Probabilistic Forecasting of the 3-H Ap Index Robert L

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004 1425 Probabilistic Forecasting of the 3-h ap Index Robert L. McPherron, George Siscoe, and Nick Arge

Abstract—Measurements of the solar wind are now available wind electric field, Ey. Almost all coupling functions depend on for nearly 40 years or four solar cycles. Several magnetic indexes the product of and transverse modulated by some function are available for all of this time, and in most cases, some time of the angle between the transverse component and the geocen- before. Particle fluxes at geostationary orbit are available for nearly two solar cycles. Such data can be used to establish em- tric solar magnetospheric (GSM) axis. A frequently used ap- pirical relations between properties of the solar wind and indexes proximation to the coupling function is the rectified dawn–dusk of magnetospheric and ionospheric activity. These relations are component of [1], [2]. Many studies have shown often expressed in terms of linear prediction filters, local linear that magnetic indexes can be predicted from coupling functions filters, neural networks, or nonlinear transformations of solar with relatively high accuracy. For example, in [3] Burton et al. wind data. The properties of these filters provide insight into the physical properties of the system and also can be used as real demonstrated that the time rate of change of the Dst index de- time predictors of future activity. All of these methods depend on pends on VBs and Dst. Integration of a simple first-order dif- real-time monitoring of the solar wind somewhere between the ferential equation from an initial value yields the time series for Earth’s bow shock and the L1 point. Consequently, they provide Dst. A long history of studies of this relation has led to models of no more than 1 h of advance warning. Longer term predictions great accuracy. A neural network was used by [4] to create a non- depend on remote sensing of the Sun or solar wind and new models that transform these observations into properties of the linear model of solar wind coupling to Dst that predicted 83% solar wind at the Earth. Thus far, the only example of such remote of the variance of hourly averages. In [5] and [6], it was estab- sensing and empirical modeling is the Wang–Sheeley–Arge model. lished that the coupling function also depends on the dipole tilt This model predicts the temporal profiles of solar wind speed, angle toward the Sun and that the decay rate of the ring current interplanetary magnetic field (IMF) magnitude, and IMF polarity depends on the strength of the convection electric field. In [7], at 1 AU. Unfortunately, geomagnetic activity indexes, and their prediction filters, also depend on the GSM Bz component of the these dependencies have been incorporated in a complex model IMF which so far is unpredictable. Fortunately, the solar wind for the hourly Dst that accounts for over 88% of its variance. often has large-scale structures that can be detected remotely. Linear prediction filters have been developed that use VBs These structures organize the properties of the solar wind in- to predict a substantial fraction ( 45%) of the variance of the cluding IMF Bz, and hence affect the probability of geomagnetic 2.5-min AL index [8], [9]. Nonlinear prediction filters [10], [11] activity at different times relative to the structure. Thus, it is possible to utilize the predictable properties of the solar wind to do considerably better predicting of order 65% of 1-min AL parameterize the probability distribution for a magnetic index and variance. Neural networks do still better predicting as much as to accurately specify the probability of high geomagnetic activity 71% of the AE variance from coupling functions and 76% from given the correct identification of a solar wind structure. the basic solar wind variables V, By, and Bz [12]. Index Terms—ap index, forecasting, geomagnetic activity, space All of these models are deterministic since they require the weather. time sequence of several solar wind variables including velocity, density, and interplanetary magnetic field (IMF) By and Bz. If I. INTRODUCTION measurements of these quantities are made at the Earth there is no advance warning other than the delay of the magnetosphere MPIRICAL models use historical databases to establish in responding to the solar wind. However, if the measurements E mathematical relations between solar wind drivers and in- are made upstream at the L1 point, then there will be 30–60 min dexes of magnetospheric activity. The relations are often physics of additional lead time. Greater lead time would require a solar based in the sense that our understanding of physical processes wind monitor further upstream. Reference [13] tested the Burton motivates the quantities used in the prediction schemes. When equation described in [3] using observations from Pioneer Venus empirical models fail, it is often evidence that our understanding when Venus was close to the Earth–Sun line. Although they is incomplete. Today, it is well known that geomagnetic activity give no quantitative measure of the quality of the fits they show is driven by a function of the dawn–dusk component of the solar graphically that much of the time the formula does well, but at times there are large differences between the predictions and Manuscript received November 1, 2003; revised June 11, 2004. the work of the observations. They attribute these differences primarily to R. L. McPherron was supported by the NSF Space Weather under Grant ATM 02-08501. All authors are part of the NSF CISM Science and Technology Center small-scale structures in the solar wind at Venus that do not im- managed by the Center for Space Physics at Boston University. pact the Earth. Possible reasons for this are that the Earth is not R. L. McPherron is with the Institute of Geophysics and Planetary Physics and connected to the monitor by a streamline or that the structures Department of Earth and Space Sciences, University of California, Los Angeles, CA 90095-1567 USA. are distorted by higher speed flows behind them. G. Siscoe is with the Center for Space Physics, Boston University, Boston, In this paper, we consider the problem of what we will call MA 02215 USA. medium-term (1–4 days) forecasting of geomagnetic activity. N. Arge is with the Phillips Laboratory, Hanscom Air Force Base, Boston, MA 01731-3010 USA. This has previously been called “short-term” forecasting [14]. Digital Object Identifier 10.1109/TPS.2004.833387 This author concludes that “forecast skill is disappointing”

