Dynamic Radiation Environment Assimilation ... - Wiley Online Library

SPACE WEATHER, VOL. 10, S03006, doi:10.1029/2011SW000729, 2012

Dynamic Radiation Environment Assimilation Model: DREAM G. D. Reeves,1 Y. Chen,1 G. S. Cunningham,1 R. W. H. Friedel,1 M. G. Henderson,1 V. K. Jordanova,1 J. Koller,1 S. K. Morley,1 M. F. Thomsen,1 and S. Zaharia1 Received 6 September 2011; revised 6 January 2012; accepted 26 January 2012; published 21 March 2012.

[1] The Dynamic Radiation Environment Assimilation Model (DREAM) was developed to provide accurate, global specification of the Earth’s radiation belts and to better understand the physical processes that control radiation belt structure and dynamics. DREAM is designed using a modular software approach in order to provide a computational framework that makes it easy to change components such as the global magnetic field model, radiation belt dynamics model, boundary conditions, etc. This paper provides a broad overview of the DREAM model and a summary of some of the principal results to date. We describe the structure of the DREAM model, describe the five major components, and illustrate the various options that are available for each component. We discuss how the data assimilation is performed and the data preprocessing and postprocessing that are required for producing the final DREAM outputs. We describe how we apply global magnetic field models for conversion between flux and phase space density and, in particular, the benefits of using a self-consistent, coupled ring current–magnetic field model. We discuss some of the results from DREAM including testing of boundary condition assumptions and effects of adding a source term to radial diffusion models. We also describe some of the testing and validation of DREAM and prospects for future development. Citation: Reeves, G. D., Y. Chen, G. S. Cunningham, R. W. H. Friedel, M. G. Henderson, V. K. Jordanova, J. Koller, S. K. Morley, M. F. Thomsen, and S. Zaharia (2012), Dynamic Radiation Environment Assimilation Model: DREAM, Space Weather, 10, S03006, doi:10.1029/2011SW000729.

1. Introduction [2] The space environment poses a number of hazards to space systems including total dose effects, surface dose/ chemistry, surface charging/discharging, bulk charging/ discharging, single event effects and others [National Space Weather Program (NSWP), 2010]. For this reason some space systems are equipped with space environment sensors. Data from those sensors can be used to: determine the causes of anomalies; specify current conditions that may affect operations or reduce performance; predict remaining reliable operational lifetime, etc. However, for a variety of reasons (including cost and resources), the majority of satellites are not equipped with space environment sensors. Satellite operators are often forced to use data obtained in one orbit (e.g., geosynchronous) to infer conditions in a completely different class of orbits. We know, though, that the satellite environment varies substantially as a function of time, orbit, particle energy, and particle species. Therefore a major challenge to the field of space weather is to use 1 Space Science and Applications, Los Alamos National Laboratory, Los Alamos, New Mexico, USA.

This paper is not subject to U.S. copyright. Published in 2012 by the American Geophysical Union

the available models and observations to accurately specify, and ultimately predict, the space environment at any satellite, in any orbit, at any time. [3] This paper describes the current status of, and results from, the Dynamic Radiation Environment Assimilation Model (DREAM). DREAM has two major objectives. The first is to accurately specify the global, time-dependent, space environment out to beyond geosynchronous orbit. The second is to better understand the physical processes that control the space environment [Reeves et al., 2005]. To date, DREAM has focused primarily on specification and understanding of the outer electron radiation belts which cover roughly the region from 3 to 7 Earth radii (RE) and energies on the order of a million electron volts (MeV). However, the basic framework of DREAM is extensible to a wide range of particle energies and regions of the inner magnetosphere. [4] Electrons with MeV energies (also referred to as relativistic electrons) are responsible for bulk charging/ discharging in spacecraft components (particularly dielectrics); they are a major source of the total dose to which spacecraft are exposed; and they produce bremsstrahlung x-rays as electrons interact with spacecraft materials.

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REEVES ET AL.: DYNAMIC RADIATION ENVIRONMENT ASSIMILATION MODEL

Relativistic electron fluxes in the outer radiation belt vary dramatically in response to solar, interplanetary, and geomagnetic conditions. The intensity at MeV energies varies over more than three orders of magnitude on time scales ranging from minutes to solar cycles [e.g., Blake et al., 1992; Reeves et al., 2011]. Using available models and observations to accurately specify the relativistic electron environment throughout the inner magnetosphere is therefore both extremely challenging and extremely important. [5] In this paper we describe the overall structure of DREAM as it exists now. We discuss the various components of DREAM and how the components are connected. We also discuss some of the results obtained during the development of DREAM. Last, we discuss methods of validation and the accuracy of the DREAM results relative to other standard radiation belt models.

