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” 0093-3813/04$20.00 © 2004 IEEE 1426 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004 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.