Fig. 1. IMF Bz from ACE at L1 has been propagated using the optimum cross-correlation delay to Wind located on Earth’s dawn terminator. Top: Correlation obtained at this delay. Bottom: Comparison of Bz traces at ACE and Wind is shown in the upper two traces. Difference shown in the bottom trace has rms fluctuations of 8.7 nT. for such forecasts. Also, she concludes that the only current hope threshold, e.g., nT %. The relevant prob- is for “nowcasting” from L1 of the type discussed above. In our ability distributions can be determined empirically from the past terminology, deterministic models using L1 data are short-term history of the solar wind and magnetic activity. To make the fore- forecasts while real-time assessments of the state of the mag- cast as precise as possible, we use the method of “air mass clima- netosphere is “nowcasting.” Despite the author’s pessimistic tology [16]–[18]. In our application of this concept, we assume outlook, we will show that probabilistic techniques should allow that the probability distributions for activity indexes depend on one to make reasonable forecasts several days in advance. We predictable solar wind parameters that can be measured or fore- begin by demonstrating that it is unlikely that any deterministic cast several days in advance. An “air mass” is some structure in model will be able to make good predictions on this time scale the solar wind such as a corotating interaction region (CIR) or a because of the dependence of activity on the stochastic variable, coronal mass ejection (CME). The underlying assumption is that IMF-Bz. We will also show that there are major problems with the structure organizes the properties of the solar wind and hence short-term ( 1 h) deterministic models of solar wind-magneto- the type of geomagnetic activity it produces. If these structures sphere coupling, even when they are physics based. One problem can be observed and their arrival at the Earth predicted, then it is that the lead time for the forecasts is too short (less than 1 h). is possible to forecast the resulting activity. Another problem is that any attempt to increase this lead time by There is a considerable body of evidence that shows that the measuring further upstream degrades the quality of the forecast. speed of the solar wind at the Earth is correlated with the coronal The fundamental problem, however, is that what we measure magnetic field [19]–[21]. Other evidence shows that the polarity upstream is not necessarily what gets to the Earth. The solar wind of the IMF is also predictable. Recently, [22] demonstrated that streamline passing through the monitor may not hit the Earth. the strength of the IMF at the Earth can be calculated as well. Then, if there are structures in the solar wind with a scale smaller Together, these three variables provide considerable information than the distance between this streamline and the Earth what that can be used to establish the climatology of a magnetic index arrives at the magnetopause will differ from what was measured. at the Earth. Another problem is that some disturbances propagate in the solar wind and may either move out or into the stream between leaving A. Problems With Deterministic Forecasting the monitor and hitting the Earth. A final problem is that we do Fig. 1 shows the problems that arise in short-term forecasting not yet have physics-based models for propagating observations as a result of propagating the observations from a solar wind from a point measurement to the Earth so there are large errors monitor to the Earth. Until recently, there were four or five kine- in arrival time, even for the relatively short distance between matical models used by various researchers to accomplish this L1 and the Earth. [23]. According to these authors, all of the common models are There is hope that probabilistic forecasting can solve some of subject to large errors ( 15 min). We illustrate this problem these problems [15]. A probabilistic forecast is one that gives with ACE and Wind measurements of the solar wind during the the probability that some measure of activity exceeds a specific Bastille Day storm (July 2000). On this day, ACE was at L1 and McPHERRON et al.: PROBABILISTIC FORECASTING OF 3-h ap INDEX 1427

for IMF Bz in GSM coordinates determined from all data in the year 1995. The smooth line is a Gaussian curve with the same rms power as the data. The data distribution is too sharply peaked close to the origin to be Gaussian and has tails that are much too high. It does, however, have a zero mean. Distributions for Bx and By are similar if IMF polarity is taken into account but are strongly biased away from zero corresponding to the spiral structure of the IMF. Fig. 2(bottom) shows the power spectra of Bz. In this case, we have selected the data for different times relative to a stream interface (see discussion). It is obvious that Bz has a power law spectrum typical of turbulent processes. There are no characteristic periodicities that would help in predicting the behavior of Bz. However, it is evident that the power spectrum depends on time relative to a stream interface. This type of information can be used to improve forecasts of geomagnetic activity as discussed below. Both panels of Fig. 2 represent problems for deterministic forecasting of geomagnetic activity. In the ﬁrst case, errors arise because of the unknown physics of propagation of discontinuities, or because the streamline containing the measured discontinuities does not hit the magnetosphere. In the second case, they arise because Bz is inherently noisy and its waveform is not amenable to long-term prediction. If the waveform of Bz is unknown it is not possible to use either empirical or physics-based models to transform Bz into geomagnetic activity.

