A new threedimensional magnetopause model with a support vector ...

JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, 2173–2184, doi:10.1002/jgra.50226, 2013

A new three-dimensional magnetopause model with a support vector regression machine and a large database of multiple spacecraft observations Y. Wang,1,2 D. G. Sibeck,1 J. Merka,1,2 S. A. Boardsen,1,2 H. Karimabadi,3 T. B. Sipes,3 J. Šafránková,4 K. Jelínek,4 and R. Lin5 Received 26 July 2012; revised 6 March 2013; accepted 10 March 2013; published 23 May 2013.

[1] We present results from a new three-dimensional empirical magnetopause model based on 15,089 magnetopause crossings from 23 spacecraft. To construct the model, we introduce a Support Vector Regression Machine (SVRM) technique with a systematic approach that balances model smoothness with fitting accuracy to produce a model that reveals the manner in which the size and shape of the magnetopause depend upon various control parameters without any assumptions concerning the analytical shape of the magnetopause. The new model fits the data used in the modeling very accurately, and can guarantee a similar accuracy when predicting unseen observations within the applicable range of control parameters. We introduce a new error analysis technique based upon the SVRM that enables us to obtain model errors appropriate to different locations and control parameters. We find significant east-west elongations in the magnetopause shape for many combinations of control parameters. Variations in the Earth’s dipole tilt can cause significant magnetopause north/south asymmetries and deviation of the magnetopause nose from the Sun-Earth line nonlinearly by as much as 5 Re. Subsolar magnetopause erosion effect under southward IMF is seen which is strongly affected by solar wind dynamic pressure. Further, we find significant shrinking of high-latitude magnetopause with decreased magnetopause flaring angle during northward IMF. Citation: Wang, Y., D. G. Sibeck, J. Merka, S. A. Boardsen, H. Karimabadi, T. B. Sipes, J. Šafránková, K. Jelínek, and R. Lin (2013), A new three-dimensional magnetopause model with a support vector regression machine and a large database of multiple spacecraft observations, J. Geophys. Res. Space Physics, 118, 2173–2184, doi:10.1002/jgra.50226.

1. Introduction [2] At rest, the magnetopause lies along the locus of points where magnetosheath and magnetospheric pressures balance. Consequently, the location of the magnetopause is a sensitive indicator of both the solar wind pressure and magnetospheric current systems from which magnetospheric pressures are derived. Accurate models for the magnetopause location are needed for many purposes, including testing numerical simulations for the solar wind1 Heliophysics Science Division, NASA/Goddard Space Flight Center, Greenbelt, Maryland, USA. 2 Goddard Planetary Heliophysics Institute, University of Maryland, Baltimore, Maryland, USA. 3 SciberQuest Inc., Del Mar, California, USA. 4 Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic. 5 National Space Science Center, Chinese Academy of Sciences, Beijing, China.

Corresponding author: Y. Wang, Heliophysics Science Division, NASA/Goddard Space Flight Center, Mail Stop 674, Greenbelt, MD 20771, USA. ([email protected]) ©2013. American Geophysical Union. All Rights Reserved. 2169-9380/13/10.1002/jgra.50226

magnetosphere interaction [Spence et al., 2004; García and Hughes, 2007], planning data acquisition for memory- and telemetry-limited missions, e.g., Time History of Events and Macroscale Interactions during Substorms (THEMIS) and Magnetospheric Multiscale (MMS), and space weather prediction [e.g., Baker et al., 1998]. [3] The task of magnetopause modeling has been approached both theoretically and empirically. Mattyn [1951] balanced solar wind and magnetospheric pressures to calculate the location of the subsolar magnetopause, while Ferraro [1960] and Mead and Beard [1964] derived analytical shapes for the magnetospheric cavity. Elsen and Winglee [1997] employed global magnetohydrodynamic models to determine the size and shape of the magnetopause as a function of solar wind dynamic pressure and interplanetary magnetic field (IMF) orientation. [4] Many studies have employed surveys of magnetopause crossings to construct empirical models for the size and shape of the magnetopause as a function of solar wind parameters. A few of them also included Earth’s dipole tilt. Over time, these studies have specified increasingly complicated geometric shapes for the magnetopause, including axially-symmetric single ellipses [Fairfield, 1971; Sibeck et al., 1991; Petrinec and Russell, 1993; Roelof and

