuracy. It can be verified from Figure 13 that, in accordance with equation (18), the upper estimate for the variance of GIC is obtained. [36] Then, it is assumed that only the maximum value of the time derivative of the horizontal magnetic field (dbt/dt) max (= max(jdb x /dtj,jdb y /dtj)) is known. As seen from Figure 10, the time of the occurrence of (dbt/dt) max coincides well with the occurrence of the maximum of GIC, the value of (dbt/dt) max being 9.5 nt/s. We use this value in equation (20) with l p = 60 s to obtain the estimate for the maximum electric field amplitude which is again used in equation (10) (now the same electric field value is used for both x and y components). The obtained GIC estimate is 33 A which is very close to the measured maximum absolute value of GIC peaking 30 A. Taking into account 11 of 15

12 Figure 11. Measured (solid line) and modeled (dashed line) GIC for the July 2000 event. GIC was modeled using the full knowledge about the temporal behavior of the ground magnetic field. The linear correlation coefficient between the two time series for the event is The bottom plot shows a blowup of the range indicated in the top plot. the quite limited information of the actual field fluctuations used, the obtained estimate can be considered surprisingly accurate. [37] Finally, we assume that only the maximum value of the magnetic field deflection from the baseline level in Nurmijärvi is known. As is seen from Figure 10, the maximum deflection of the x component does not coincide with the time of the maximum GIC. However, the value of the maximum deflection can be used to give estimate for the maximum potential of the storm to generate certain level of GIC. Thus we will use both the maximum deflection of the entire storm (1400 nt) and the maximum deflection during the time instant of the maximum measured GIC (600 nt) to estimate the level of GIC. From the bottom left plot of Figure 4, we see that the corresponding levels of (dbt/dt) max with the highest probability are about 3 nt/s and 1.5 nt/s. These values are again used in equations (20) (with l p = 60 s) and (10) (again the same 12 of 15

13 Figure 12. Power spectra of the measured (solid line) and modeled (dashed line) GIC fluctuations during the July 2000 event. The spectra were smoothed by computing a moving average over 15 frequencies. electric field value is used for both x and y components) to obtain GIC estimates of 10 A and 5 A. Both estimates are again well within an order of magnitude accuracy (measured peak GIC was 30 A) which, considering the very limited amount of information about the field fluctuation going into the analysis, can be considered quite satisfactory. 6. Discussion and Summary [38] The aim of the work described here was to obtain practical schemes for estimating GIC levels from data Figure 13. Estimated (dashed line) and measured (solid line) variance of GIC fluctuations for 30-min time windows during the July 2000 event. 13 of 15

14 containing different levels of knowledge about processes behind the geomagnetic induction driving GIC. It was seen that by using the basic statistical properties of the auroral geomagnetic environment, the gaps in our knowledge can be filled with more intelligent approaches than mere arbitrary guesses. Namely, here the gaps were filled with knowledge about certain level of stability of the magnetic field spectra (when Fourier transforms are applied in a piecewise fashion), typical pulsed behavior of the time derivative of the magnetic field and the statistical coupling between the amplitude of the auroral currents and the time derivative of the magnetic field. Obviously, as more weight is put on the statistical and approximate behavior of the field variations, less deterministic estimates can be obtained using the methods described above. [39] Below we summarize, in the order of decreasing knowledge about the geoelectric or magnetic field variations, the methods that can be used to estimate the GIC levels in the system of interest. Although ideally also spatial information is used, we discuss only situations where the driving geoelectric field can be assumed to be spatially uniform. [40] 1. The actual large-scale (100 km) geoelectric field driving GIC is measured. This is naturally an ideal situation that enables accurate calculation of the GIC flow if the topology and the electrical parameters of the studied system are known. In practice, measurements of the large-scale geoelectric field are very rare. [41] 2. Temporal variations of the magnetic field at the surface of the Earth are measured. If also the underlaying one-dimensional ground conductivity structure is known, the plane wave method can be applied to calculate the geoelectric field and GIC. Note that if both the magnetic field variations and GIC are measured and the network parameters (a and b in equation (10)) are known, the effective ground conductivity can be solved using relation (10). [42] 3. Instead of the exact temporal behavior of the magnetic field variations, only the power spectrum of the field variations is known. In this case, because of the absence of the phase information, the temporal behavior of the geoelectric field and GIC cannot be reconstructed. Power spectra of the geoelectric field and GIC can be solved if the underlying ground conductivity structure is known. The power spectrum of GIC enables to estimate the total ohmic work done by GIC. [43] 4. If both the static basis spectrum and the variance of the field variation are known, one can estimate the variance of GIC. It was shown that there are situations when the shape of the db/dt spectrum is relatively stable thus enabling the usage of the proposed scheme. [44] 5. If just a single value of the time derivative of the magnetic field is known, strong assumptions about the field variations need to be made. By assuming a sinusoidal shape of the field variations, one can use the plane wave method to estimate the peak geoelectric field value. Although no unambiguous rule for the selection of the appropriate frequency associated with the pulse can be made, it was shown that the computed mean value l p for pulse lengths provided reasonable estimates for the geoelectric field strength. [45] 6. In a rather poor situation when only one single value of the magnitude of the magnetic field deflection from the baseline level is known, one needs to rely on statistical coupling between the strength of the auroral ionospheric current and the time derivative of the magnetic field. It was shown that such a coupling does exist and that the statistics provide an unambiguous rule for the selection of the appropriate value of db/dt. The chosen db/ dt value can then be used in scheme 5 to estimate the geoelectric field strength and finally GIC. [46] In addition to the schemes above, the effect of the varying sampling rate of the magnetic field variations was studied. It was shown that although the time derivative is very sensitive to temporal averaging, the geoelectric field can be estimated relatively accurately down to sampling rates of about 60 s. [47] Finally, we acknowledge that although our explicit example on the usage of the methods indicated that quite satisfactory results can be obtained, more detailed information about the accuracy of the presented methods is needed. However, rigorous, and laborious, validation of each individual method is out of the scope of this paper. More rigorous quantification of the accuracy of the methods will be an important followup of the paper at hand. [48] Acknowledgments. The research was performed while A.P. held a National Research Council Associateship Award at NASA Goddard Space Flight Center. The work by A.P. was supported also by the Academy of Finland. Valuable discussions with A. Klimas, V. Uritsky, and D. Vassiliadis are gratefully acknowledged. We are indebted to Gasum Oy company for their support and great interest in our GIC studies. References Boteler, D. H. (1990), Prediction of extreme disturbances with application to geomagnetic effects on pipelines and power systems, in Solar-Terrestrial Predictions: Proceedings of a Workshop at Leura, Australia, October , 1989, vol. 1, edited by R. J. Thompson et al., pp , Environ. Res. Lab., NOAA, Boulder, Colo. Boteler, D. H., and R. J. Pirjola (1998), The complex-image method for calculating the magnetic and electric fields produced at the surface of the Earth by the auroral electrojet, Geophys. J. Int., 132(1), Boteler, D. H., R. J. Pirjola, and H. 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