B. Structures in the Solar Wind There are a variety of structures in the solar wind that can potentially organize the properties of the wind in the vicinity of the structure. The most obvious are CIR and CME. These can be subdivided into high- and low-speed streams, clouds, flux ropes, Alfven wave trains, etc. If it is true that these structures organize the behavior of the solar wind then they should also organize the Fig. 2. Top panel shows the probability density function for the IMF Bz measured by Wind during the year 1995. Bottom panel shows autospectrum of geomagnetic activity caused by this wind. The basic problem Bz at four different times relative to a high-speed stream interface. RMS power is thus reduced to predicting when one of these structures will (middle left) in each spectrum is shown in the same order as the traces defined arrive at the Earth, and what its properties will be when it arrives. by the legend. This is an easier problem than predicting the precise waveform of Bz. The authors of [26] have already applied this approach to Wind was at the Earth’s dawn terminator at a distance of 60 Re. In CMEs. In this paper, we apply it to CIRs. Fig. 1(bottom), the ACE data have been advanced using the op- timumcrosscorrelationdelaytothelocationofWind.Thesecond curve with the same base line is the Wind measurements, and the C. Data Base offset curve is the difference between the Wind and propagated The database used in this study includes solar wind plasma ACE measurements. For this interval and technique there was a data and magnetic field data from the Wind spacecraft at approxi- 9 nT root mean square (rms) difference between propagated Bz mately 88-s resolution. Also used are indexes of magnetospheric and the observations. Other methods gave even larger rms power activity including 3-h ap, 1-min Sym-H, 1-h synchronous elec- in the difference. These propagated series often show optimum tron flux [27], and a 1-h ground Pc 5 wave activity index [28]. The correlations with the observations at a nonzero lag. solar wind data were edited, transformed to GSM coordinates, A new technique for kinematical propagation of solar wind and interpolated to a uniform grid of 90-s resolution. Gaps in data has recently been described in [24] and [25]. This new plasma parameters less than 10-min duration were filled by in- approach uses minimum variance analysis to continuously de- terpolation. For Bz only single missing points were filled. termine the orientation of solar wind discontinuities and hence their arrival time at the Earth. The method is said to significantly D. Data Analysis reduce the rms difference between the propagated and measured field. We begin by showing that the median behavior of the 3-h Another problem is illustrated in Fig. 2. The heavy line with index depends on the solar wind. Fig. 3 presents a map crosses in the top panel shows the probability density function of the joint dependence of median on solar wind velocity 1428 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004

Fig. 3. Joint dependence of median u on the solar wind velocity (abscissa) and on the magnitude of the IMF (ordinate) is shown by contour lines and shading. Each contour line corresponds to a discrete value of u .

Fig. 4. Complementary cumulative probability distribution of u for points along the diagonal of the joint distribution shown in Fig. 3. Medians are denoted by the intersection of a dashed horizontal line with each of the curves.

(abscissa) and IMF magnitude (ordinate). increases as ei- The integral of this distribution from to a specific value ther variable increases, but the strongest disturbances are found of is called the cumulative probability distribution (cdf). It along the diagonal of the map. The highest median index gives the probability that will be less than a specified value. (8–9) is found for km/s and nT. A more com- The complementary cumulative distribution function (ccdf) is plete characterization of this relation is provided by the comple- one minus the cdf. It gives the probability that will exceed mentary cumulative distribution function. For each bin in the the specified value. Fig. 4 presents a set of ccdfs taken along the joint distribution is distributed over various values around diagonal of Fig. 3. In this figure, the ordinate is the probability the median value shown in the plot. We can calculate the proba- that the index will exceed the value of on the abscissa. For bility density function (pdf) for this local distribution of values. weak IMF and slow wind we obtain the left-most distribution. It McPHERRON et al.: PROBABILISTIC FORECASTING OF 3-h ap INDEX 1429

Fig. 5. Speed, density, temperature, and azimuthal flow angle variations for three high-speed stream interfaces. Fiducial times for these interfaces are indicated by vertical dashed lines at zero crossings of the azimuthal flow angle. can be seen that increases in both velocity and IMF magnitude the quartiles of the ensemble of events and superpose them as drastically change the distribution. heavy lines outlined in black. Fifty percent of all data lie be- The preceding two figures are averaged over all observations tween the upper and lower traces. of regardless of the type of structure or time in a structure. It can be seen that the solar wind velocity reaches a minimum We can be more precise in our predictions if we take these into several hours before the interface and is increasing rapidly at account. Fig. 5 shows how we can do this for a corotating in- the time of the interface. The minimum median value before teraction region. The four panels exhibit three typical stream the interface is about 350 km/s, and after the interface the max- interfaces between slow- and high-speed streams designated by imum median value is about 570 km/s. There is considerable vertical dashed lines. The interfaces are identified on higher res- variability in individual streams as seen from the large spread olution plots by first finding a minimum in the velocity and then of the traces about the median curve. We have already shown determining the time of a nearby zero crossing in the azimuthal that geomagnetic activity depends on the speed so we should flow angle. It can be seen that these times organize the behavior take this variability into account in any forecast scheme if it is of the solar wind. Before the interface, the velocity is a minimum possible to do so. and the density a maximum. After the interface, the velocity and The results of superposed epoch analysis of IMF data and temperature increase while the density decreases. Prior to the magnetospheric response for all stream interfaces in 1995 are interface the azimuthal flow is deflected about 5 west of radial shown in Fig. 7. From the top down, the panels show the velocity, and afterwards it is deflected about 5 east. GSE dawn-dusk electric field, the ap index, the Sym-H index, The time of the interface can be used as epoch zero in a super- the log of Pc5 power, and the noon flux of relativistic electrons posed epoch analysis of various solar wind parameters. Fig. 6 at synchronous orbit. All parameters are well ordered by time illustrates this analysis for the solar wind velocity. In the year relative to the interface. The red center line in each panel is the 1995, we identified 34 stream interfaces. Ten-day segments of median and the two blue lines are the quartiles. Note that there are data centered on the time of the interface were selected from the a number of important features in these curves that are relevant original time series and placed in successive rows of a two-di- to the generation of geomagnetic activity. The speed is low mensional array. For each column of this array, we determined before the interface meaning that magnetic flux is transported 1430 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004

Fig. 6. Ensemble of stream interfaces selected in 1995 is shown by 34 overlapping traces. Epoch zero was deﬁned as the time of a zero crossing in the azimuthal ﬂow angle just before a large rise in solar wind velocity. Heavy lines show the quartiles of the cumulative probability distribution at each sample time relative to epoch zero.