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Figure 1. (From left to right) Distributions of the magnetopause crossings in our database in the GSM XZ, XY, and YZ planes. Sibeck, 1993], two second-order fits for the dayside and nightside [Formisano et al., 1979], a modified ellipse with a magnetotail flaring angle [Shue et al., 1997, 1998], and more complicated shapes specified to account for the cusp indentations [Boardsen et al., 2000; Lin et al., 2010]. All these studies demonstrated that the location and shape of the magnetopause depend strongly on solar wind dynamic pressure, IMF Bz , and Earth’s dipole tilt. [5] Although it may be less difficult to work with specified analytical forms, they fall short in describing the complex shape of the magnetopause, including its cusp indentations and asymmetries. Also, some or all of the major control parameters must be assumed to be independent from one another for their individual analytical trends. Machine learning techniques may prove helpful. Dmitriev et al. [1999] used an artificial neural network to construct a three-dimensional magnetopause model without assuming analytical shapes. However, the model could only be used to describe the dayside magnetopause and exhibited significantly larger fitting errors than other models [Suvorova et al., 1999]. Another major difficulty for all the previous magnetopause models is that each of them has only one global error for the whole magnetopause under all conditions. This is far from ideal in the statistical point of view because of the usually very uneven distribution of control parameters and crossing locations. As such, these errors are very limited in evaluating model accuracy for different situations. [6] In this study, we constructed a large magnetopause crossing database and used a support vector regression

machine (SVRM) to explore the magnetopause and its dependence on some major solar wind and geophysical parameters. The resulting new three-dimensional magnetopause model resolves all the major difficulties mentioned above for empirical magnetopause modeling. Specifically, it does not assume an analytical magnetopause shape or uncorrelation among different control parameters, avoids single parameter trend analysis, has superior fitting accuracy to both used and unused observations in model construction, has a nonlinear response to all control parameters, and provides a systematic approach to estimate model error at each magnetopause location for each set of conditions. The new model reveals magnetopause, including cusp structures, with complex nonlinear dependence on different control parameters.

2. Data [7] Geocentric Solar Magnetospheric (GSM) coordinates were used in this study. We built a database of 15,089 magnetopause crossings whose spatial distributions in the GSM XZ, XY, and YZ planes are shown in Figure 1. Our data consist of magnetopause crossings at locations with Xgsm –15 Re with valid solar wind conditions. These magnetopause crossings were observed by the 23 satellites listed in Table 1. In the database, crossings from Cluster1 are newly identified, those from Geotail, Interball 1, Magion4, and THEMIS A–E are from our co-authors in the Czech Republic, and the remaining ones are a collection of magnetopause crossings from various authors

Table 1. Number of Magnetopause Crossings and Their Time Span From Each Satellite Satellite

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AMPTE CCE (1984 August–1988 October) Cluster1 (2001 January–2004 December) Geotail (1992 October–1997 June) Hawkeye (1974 June–1977 July) IMP4 (1967 June–1968 December) IMP8 (1974 May–1978 September) Interball 1 (1995 August–1998 October) OGO-5 (1968 March–1969 April) Prognoz7 (1978 November–1979 February) THEMIS-A (2007 June–2008 November) THEMIS-C (2007 June–2008 November) THEMIS-E (2007 June–2008 November)

29 2556 1352 1973 87 8 1771 56 40 1183 1984 820

AMPTE IRM (1984 August–1986 January) Explorer33 (1967 January–1968 May) HEOS2 (1972 December–1973 July) IMP3 (1966 November–1966 November) IMP6 (1972 October–1973 December) ISEE (1977 October–1979 November) Magion4 (1996 March–1997 August) Prognoz10 (1985 May–1985 November) Prognoz8 (1981 January–1981 September) THEMIS-B (2007 June–2008 November) THEMIS-D (2007 June–2008 November) Total

36 20 6 3 21 353 119 31 71 1693 877 15089

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Figure 2. Crossing location, in GSM longitude and latitude, and other control parameter distributions of the database after east-west and north-south duplication. using older satellite observations between 1963 and 1998 (http://ftpbrowser.gsfc.nasa.gov/magnetopause.html). During one satellite passage, there can be multiple magnetopause crossings because of the motion of the magnetopause under changing solar wind conditions. In our database, some data sets count multiple magnetopause crossings individually, e.g., those from Hawkeye and Cluster1, and some others only count one from multiple crossings during one passage, e.g., those from ISEE. Ideally, different data points from these data sets should carry different weight depending on how many actual crossings they represent. However, it is not practical to find them out since many data sets were obtained a long time ago. Later results will show that our uniform treatment of these data does not prevent us from getting good fitting of the model to the observations. Our ongoing large-scale multiple spacecraft magnetopause crossing database construction with uniform criteria will eventually help us eliminate such data inconsistency in future modeling. We chose solar wind dynamic pressure (Pdyn ) and IMF Bz as the key control parameters that have been widely used by previous models. In addition, we also chose Earth’s dipole tilt () which is generally believed to have major effect on magnetopause shape but has only been used in a few magnetopause models [e.g., Formisano et al., 1979; Boardsen et al., 2000; Lin et al., 2010]. In our database, the older magnetopause crossings come with hourly-averaged solar wind Pdyn and IMF Bz from various spacecraft. For the other crossings, we use instantaneous 1min OMNI database values that are shifted to the nose of the bow shock [King and Papitashvili, 2005] for solar wind control parameters. When calculating solar wind dynamic pressure, we assumed 5% alpha particles and 95% protons in the solar wind. [8] To mitigate the uneven dawn-dusk distribution of crossings seen in Figure 1, we assume dawn-dusk symmetry of the magnetopause and duplicate points by reversing

their y locations. To address the uneven north-south distribution of crossings, we reverse Zgsm and , assuming that the magnetopause is a mirror image of itself across the equatorial plane for the same toward and away from the Sun. There are factors that can cause magnetopause asymmetry, i.e., transversal solar wind speed and the Earth’s aberration, which can have complex effects and will be left for a future study. Figure 2 shows the distributions of the final magnetopause crossings used in this study after the above duplications. We see a substantial number of observations for Pdyn between 0.5 and 7 nPa, IMF Bz between –8 and 6 nT, and between –35ı and 35ı . Besides, the spatial coverage of the crossings is also very good except farther down the tail. These ranges are the valid parameter ranges of our new model. Model extrapolation beyond these ranges is allowed by the model, but caution should be used, especially when control parameters are far away from these valid ranges. [9] To allow maximum flexibility of the new model for complex and unexpected magnetopause structures, we could use the radial distance from the Earth, r, as the parameter to be fitted, with GSM longitude, (0ı on the +X axis and positive/negative toward duskward/dawnward), and latitude, (0ı on the +X axis and positive/negative toward northward/southward), as additional control parameters. However, this choice leads to singular points near the poles where locations close in space, which should have similar r values, can be far apart in longitude and have quite different fitted r values. To avoid this problem, we used an n-vector, a three-parameter non-singular position representation, to replace and : nx = cos( ) cos(), ny = cos( ) sin(), and nz = sin( ).

3. Method [10] The Support Vector Machine (SVM) is a supervised machine learning technique in the framework of statistical

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learning theory. First developed in the 1960s [Vapnik and Lerner, 1963], SVM has exhibited considerable promise as a classification technique [Noble, 2006]. SVRM is an extension of SVM and was first proposed by Drucker et al. [1997]. The basic idea of SVRM is to map multi-dimensional data into a high-dimensional feature space, seven-dimensional in this case including both control and fitted parameters, via nonlinear mapping through a selected kernel function, and to perform a linear regression in this space. The result is a smooth surface in this multi-dimensional space that best fits the data within some tolerance range and can further be used for prediction at other locations in this space. The key mathematical challenge of SVRM is to minimize the sum of the model errors for all the data points and a complexity term that enforces the flatness of the surface in the high-dimensional feature space. This leads to solving a quadratic programming problem, which is uniquely solvable [Vapnik, 1995]. The solution is represented by a subset of the training data as support vectors that are automatically selected by SVM/SVRM algorithm and can be used to uniquely determine the fitted surface. For a more comprehensive introduction to SVRM, please refer to Smola and Schölkopf [2004]. [11] LibSVM is a freely available and widely used implementation of SVRM which handles all the complex technical details for this technique [Chang and Lin, 2001]. Version 2.86 of LibSVM was used in this study for model fitting and predictions. In LibSVM, different SVM types and kernel functions are supported. Here, we chose commonly used 2 -SVR and Radial Basis Function (RBF), e– |u–v| (the function value around a point, v, is dependent on the distance from this point), respectively as recommended by the authors of LibSVM because this combination is very flexible and robust in handling many problems and is free from many difficulties of some other combinations. Specifically, RBF is used first to map our data to high dimensional feature space, then in this feature space, -SVR is used to obtain a smooth model surface. As we will show later in the paper, they also work very well for our problem. [12] There are three major steps to create an SVRM model. First, we need to create a specific LibSVM input data file, train.dat, which is a simple text file. In train.dat, the first value of each line is the fitted value, and the control parameter values go after it. Before each control parameter value, there is a string, like “1:”, to mark which control parameter it belongs to. Second, we need to scale the control parameters to within –1 and 1 for better accuracy using LibSVM command: svm-scale -s scaling_parameters.dat train.dat > scaled_train.dat. The scaled data file is scaled_train.dat and the scaling parameter file, scaling_parameters.dat, is also saved which will be needed for later model prediction. Finally, we can create the model using LibSVM command: svm-train -s 3 -t 2 -p * -c * -g * scaled_train.dat model.dat, here “-s 3” selects -SVR, “-t 2” selects RBF, “-p *” sets p which is the distance from the fitted surface within which no fitting error is counted for a data point, “-c *” sets the error factor, C, for data points falling outside of distance p, and “-g *” sets the value in the RBF which controls how far a discrete data point is effective in constructing continuous fitted parameter function for fitting in the high-dimensional feature space. Note that both p and C are introduced through -SVR. In the resulting model file, model.dat, there are first