Fig. 7. Quartiles of the speed and dawn–dusk electric ﬁeld in the solar wind and the magnetospheric response including 3-h ap index, 1-min sym-H index, log of maximum Pc5 power on the ground, and the hourly noon ﬂux at synchronous orbit as a function of time relative to a stream interface. Three sets of lines in each panel represent quartiles of the 34 stream interfaces selected in 1995. McPHERRON et al.: PROBABILISTIC FORECASTING OF 3-h ap INDEX 1431

Fig. 8. Complementary cumulative distribution (ccdf) for the 3-h ap index is Fig. 9. ccdf of the 3-h ap index for 2-day intervals before and around stream plotted for two time intervals relative to a stream interface. Solid curve in the interfaces (see caption for Fig. 8). Strength of the IMF has been used to separate upper right shows the ccdf of all observation for an active 2-day interval centered the data into two subsets in each interval. Activity depends strongly on time on the interface (1toC1). Solid curve on the left shows the ccdf for a 2-day relative to the interface and to lesser extent on the strength of the IMF. quiet interval before the interface (3to1 days). Dotted curves above and below these curves represent the ccdfs for the subsets of data corresponding to a solar wind speed greater or less than its median. that at any particular time relative to the interface the ccdf will depend on these quantities. The truth of this is demonstrated in slowly in the slow speed wind. After the interface it becomes Fig. 8. Two sets of curves are plotted in the figure. The right high, increasing the rate of transport so that reconnection at the set presents ccdfs for a two-day interval centered on the stream magnetopause should be stronger. The electric field VBz in the interface. Since there are 16 three-hour samples in two days second panel is also organized by the interface. Before the in- and the 1995 ensemble contained 34 interfaces, there were 544 terface, its median is approximately zero and the fluctuations values used to calculate the ccdf depicted by a solid curve. The about zero are small. Thus, there is very little electric field to two ccdfs identified by dashed lines were each calculated from drive geomagnetic activity. For about 12 h on either side of the half as many values. The dashed curve to the left of the solid interface, median VBz is biased negative and the fluctuations are line is calculated for all solar wind speeds in this interval less very large. Thus, there is strong geomagnetic activity in this in- than the median speed for the interval. Similarly, the curve on terval. Subsequently, there is a weak negative bias and the fluc- the right is calculated from all events with speed greater than tuation level decreases gradually over many days. In a figure the median. It is obvious that solar wind speed influences the not shown here, we use the GSM VBz and find that the negative probability of activity during this interval. For example, if the bias is much enhanced after the interface as a consequence of the speed is low then there is only a 4% chance that nT. In Russell–McPherron effect [29]. Because of the negative bias in contrast, if the speed is high the probability is 9%. VBz, geomagnetic activity continues for many days even in the The set of curves on the left side of Fig. 8 presents the situa- absence of negative excursions in Bz. Because the solar wind tion for a two-day interval starting three days before and ending speed is high after the interface, the effects of the bias and Bz one day before the interface. The solid curve indicates a dramat- fluctuations are amplified. As a consequence of these two fac- ically different situation than for the 2 days centered on the in- tors, geomagnetic activity should continue for many days. The terface. Magnetic activity is extremely low before the interface. truth of these expectations is demonstrated in the third panel The probability that nT is zero! Nonetheless, the speed for the 3-h ap index where ap peaks at the time of the interface of the solar wind on this day is important as can be seen from the and continues for many days after. Sym-H in the fourth panel fact that the lowest speed wind in this interval did not produce shows that stream interfaces cause weak but systematic changes nT while high-speed wind produced nT. in Sym-H. These disturbances would not qualify as magnetic Fig. 9 demonstrates that magnetic activity also depends on the storms, but they display exactly the same behavior as storms. strength of the IMF at a given time relative to a stream interface. There is an initial phase of elevated Sym-H followed by the de- High IMF B on the days around an interface produces much velopment of a 30 nT main phase. Sym-H is depressed but re- higher ap activity than does low IMF B. This is also the case on covering for many days following the interface. This is a result the two days preceding the interface. Although activity is very of the bias in VBz and the large fluctuations in VBz caused by low on these days the strength of the IMF is important. the high-speed stream. The bottom two panels show that stream interfaces also organize Pc5 wave activity and relativistic elec- E. Prediction of the Solar Wind Speed Profile at 1 AU trons. Since these are not the topic of this paper, they are not The speed of the solar wind at the Earth depends on the Sun’s discussed further. coronal magnetic field [19], [20]. This field cannot be measured We demonstrated above that magnetic activity depends on directly so it must be estimated from photospheric observations. the solar wind speed and IMF magnitude. We expect, therefore, This is achieved by converting full disk solar magnetograms 1432 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004