some basic model fitting parameters including SVM type and kernel, followed by support vectors which are a subset of the training data selected by LibSVM. For our new magnetopause model, there are 60356 support vectors. Using this model file, LibSVM can later reconstruct the fitted model surface in high-dimensional feature space and provide model values for user input control parameters. [13] There are three major steps to make model prediction using the above generated model. First, we need to create a specific LibSVM input data file, predict.dat. This input data file has the same format as train.dat except that the fitted values can be set to zero since they are to be calculated. Second, we need to scale the control parameters in predict.dat to within –1 and 1 using the scaling parameter file produced above with LibSVM command: svm-scale -r scaling_parameters.dat predict.dat > scaled_predict.dat, where scaled_predict.dat is the resulting scaled input file. Finally, we can make model prediction using LibSVM command: svm-predict scaled_predict.dat model.dat result.dat where model.dat is the model file that we obtained above, and result.dat saves the model values under the conditions in predict.dat. To help people make their own magnetopause calculations, we will provide our new magnetopause and its error models as model and scaling parameter files. Users can download LibSVM and follow the steps above to make their own model calculations. A very easy web interface will also be provided so that users can simply provide their interested conditions to obtain model magnetopause location. [14] Since a finite p leads to the fitted surface systematically deviating from the center of the data distribution, which is not what we want, we set p to 0, thereby requiring the fitted surfaces to go through the center of the data distributions as much as possible. Larger leads to a smaller data effective range and model reflects more local observations. Larger C leads to tighter fitting of the data points which can adversely affect model smoothness. As a result, a smaller pair of ( , C) leads to a smoother but less accurate fit to the data, while a larger pair of ( , C) leads to more bumpy but more accurate fit to the data. A major challenge for our model construction, which is also a key human supervision part of the SVRM modeling in this study, is to find a balanced pair of ( , C) for both fitting accuracy and model smoothness. [15] A standard method to perform a grid search (C = 2–5 , –3 2 , : : : , 215 , and = 2–15 , 2–13 , : : : , 23 ) for the best fit and C was provided by Chang and Lin [2001]. After getting the best fitting pair of (C, ), a refined uniform grid search around it can be done to get a better pair of (C, ). However, such a method does not work for our problem because it leads to an overfitted bumpy magnetopause that overly emphasizes local observations and neglects global smoothness of the magnetopause. In our study, we searched a unit grid of from 1 to 5 and C from 1 to 30. The ending search values of and C were determined after trials. For each pair of and C, we performed SVRM model fitting of our data set, then plotted model results for a set of standard control parameters of Pdyn , IMF Bz , and in the GSM XZ, XY, and YZ planes, together with global root-mean-square errors (RMSEs). We searched from the smallest and C values to the largest and C values until we began to see unusual bumpy structures clearly visible to the eyes that we believe reflect local small-scale observations rather than global patterns. This led to the best pair of = 1 and C = 20 for our

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Figure 3. (a and b) Model magnetopause location in the GSM XY plane for the final model and a model with a larger pair of ( , C) for Pdyn = 1, 2, 4 nPa, IMF Bz = 0 nT, and = 0ı . (c and d) Magnetopause error model values in the GSM XY plane for the final error model and an error model with a larger pair of ( , C) for Pdyn = 1, 2, 4 nPa, IMF Bz = 0 nT, and = 0ı . final magnetopause model. Later results with global magnetopause structures free from localized small-scale structures, high fitting accuracy to both known and unknown observations, and consistency with previous models and theoretical predictions show that it is OK for the best pairs of and C to be at the boundary of their search grids, and there is no need to perform further search for non-integer values for and C. Panels a and b of Figure 3 show model magnetopause location in GSM XY plane with Pdyn = 1, 2, 4 nPa, IMF Bz = 0 nT, and = 0ı for the final model with the chosen pair of ( , C) and a model with a larger pair of ( , C). Clearly, we see more bumpy magnetopause structures with the larger pair of ( , C), which are more likely reflecting localized observations. The RMSE of r from our final model

is 1.12 Re. The RMSE for the dayside crossings, about 2/3 of the total crossings, is 0.87 Re, and the RMSE for the nightside crossings, about 1/3 of the total crossings, is 1.52 Re. As such, the significant amount of nightside crossings in our database are a major contributor to the global error of the model. This is easy to understand because the location of the nightside magnetopause is more likely affected by the Earth’s aberration and transverse solar wind velocity, which are not treated in this study for simplicity. Later in the paper, we will only show results with Xgsm –5 Re, so we can concentrate on the more accurate sunward side of the magnetopause. [16] Table 2 lists the RMSEs from our model and some previous models, and shows that our model is one of the most accurate among them. These RMSEs are calculated from all the data used, respectively, in these studies. As such, they do not provide an adequate means of estimating how these models perform when predicting unseen observations. To ensure that our new model does not overfit and that it can successfully predict unseen observations, we followed the approach of Karimabadi et al. [2009] to randomly employ 2/3 of our database in building a testing model, and then used this testing model to predict the remaining 1/3 of the database that the testing model had never seen. We obtained an RMSE of 1.14 Re, which is very similar to that with all the data, 1.12 Re. This indicates that our new model can guarantee a similarly high accuracy for both known and unknown observations. [17] In reality, a uniform model error for all locations and all control conditions is far from ideal and cannot possibly reflect uneven observations. To improve this situation, we used a procedure similar to the one above to construct an error model for the new magnetopause model. Now, instead of fitting magnetopause radial distance, we fit the absolute errors between the model magnetopause radial distance and the observed radial distance for the same control parameters. We assumed p = 0 and searched a unit grid of from 1 to 2 and C from 1 to 20. Again, the ending search values of and C were determined after trials. We checked model results for the same set of standard control parameters of Pdyn , IMF Bz , and in GSM XZ, XY, and YZ planes and chose = 1 and C = 1 for the final error model to avoid very small or even negative errors. Figures 3c and 3d show magnetopause error model values for Pdyn = 1, 2, and 4 nPa, IMF Bz = 0 nT, and = 0ı in the GSM XY plane from the error models with the selected pair and a larger pair of ( , C). Here, the direction of each point on the curves is determined by the model magnetopause location, and its distance from the origin is the error model value. Clearly, the larger pair leads to overly small fitting errors under certain conditions

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Table 2. Model Error Comparison Model

RMSE (Re)

Roelof and Sibeck [1993] Shue et al. [1998] Kuznetsov and Suvorova [1998]a Dmitriev et al. [1999]a Boardsen et al. [2000] Lin et al. [2010] SVRM

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in these regions, not because model errors are truly small, but because there are not enough observations for a statistically valid error estimation. To avoid these misleading small errors and mark the parameter space where there are too few observations, we set a lower threshold of 0.2 Re for the error model. All the error model values smaller than this threshold, as well as their corresponding model magnetopause locations, are less trustworthy because of the likely insufficient observations. [19] To finally confirm that our new magnetopause model and its error model fit the observations well, we use the following conditions as an example: Pdyn0 = 2 nPa, IMF Bz0 = 0 nT, and 0 = 0ı . Figure 5 from left to right shows the model magnetopause (solid line) and its error bounds (dotted lines) in the GSM XZ, XY, and YZ planes for this set of conditions. Magnetopause crossings within 1 Re of each plane, Pdyn0 ˙ 0.4 nPa, IMF Bz0 ˙ 1 nT, and 0 ˙ 5ı , are shown as black dots. Only a few points satisfy these conditions, and overall they fit very well with the magnetopause model. The error bounds from the error model also capture the errors at different locations very well. In the right panel, the oval model shape captures the east-west elongated distribution of magnetopause crossings, which would certainly not be captured with axisymmetric models, and is qualitatively consistent with earlier results by Sotirelis and Meng [1999], Boardsen et al. [2000], and Lin et al. [2010].

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(sometimes very close to zero), while errors for the chosen pair behave much better. Figure 4 shows samples of the new magnetopause model in the GSM XZ plane with Pdyn = 2 (solid line) and 4 (dashed line) nPa, IMF Bz = 0 nT, and = 0ı , as well as their error bounds from the magnetopause error model (dotted lines). Clearly, there are different errors at different locations under different conditions. [18] In parameter space near the boundary of the data distribution, there are regions with very sparse data points. The new magnetopause model tends to fit very close to these points, which is exactly what the model should do. However, this could lead to very small error model values

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4. Results [20] Figure 6 shows the model magnetopause dependence on Pdyn in the GSM XZ, XY, and YZ planes for Pdyn = 1, 2, and 4 nPa, IMF Bz = 0 nT, and = 0ı . In this figure and Figures 7 and 8, we do not show error bounds because they would obscure trends in model magnetopause locations. In the figure, we see a compression of both the dayside and nightside magnetopause with increasing Pdyn . Clear cusp indentations are seen in the XZ plane for all the Pdyn values, and they move inward with the magnetopause for increasing

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Figure 5. (From left to right) Model magnetopause (solid line) and its error bounds (dotted lines) in the GSM XZ, XY, and YZ planes for Pdyn0 = 2 nPa, IMF Bz0 = 0 nT, and 0 = 0ı . Magnetopause crossings within 1 Re of each plane, Pdyn0 ˙ 0.4 nPa, IMF Bz0 ˙ 1 nT, and 0 ˙ 5ı are shown as black dots. 2178