Fig. 10. Comparison of the predicted solar wind velocity (squares) with the observed velocity (thin noisy line) in May—June 1995. to a synoptic map of the central meridian radial field observed source surface map. At this location, the speed of the fastest during a 27-day solar rotation. This field is then expanded as parcel is adjusted according to a potential field to a source surface at 2.5 solar radii using the synoptic map as the lower boundary condition, an assumption of no currents in the corona, and an outer boundary condition of completely radial field on the source surface. The flux tube expansion factor is then calculated at every point on the source If the following parcel is faster than the preceding parcel, its surface according to the formula time is adjusted to be slightly smaller than the speed of the preceding slower particle. The parcels are then again propagated to 2/8 AU and the process of adjustment of times and speeds repeated. This process is continued until the parcels arrive at 1 AU with a distorted time sequence that is interpolated to a reg- where is the magnetic field extrapolated to the source ular time base. This time series is then the 2–5-day advance surface and is the photospheric field at the footprint warning of the solar wind expected to arrive at the Earth. of the field line passing through the point . The expansion An example of such predictions made for the year 1995 is factor at the projected position of the Earth on the source surface shown in Fig. 10. The curve identified by square symbols is the is then correlated with solar wind speed measurements at the predicted time series that is compared to the observed solar wind Earth. To do this, it is necessary to back propagate the observed speed shown by the more rapidly varying thin line. It is apparent speed of a solar wind parcel to determine the time it left the Sun that the prediction scheme captures the major features in the and hence the point on the source surface from which it came. variation of the solar wind. However, the predicted time of ar- This procedure has established the empirical relation [30] rival for the stream at the first of June is too early by one sample point ( 8 h). Also, the actual speed and time of maximum speed km s near May 25 differ from the predictions by a significant amount. Obviously, these problems will contribute to errors in the prob- With this formula and the maps of the expansion factor on the abilistic forecast. source surface, it is possible to create a new map giving the speed of solar wind emanating from every point on the source F. Detection of Stream Interfaces surface. The trajectory of the sub-Earth point can then be plotted Our probabilistic forecast algorithm depends on time relative on the map. Solar wind is emitted toward the Earth from suc- to a stream interface so we must identify the interface in the cessive points along this trajectory. If the solar wind speed in- predicted time series of solar wind speed. However, to deter- creases along the trajectory, later parcels will catch up and in- mine the statistical properties of magnetic activity relative to teract with earlier parcels. In the approach described in [30], interfaces we used both the solar wind speed and flow angle to successive parcels are allowed to interact. The procedure moves identify the time of the interface (see earlier discussion). Since the th solar wind parcel to 1/8 AU at the speed specified by the only the speed is predicted, and this with low time resolution, McPHERRON et al.: PROBABILISTIC FORECASTING OF 3-h ap INDEX 1433

Fig. 11. Detection of a stream interface from the time series of solar wind speed. Top left shows the ensemble of stream interfaces in 1995 and quartiles of this ensemble. Top right panel displays the median profile for the 2-day interval centered about the interface highlighted by a heavy trace. This 2-day segment is repeated at bottom left with a large cross showing the time of the interface. Bottom right panel presents the result of convolving the 2-day segment in the left panel with the 8-day median curve shown above. we must develop an automatic procedure for detecting a stream lines are the three quartiles. The upper right panel presents only interface using only the speed profile. We have done this using the median of the speed ensemble. The thin line is the original the procedure described next. data at 90-s resolution. A thicker and smoother line is the same Let be the observed time series of solar wind velocity. data smoothed with a 4-h low-pass filter. A thick line highlights Determine manually the times of stream interfaces and a 2-day segment centered on the interface which is identified construct an ensemble array with rows containing segments of by the heavy cross. This 2-day segment plotted in the lower left data of length centered on the time of each interface. panel is the interface detector. The numerical values of this curve Create the ensemble average variation denoted with angular have been normalized so that the detector has zero mean and the brackets by averaging all columns of the ensemble array ob- sum of the absolute values of the weights is 1.0. This normal- taining the velocity variation where the ized curve (in reverse time order) is convolved with the predicted epoch time is defined by . Remove the temporal speed profile standardized by removing the mean and dividing mean from the ensemble average series to obtain deviations by the standard deviation. The result of convolution with the . Normalize this series 8-day median profile is presented in the lower right panel. The of deviations by the absolute sum of the series obtaining the output of the detector is largest and near 1.0 at the time of the stream interface velocity pattern given by interface. We tested this detector on the 1995 solar wind speed profile (1) measured by the Wind spacecraft, comparing the times automatically determined from this profile to those obtained manually from the azimuthal flow angle. The results (not shown) are Next, remove the mean of the observed (or predicted) solar wind highly satisfactory. There was only one false alarm defined as an velocity series and divide by its standard deviation to obtain a automatic detection not selected manually. This event was the “standardized” velocity time series . second velocity peak in a complex stream that had been delib- Convolve the interface detector pattern with this series to obtain erately suppressed in the manual selection. There were five fail- the response function ures to autodetect a manual interface. Two of these were a consequence of missing velocity data. The remaining three would (2) have been selected had the threshold for detection been lowered to 0.4. In total, the automatic procedure detected 21 of the 26 Fig. 11 shows the high-resolution stream interface detector con- manually selected interfaces. structed by the above procedure. The top left panel presents the We have also determined the accuracy of the automatically 1995 ensemble of solar wind speed superposed relative to the detected interface time. The histogram of differences time of a zero crossing in the azimuthal flow angle. The heavy was biased toward being late with a mean value of 0.4 1434 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004