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Figure 6. (From left to right) Model magnetopause dependence on Pdyn in the GSM XZ, XY, and YZ planes for Pdyn = 1, 2, and 4 nPa, IMF Bz = 0 nT, and = 0ı . Pdyn . Note that the cusp structures in this study are the outer cusp boundary which is the boundary between the cusps and the magnetosheath. The magnetopause in the XY and YZ planes is not only much less structured but also shows clear compression with increasing Pdyn . In the XY plane, the subsolar magnetopause becomes blunter at Pdyn = 4 nPa than at 1 and 2 nPa. In addition, the magnetopause crosssection in the YZ plane is east-west elongated for all the Pdyn values. For Pdyn = 2 nPa, the radius of the magnetopause along the Y axis is 16.8 Re, while it is only 14.4 Re along the Z axis. [21] Figure 7 shows the model magnetopause dependence on IMF Bz in the GSM XZ, XY, and YZ planes for IMF Bz = 0 and ˙4 nT, Pdyn = 2 nPa, and = 0ı . In the XZ plane, the subsolar magnetopause lies only slightly further radially outward from Earth for Bz = 4 nT than for Bz = 0 nT; IMF Bz= -4 nT IMF Bz= 0 nT IMF Bz= 4 nT

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perhaps, because northward magnetosheath magnetic field lines are appended to the dayside magnetopause by reconnection poleward of the cusps. By contrast, the high-latitude magnetopause flares less and lies closer to the Earth for Bz = 4 nT than for Bz = 0 nT; perhaps, because reconnection poleward of the cusps transforms open lobe magnetic field lines into closed flank magnetic field lines. Further, we see clear subsolar magnetopause erosion during southward IMF in comparison with zero IMF Bz , which is likely caused by subsolar reconnection. Also, a clear expansion of the high-latitude magnetopause with larger magnetopause flaring angle is observed, which is likely caused by the addition of newly reconnected field lines generated by merging on the equatorial magnetopause. Again, clear cusp indentations are seen for all these IMF Bz values, and we notice that the cusps shift toward the equatorial plane as IMF Bz turns southward.

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The magnetopause in the XY plane clearly demonstrates that the outward expansion of the magnetopause during IMF Bz = 4 nT in comparison with zero IMF Bz only occurs in a narrow region close to the subsolar point. Outside that region, magnetopause compression is seen. By comparison, magnetopause erosion effect during IMF Bz = –4 nT is seen over a broad range of local times. In the YZ plane, IMF Bz has a clear influence on the high-latitude magnetopause but little affect on the low-latitude magnetopause flanks. In addition, the magnetopause under all these IMF Bz shows a clear east-west elongation. [22] Figure 8 shows the model magnetopause dependence on Earth’s dipole tilt in the GSM XZ, XY, and YZ planes for = 0ı , 15ı , and 25ı , Pdyn = 2 nPa, and IMF Bz = 0 nT. In the XZ plane, we see drastic changes of magnetopause shape with increasing , including a breakdown in the north-south symmetry and a big displacement (more than 2 Re for 15ı tilt and 4 Re for 25ı tilt) of the magnetopause nose, the point closest to the Sun. In addition, the cusps see similar drastic changes, including big displacements and changes in depth with increasing . Clear cusp indentations are seen for all values but 25ı when the northern cusp is almost flattened. The magnetopause in the XY plane is much less

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lines) and its error bounds (dotted lines) on Pdyn , IMF Bz , and , respectively, for various sets of control parameters. This distance is an indication of high-latitude magnetopause size and can be used to estimate its erosion/field line accumulation. The figure clearly shows the dependence of this distance on all the control parameters. The left panel shows that distances decrease with increasing Pdyn for all the combinations of the other two control parameters. The middle panel shows that the location of the high latitude magnetopause shows little change when IMF Bz increases from –8 nT to –5 nT, but it shrinks steadily when IMF Bz increases from –5 nT to 6 nT. The right panel shows that the distance to the northern polar increases steadily with from 0ı to 35ı . [25] Figure 11 from left to right shows the dependence of the magnetopause equatorial dusk flank, model intercept on the +Y axis, (solid lines) and its error bounds (dotted lines) as a function of Pdyn , IMF Bz , and , respectively, for various sets of control parameters. Once again, all parameters have a strong influence on the magnetopause flank location. The left panel shows that distances to the polar magnetopause decrease with increasing Pdyn for all combinations of the other two control parameters. The middle panel shows that this distance has weak peaks close to IMF Bz = 0 nT