Fig. 12. Stream interface detector with 8-h resolution is shown. Detector (heavy curve with C symbols) is a derivative operator that detects the time of maximum slope on the speed profile. Result of convolving this detector with the speed profile shown by a thin line with crosses is given by the thin line with open circles. days, but it had its peak at zero offset. The rms deviation of the We have compared the interface times determined from the distribution was 0.6 days. Examination of individual events such Wang–Sheeley predicted solar wind velocity curve to the times as the one at the end of February 1995 shown in Fig. 5 suggests selected manually. We find that the mean difference in times that this distribution is a result of manually selecting the first is 0.9 days with a standard deviation of 1.1 days. From this, zero crossing of azimuthal flow whenever there were multiple it appears that the Wang–Sheeley model is predicting an early crossings. arrival of the solar wind at the Earth and that there is a fairly To apply this detector to the lower resolution prediction time large scatter in the predicted arrival times. series, we resampled the interface detector curve at 8-h intervals centered on the interface. The final detector plotted in Fig. 12 G. Forecast of ap Disturbance with a heavy curve identified by symbols consists of seven Our first ap forecast was performed with a program designed points. The result of convolving this curve with a low-resolution to test the procedure, but not to function in a real time situation. time series of median speed (thin line with small x) is shown by The results are presented in Fig. 14. The top panel compares the thin line identified by circles. (Note the response should be the observed and predicted solar wind speed. The bottom panel multiplied by 1.0). compares the observed ap (blocky line) with the predicted upper We have applied this detector to the time series of predicted quartile of ap (heavy dark line) and with the predicted lower speed during 1995. The results are presented in Fig. 13. The quartile (thick gray line at bottom). The result was obtained in output of the interface detector is plotted in the top panel. All the following manner. We created a two-dimensional column time intervals with absolute detector output greater than 0.5 array filled with data flags. This array had one column for each are taken as an event. The time of the event is defined as the automatically detected interface and a row for each 8-h sample center of these intervals. These are indicated in the figure by in the year 1995. The upper and lower quartiles of the 1995 en- heavy circles. These times are plotted on the graph of predicted semble of ap was generated for a 28-day interval centered on the solar wind speed in the bottom panel. For comparison, the manually selected interfaces times. This curve was filtered and heavy crosses identify the interfaces selected manually. The decimated to 8-h resolution. Then for each interface time auto- row of crosses and circles along the bottom correspond to matically selected, we placed the low resolution quartile curves events that were identified by one of the techniques, but not the into columns of the prediction array, centered at the appropriate other. According to this plot there were three false alarms (auto time. We then projected this array horizontally taking the max- without manual) and eight failures to detect (manual without imum value of ap in each row. These envelopes are plotted in auto). Close examination shows that some of the failures to the bottom panel of Fig. 14 with heavy lines. detect (4) are a result of missing data in the predicted time series It is expected that the observed data lies between the upper (this is a result of bad weather at the solar observatory). The and lower quartiles 50% of the time. The observations should three false alarms appear to be failures of the Wang–Sheeley exceed the upper quartile 25% of the time and should be less model to accurately predict the form of the speed variation. In than the lower quartile 25% of the time. Let us define an ap the predicted speed curve one event in February and two in July value above the upper quartile as a failure to predict, a value clearly differ from the observations (not shown). below the lower quartile as a false alarm, and any value between McPHERRON et al.: PROBABILISTIC FORECASTING OF 3-h ap INDEX 1435

Fig. 13. Profile of predicted solar wind speed during 1995 is plotted in the bottom trace. Large “X” show the times identified interactively as stream interfaces. Circles show the time selected by the automatic detector. Top panel presents the output of the detector with circles identifying the time its absolute value exceeds 0.5. Row of Xs at bottom show manually identified interfaces not detected by the automatic procedure. Circles represent auto detection of an interface not selected manually.

Fig. 14. Comparison of predictions with observations in 1995. Top panel compares observed (thin line) and predicted (squares) solar wind speed. Bottom panel compares the observed ap index (step-like curve) with predicted upper quartile (upper heavy line) and the lower quartile (lower heavy gray line). Circles denote the interface times selected manually and Xs the automatically selected times. Observed ap index should lie above the upper quartile 25% of the time. It should be below the lower quartile 25% of the time. the two quartiles as a success. We find that we failed to predict There are a number of obvious problems with this preliminary high values 21% of the time, that we had false alarms 31% of prediction scheme. Most important is the fact that the predicted the time, and success 48% of the time. These results are almost quartiles do not depend on solar wind velocity, IMF magnitude, exactly what we expect statistically. or IMF radial polarity. Because of this, the predictions are iden- 1436 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004 tical for weak streams (low velocity and IMF B) and for strong peaks at the interface, and then drops rapidly in the next 12 h streams. In principle, it should be possible to take these into ac- after the interface. However, for many days after the interface count, but in the year 1995 there were only 34 stream interfaces. activity is elevated and slowly decaying because of the high ve- It is not feasible to subdivide this ensemble into multiple subsets locity of the solar wind and a persistent negative bias in the mean each with enough data to define the cdfs required for the predic- value of Bz caused by the Russell–McPherron effect [29]. Also, tion. Even if we use the three years 1994–1996, we have only at any specific time relative to the interface the level of activity 76 stream interfaces and this is still insufficient. Observations of depends on the magnitude of the speed and the IMF. This be- the solar wind prior to 1995 provide only about 45% coverage havior makes it possible to predict the probability that activity of the solar wind and it is difficult to detect stream interfaces will exceed some threshold provided one can predict the tem- in these data. As yet, there are insufficient data from the latest poral waveform of these variables. These are exactly the param- solar minimum to improve out statistics from current data. eters predicted by the WSA model (WSA) [30]. Another problem is variable length streams. The data suggest It should be noted that it is possible to include probabilistic es- that even when stream interfaces are close together geomagnetic timatesas part of adeterministic forecast. In [33], local linear pre- activity dies to small values immediately before the arrival of diction filters are used to forecast the 2.5-min AL index from VBs the next interface. Our ensemble average curves are based on immediately upstream of the Earth. The filter used to advance the stream interfaces that are about 14 days apart. Consequently, prediction is determined instantaneously from the past history of our predicted activity decays more slowly after an interface that VBsandAL[10].Also,thecumulativeprobabilitydistributionfor it should following a stream of short duration. We have partially theerrorinthepredictionisdeterminedasafunctionofthesystem compensated for this effect by forcing the ensemble quartiles state. This distribution is used to generate contours of probability to gradually transition from the portion of the curves after the that the observed index will be greater or less than the predicted interface to the curves before the interface. index. This scheme differs from ours in several ways. First, it de- Other problems are a result of errors in the Wang–Sheeley pre- pends on upstream measurements close to the Earth and in partic- dicted speed profile. Some peaks in speed are entirely missed; ular on knowing the waveform of VBS. Consequently, the tech- some predicted peaks do not arrive at the correct time; and some nique cannot make medium-term forecasts. Second, this scheme peaks do not have the correct speed. One improvement would be usesthetimehistoryoftheinputandoutputtodefinethestateofthe to combine solar magnetograph data from all available ground system. Our technique uses time relative to a predicted arrival of observatories rather than use only a single observatory which a stream interface and predicted values of several important solar may be compromised by bad weather. Also, considerable im- wind quantities at this relative time. provement in the propagation algorithm moving solar wind from In the WSA model, astronomical observations of the Sun are the solar source surface to the Earth appears to be needed to im- converted to a time sequence of solar wind speed and IMF po- prove accuracy of the predicted interface times. Alternatively, larity that will arrive at the Earth in 3–5 days. (The real time pre- some method is needed for using real time measurements at L1 diction of IMF magnitude has not yet been implemented but it is to refine estimates of interface arrival times. For example, pre- thought to be straightforward and [22]). The time series of speed cursors such as the gradual increase in density and deflection of can be processed to obtain the time an interface will arrive at the the flow direction might be useful. Real-time detection of stream Earth several days later. This time is then used as the entry in a interfaces would allow one to offset the predicted solar wind pro- lookup table giving the probability that an activity index will ex- file to agree with observations. This should improve the quality ceed a specified threshold. In the example discussed, we did the of predictions in the days after a stream passes. inverse predicting the thresholds above and below which there is a 25% probability of occurrence. If the table is also a func- H. Discussion tion of solar wind speed then a more precise prediction should Deterministic models forecasting magnetic activity on the be possible. Refinement of the table to include IMF polarity and Earth require time series of the solar wind velocity and GSM Bz magnitude should further increase the quality of the prediction. component of theIMF.At the present time, there isno knownway Thus far, we have only a crude implementation of the fore- of accurately predicting IMF Bz for more than 30–60 min ahead cast procedure advocated here. In this paper, we used the time except inside of magnetic flux ropes [31], [32]. The component series of the upper and lower quartiles to predict the value of of the IMF is usually a stochastic variable produced by the arrival 3-h ap between which 50% of the values are expected to lie. of waves and discontinuities at the Earth. It is not likely that we We found that it did fall in this region 48% of the time. How- can remotely sense its waveform at the Sun, nor is it likely that ever, there were notable exceptions that can be attributed to that a waveform measured this far upstream of the Earth will hit inaccuracies in the time series of predicted solar wind speed. the Earth or retain its shape in traveling from a remote location For our simple procedure, the primary problem is that the pre- to the Earth. This situation would appear to preclude accurate dicted interface is either too early or too late as compared to long-term forecasts of space weather at the Earth. the real solar wind. Another serious problem is missing data. Fortunately, the statistical properties of the solar wind in- Since the procedure depends on observations of the photo- cluding V and Bz are often organized by structures in the solar spheric magnetic field, bad weather on the Earth causes gaps wind. For example, in this paper, we demonstrated that stream in the synoptic map that propagate to gaps in the time series interfaces (CIRs) caused by the collision of a high-speed stream of predicted speed at the Earth. These gaps interfere with our with a slow speed stream produce such organization. Geomag- simple stream interface detector introducing even longer gaps netic activity begins to increase about 12 h before the interface, in the ap prediction series. McPHERRON et al.: PROBABILISTIC FORECASTING OF 3-h ap INDEX 1437