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we add such a curve with the proper factor to align it with model results to the left panel of the figure for reference. Model curves follow the theoretical trend quite closely for Pdyn > 6 nPa. For smaller Pdyn , the new model’s subsolar magnetopause distance generally increases faster than the analytical prediction with decreasing Pdyn until Pdyn = 1.5 nPa when the analytical curve increases dramatically. Since we have demonstrated that our model reflects observations very well, the above discrepancy could be caused by over simplification of the analytical formula that misses many critical factors, including the complex geometry of the magnetopause and its effects on solar wind flow. The middle panel of Figure 9 clearly shows rapid inward erosion of the magnetopause as the IMF turns southward, but little change in position as the IMF turns northward. The erosion is much stronger for Pdyn = 1 nPa and much weaker for Pdyn = 4 nPa. The right panel of Figure 9 shows that the dependence of the subsolar location on is highly regulated by Pdyn . When Pdyn is not very large, subsolar magnetopause distance generally decreases with increasing . However, when Pdyn = 4 nPa, subsolar magnetopause distance changes little with . [24] From left to right, Figure 10 shows the dependence of the model magnetopause intercept on the +Z axis (solid


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Magnetopause nose deviation from Sun-Earth line (Re)


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for all sets of conditions except for Pdyn = 2 nPa and = 25ı . The right panel indicates that the distance to the flank generally decreases with increasing for all combinations of the other two control parameters except when Pdyn = 4 nPa and = 0ı . [26] Most previous magnetopause models assume axisymmetric analytical forms that put the subsolar point on the Sun-Earth line. However, as the results shown in the left panel of Figure 8 indicate, this is not valid when Earth’s dipole tilt is nonzero. Figure 12 shows how far the nose of the magnetopause lies from the Sun-Earth line with increasing Earth’s dipole tilt for several typical sets of Pdyn and IMF Bz values. The nose of the magnetopause starts to deviate from the Sun-Earth line as soon as becomes nonzero and its distance from the Sun-Earth line increases nonlinearly with . When = 35ı , the deviation can range from 3.7 Re for high Pdyn and no IMF to as much as 4.9 Re for low Pdyn and no IMF. IMF Bz clearly has strong effect on the nose shift too, but not as strong as Pdyn . As a comparison, Sotirelis and Meng [1999] used T96 model [Tsyganenko, 1995] and pressure balance to study the shape of the magnetopause. They found that the vertical offset of the nose from the Earth-Sun line varied linearly with dipole tilt angle, reaching 3 Re for maximal tilt and having a weak dependence on solar wind dynamic pressure.

[27] In this study, we built a new three-dimensional magnetopause model with multi-spacecraft observations and a support vector regression machine. No pre-assumed analytical forms were used in the model and the shape of the magnetopause, including its cusps and complex dependence on various control parameters, is automatically determined after we find a balance between good fit to the observations and global smoothness of the model magnetopause. We also introduced a systematic error analysis approach that allows the error to be determined at each location for each set of conditions and makes it very convenient to assess magnetopause model accuracy as never done before. The new model is one of the most accurate among some most widely used and/or most complex previous models that we have checked for known observations based on RMSE. Furthermore, our new model has similarly high prediction accuracy for unknown data which is not guaranteed by the previous models examined. Data and model comparison indicates that the new model represents observations very well. [28] Debates continue about the existence of the cusp indentations [e.g., Lavraud et al., 2004]. Our model predicts clearly defined outer cusp indentations for most conditions tested. On the other hand, it also predicts that the cusp indentation can disappear during large dipole tilt. We also find that the location and depth of the cusp indentations are controlled by Pdyn , IMF Bz , and Earth’s dipole tilt. In addition, our model results are consistent with theoretical and simulation predictions that the magnetopause moves inward with increasing solar wind dynamic pressure. We also see that the dayside magnetopause moves inward with negative IMF Bz , a clear indication of magnetopause erosion caused by reconnection. However, such an erosion is highly controlled by solar wind dynamic pressure, with more decrease of magnetopause size for smaller Pdyn and less decrease of magnetopause size for larger Pdyn for the same drop of IMF Bz . This can be easily explain by the fact that the amount of magnetic flux removed by reconnection is less significant compared to the local magnetic field when the magnetopause is highly compressed by higher solar wind dynamic pressure. [29] Table 3 shows model subsolar magnetopause locations and their errors for Pdyn = 1, 2, 3, and 4 nPa, IMF Bz = 0 nT, and = 0ı from our new model and several previous models: Roelof and Sibeck [1993], Shue et al. [1998], Boardsen et al. [2000], and Lin et al. [2010]. The errors of the previous models are the RMSEs which are

Table 3. Comparison of Subsolar Magnetopause Locations and Their Errors From Several Models for Pdyn = 1, 2, 3, and 4 nPa With Zero IMF Bz and Earth’s Dipole Tilt Pdyn

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taken from their papers, the errors of our model are from our new magnetopause error model. The new model’s subsolar magnetopause location agrees reasonably well with that of the previous models when errors are considered. But our new model has more expanded subsolar magnetopause for Pdyn = 1 and 2 nPa and more compressed subsolar magnetopause for Pdyn = 4 nPa. Note that the new model has different and smaller errors than the previous models, each of which has the same larger error for different Pdyn . In addition, the errors for the new models are larger for smaller Pdyn and smaller for larger Pdyn . Some previous models have similarly small global fitting errors near the subsolar magnetopause as our model errors in Table 3 [e.g., Boardsen et al., 2000; Lin et al., 2010]. However, those models cannot systematically provide error evaluation for different locations and conditions as we do. As such, their single global errors can hide their actual lower errors at some locations while ours will not. [30] To obtain analytical descriptions, conventional methods usually treat different parameters independently, assuming that the other parameters are uncorrelated. Sometimes this treatment can be significantly flawed because some supposedly uncorrelated parameters can be highly correlated. For example, the longitude and Earth’s dipole tilt of the Cluster1 magnetopause crossings in this study are highly correlated because of Cluster1’s orbital bias (crosscorrelation coefficient –0.88). On the other hand, treating less correlated parameters as uncorrelated is not accurate either. Throughout the entire process of our new model construction, we avoided considering individual parameter trends and did not need to worry about control parameter correlation at all. In addition, not assuming any analytical shape makes it possible to study the true dependence of magnetopause location on different control parameters. This is especially convenient when there is no physically sound way to guess an analytical relation, e.g., the deviation of the magnetopause nose from the Sun-Earth line for nonzero Earth’s dipole tilt. Moreover, with a larger database, this technique can easily include new more refined and more accurate modeling of the observations than that can be done using smaller database. In this way, new model can be easily regenerated to make use of new data without being restricted by the limitations of specified analytical forms. [31] In this study, we explicitly introduced a new method not to solely consider the accuracy of the fitting but to balance the fitting accuracy and global smoothness of the model. In previous analytical models, functional forms of the shape are already set and smoothness is implicitly assumed, except for a very few models with discontinuous regional boundaries. As such, our requirement of model smoothness is not a disadvantage compared to previous analytical models. Instead, this requirement, together with non-analytical formula restrictions, allows us to explore the real dependence of the model on different control parameters while avoiding wiggles and rigid patterns from overfitted local observations.

6. Summary and Conclusions [32] In this study, we used advanced machine learning technique, a support vector regression machine, and an unprecedented large magnetopause crossing database

to construct a new 3-D magnetopause model that does not assume pre-defined analytical forms and uncorrelation among control parameters. Below are the major results from this study: [33] 1. We constructed a large multi-spacecraft database consisting of 15,089 magnetopause crossings by 23 spacecraft which have great spatial and other control parameter coverage. [34] 2. The Support Vector Regression Machine technique with a newly proposed method to balance model smoothness and fitting accuracy has proven to be very successful in producing a new 3-D magnetopause model which has one of the best fit to observations among previous models. The new model fits not only known but also unknown observations accurately, and it reveals true trends from data distributions instead of being restricted by pre-defined analytical forms. [35] 3. We propose a new error analysis technique based on SVRM which produces a magnetopause error model that provides different model error evaluations at different locations and under different sets of conditions. This technique reflects model error distributions much better and provides much more refined evaluation to model error than previous single error techniques. [36] 4. Axisymmetry is far from a good approximation for the magnetopause, and an east-west elongated magnetopause is found for all sets of conditions tested. [37] 5. Earth’s dipole tilt has a significant effect on the magnetopause shape. It can lead to major north-south asymmetries and cause the magnetopause nose to move off the Sun-Earth line nonlinearly by as much as 5 Re. Also, it affects the location and depth of the cusps significantly, i.e., the northern cusp is significant flattened for large tilt. [38] 6. Magnetopause erosion effects are seen at the subsolar point during southward IMF, which are stronger when the solar wind dynamic pressure is weaker. When IMF Bz is positive, the subsolar magnetopause is nearly stationary for all the conditions tested, and we find significant shrinking of the high-latitude magnetopause with smaller magnetopause flaring angle under moderate northward IMF. [39] The new systematic SVRM model building and error evaluation techniques introduced here not only can solve many long lasting problems in magnetopause empirical modeling but also have great potential to significantly advance empirical modeling of problems of similar nature, namely unknown and/or complex spatial structures controlled by various correlated or uncorrelated parameters. The new approach can easily handle large amounts of data with many control parameters and will find a smooth model to best fit the known data and provide similar prediction accuracy for unknown observations within valid ranges of the model. [40] Acknowledgments. We thank both reviewers for their valuable comments and constructive suggestions. The work at GSFC was supported by NASA Geospace SR&T Program under grant NNX09AP16G. Work at SciberQuest, Inc. was supported by NASA’s Heliophysics Data Environment Enhancements Program. We thank the Cluster FGM, CIS, and PEACE instrument teams and ESA Cluster Active Archive for Cluster data. We thank Space Physics Data Facility (SPDF) and National Space Science Data Center (NSSDC) for providing the OMNI data set.

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