There are a number of problems that must be solved to One might hope that the coming cycle of high-speed streams implement our procedure in a real-time environment. First is that is just beginning in the summer of 2003 could also be the problem of disparate sample rates. Solar wind data used in used. It is not obvious that this will be the case. Although not the development of the ensemble quartiles have roughly 90-s demonstrated in this paper, we find that virtually all of the major resolution, the ap index has 3-h resolution (but was resampled equinoctial stream interfaces in 1994–1996 were associated at 90-s), and the series of predicted solar wind speeds and polar- with an IMF that is geoeffective during high-speed wind after ities has approximately 9-h resolution. Note that this resolution the interface. By geoeffective we mean that the IMF obeyed is not compatible with the eight 3-h indexes obtained each day the Russell–McPherron rule: “Spring to and Fall away” which and it would be better to force this value to be precisely 8 h. is the IMF orientation near equinox that produces a southward In our ensemble statistics for the ap index, the 3-h resolution GSM field from the Parker spiral. This is almost certainly a is disguised by the fact that the fiducial time of the analysis is consequence of the polarity of the Sun during Cycle 22. In the uniformly distributed relative to the 3-h breaks in the index. current cycle the solar field has reversed and we might expect Consequently, the quartile curves appear continuous rather than that the IMF will be geoeffective before the interface when the step-like. speed is low. This should lead to slightly stronger activity before The stream interface detector used here can certainly be im- and weaker activity after the interface than in the last cycle. Despitetheproblemsdiscussedabove,theproceduredescribed proved. First, it would be more appropriate to use running es- in this paper is capable of making forecasts of the 3-h ap index timates of the mean and standard deviation to standardize the severaldaysinadvance.Nosuchprocedurecurrentlyexists.Even velocity time series. In our preliminary work, we used constant a crude implementation should be of considerable value. We values determined from a year of data. This change would allow are currently planning a real-time algorithm to implement these weak stream interfaces to produce a larger response and thereby ideas and hope to test the procedure in the coming year. exceed our threshold for detection. It may also be advantageous to recenter the interface detector at the minimum of the velocity ACKNOWLEDGMENT profile gaining both 12 h of advance warning and producing a more distinctive pattern. An additional improvement might be The authors would like to thank the NSSDC and the prin- development of a binary detector utilizing a neural network. cipal investigators of the Wind plasma and magnetic field exper- Another problem is the lack of sufficient data to define the iments for making their solar wind data available for this study. ccdfs for the ap index. In each year there are roughly 13 solar ro- They also thank the Wilcox Solar Observatory for the solar mag- tations. If there are two high-speed sectors per rotation we have netic data used to predict the speed profiles. They would also about 26 interfaces per year. Also, during the declining phase of like to thank R. Ulrich and the staff at Mount Wilson solar ob- the solar cycle there are only about three years in which clear servatory for providing access to the photospheric field data. high-speed streams are present. Thus, in the years 1995–1996 we expect to have about 75 interfaces. This is almost exactly REFERENCES what we found. The 75 events in the ensemble are barely suf- [1] W. D. Gonzalez, “A unified view of solar wind-magnetosphere coupling functions,” Planetary Space Sci., vol. 38, pp. 627–632, 1990. ficient to define a ccdf dependent on a single variable such as [2] W. D. Gonzalez, B. T. Tsurutani, A. L. C. Gonzalez, E. J. Smith, F. epoch time relative to the interface. They are insufficient to Tang, and S. I. Akasofu, “Solar wind-magnetosphere coupling during divide the dataset into subsets corresponding to high and low intense magnetic storms (1978–1979),” J. Geophys. Res., vol. 94, pp. 8835–8851, 1989. levels of the speed and IMF magnitude and IMF polarity. This [3] R. K. Burton, R. 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[14] J. A. Joselyn, Geomagnetic Activity Forecasting: The State of the Art, Robert L. McPherron received the Ph.D. degree from the University of Cali- vol. 33, pp. 383–401, 1995. fornia, Berkeley, in 1968. [15] R. L. McPherron and G. L. Siscoe, “Probabilistic forecasting of geomag- He is a Professor of space physics with a joint appointment at the Institute netic indices using solar wind air mass analysis,” Space Weather, vol. 2, of Geophysics and Planetary Physics and the Department of Earth and Space p. S01001, 2004. Sciences, University of California, Los Angeles. His research interests include [16] T. B. Low and D. R. Hudak, “Development of air mass climatology anal- the study of ultralow-frequency waves, magnetospheric substorms, magnetic ysis for the determination of characteristic marine atmospheres. I. North storms, solar wind, magnetosphere coupling, and space weather forecasting. He Atlantic,” Theoretical Appl. 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Goldstein, “Solar wind stream interactions and the wind speed-expansion factor relation- George Siscoe received the Ph.D. degree in physics ship,” Astrophys. J. Lett., vol. 488, pp. L51–L54, 1997. from Massachusetts Institute of Technology, Cam- [22] Z. Xuepu and J. T. Hoeksema, “Prediction of the interplanetary magnetic bridge, in 1964. field strength,” J. Geophys. Res., vol. 100, pp. 19–33, 1995. He has held research and faculty positions at [23] A. J. Ridley, “Estimations of the uncertainty in timing the relationship the California Institute of Technology, Pasadena, between magnetospheric and solar wind processes,” J. Atmospheric Massachusetts Institute of Technology, Cambridge, Solar-Terrestrial Phys., vol. 62, pp. 757–771, 2000. and the University of California, Los Angeles. He [24] D. R. Weimer, D. M. Ober, N. C. Maynard, W. J. Burke, M. R. Collier, is currently a Research Professor in the Center for D. J. McComas, N. F. Ness, and C. W. Smith, “Variable time delays in Space Physics, Boston University, Boston, MA. His the propagation of the interplanetary magnetic field,” J. Geophys. Res., professional activities include: Editor of the Journal vol. 107, pp. SMP 29-1–SMP 29-15, 2002. of Geophysical Research (1978-1981), Member of [25] D. R. Weimer, D. M. Ober, N. C. Maynard, M. R. Collier, D. J. the Advisory Committee for Atmospheric Sciences, National Space Foundation McComas, N. F. Ness, C. W. Smith, and J. Watermann, “Predicting (1978-1981), Member of the Committee on Planetary and Lunar Exploration, interplanetary magnetic field (IMF) propagation delay times using the National Research Council Committee (1981-1983), Chair of the Department minimum variance technique,” J. Geophys. Res., vol. 108, pp. SMP of Atmospheric Sciences, University of California, Los Angeles (1983-1988, 16-1–SMP 16-12, 2003. 1991-1993), Member of the National Research Council Space Studies Board [26] J. Chen, S. Slinker, J. A. Fedder, and J. 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