2. The DREAM Numerical Framework [6] The Dynamic Radiation Environment Assimilation Model does not consist of a single code or even a single language. DREAM uses a computational framework that is extremely flexible and adaptable to new developments, new data sources, and new physical understanding. Although it is a heterogeneous system, it is fully functional and has been run for a variety of intervals of scientific interest as well as for routine production of model outputs spanning several years of radiation belt dynamics. Currently, the various components of DREAM communicate through file transfer. Two newer implementations of DREAM are also under development. The “full-service” implementation includes all of the components that are available at any given time. It allows both routine analysis using components that have been tested and validated or, separately, the development and debugging of modules with new or advanced capability. The second implementation is designed to run on workstation-level platforms using real-time space weather observations and an interactive user interface. It uses the same modules as the full-service implementation but integrates them into a stand-alone, platform-independent code. The real-time implementation uses a more limited subset of code modules. It is also used to provide web services through standardized Service Oriented Architectures (SOAs). [7] In this section we describe the overall organizational structure of DREAM and describe its basic components and operation. Later sections describe the various stages of the calculations in greater detail, illustrate how the components of DREAM work in specific applications, and discuss some of the results to date. [8] Figure 1 shows how the Dynamics Radiation Environment Assimilation Model is organized. It consists of five basic components: (1) radiation belt observations and data pre-processing; (2) global magnetic field and inner magnetosphere modeling; (3) physics-based models of the radiation belts (including artificial belts); (4) data assimilation, and (5) external user requirements and interfaces.

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[9] Data assimilation is a term that is used quite loosely to mean anything from specifying boundary conditions to sophisticated techniques such as Kalman Filtering, 3DVAR, etc. DREAM currently uses Kalman Filters [Kalman, 1960] including simple or ensemble methods [e.g., Welsh and Bishop, 1995; Evensen, 2003]. All three use observations and physical models, along with the uncertainties in each, to produce an optimized global “state space.” Typically the state-space is not fully consistent with either data or model but rather a “compromise” that best represents the “true state” of the system. The true state is estimated by minimizing an error function. [see Koller et al., 2006, 2007a, 2007b, and references therein]. The results are validated using an independent set of external observations. 2.1. Observations and Preprocessing [10] Both real-time and archival data sources require some amount of pre-processing as illustrated by the yellow module in Figure 1. Any pre-processing has to account for the capabilities and limitations of the initial measurements. Some satellites include comprehensive observations that provide differential flux over a broad range of energies and include magnetic field measurements that can be used to convert angular-resolved measurements into pitch angle distributions. Other satellites include only targeted observations from instruments that may provide limited angular resolution, omni-directional or hemispherical measurements, and/or lack a magnetometer to convert inertial coordinates to pitch angles. They may also have limited energy coverage and/or limited energy resolution, including dosimeters that measure all particles with sufficient energy to penetrate a given thickness of shielding. (The uses of, and limitations of, the different classes of detectors is discussed by O’Brien et al. [2008].) [11] Pre-processing of radiation belt data can produce results with either small or large uncertainties. The precision and accuracy of the original measurements are typically known (to some degree) prior to launch. However, detailed on-orbit cross-calibration and modeling of instrument response can refine and improve the quality and reliability of the observations. Often, assumptions about the shape of the energy spectrum or pitch angle distribution must be applied. For example, Ni et al. [2009b] present a methodology for calculating phase space density from ominidirectional Akebono data with good overall comparison to results using pitch angle resolved data from the Combined Release and Radiation Effects Satellite (CRRES). [12] Typically, ancillary data and/or models also need to be applied – for example magnetic field models, estimated spacecraft trajectories, estimation of backgrounds, etc. Ideally, some independent measure of the total uncertainty from all sources can be applied. One such technique is testing internal consistency within the assimilative model as discussed in section 2. We also note that equal care must be applied to data sets that are not assimilated, but rather are used to validate the results. This too is included in the radiation belt observations module in Figure 1.

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Figure 1. A flow chart illustrating, schematically, the computational framework used by the Dynamic Radiation Environment Assimilation Model (DREAM). The five main components and their operation are described in section 2.

2.2. Global Magnetic Fields [13] The motion of charged particles in the magnetosphere is organized by the large-scale electric and magnetic fields. For particles with energies greater than tens of keV (E ≳ 104 electron volts) the EB drift motion can be neglected and magnetic drift dominates. Magnetic drift can be organized around three periodic motions each with an associated adiabatic or “magnetic” invariant: gyration around the magnetic field, bounce along the field between magnetic mirror points, and longitudinal drift around the Earth along a drift shell (or “L-shell”). While local magnetic field measurements are available from a number of satellites, there is no way to directly observe the entire, global magnetic field. Therefore, radiation belt modeling also requires a model of the global geomagnetic field. Those codes and calculations are represented by the red module in Figure 1. [14] The simplest assumption of a tilted dipole field is grossly inadequate to describe the distorted, dynamic geomagnetic field. Stretching and compression of the field

changes both the local field vector and the pitch angle distribution of particles on the field line. The storm-time ring current diamagnetically “inflates” the field and adiabatically distorts particle drift orbits while simultaneously changing the energy and pitch angles of the electrons on those drift shells [e.g., Kim and Chan, 1997]. These are not small effects and radiation belt models for space weather applications must include them in a physically realistic way to achieve even minimal accuracy. DREAM can, in principle, use any representation of the geomagnetic field ranging from static models like Olsen and Pfitzer [1974] to global MHD models. But DREAM, in its current form, uses dynamic statistical models such as the Tsyganenko models (discussed below) or the event-specific Ring currentAtmosphere interactions Model with Self-Consistent B fields (RAM-SCB) model [Jordanova et al., 1997; Zaharia et al., 2006]. RAM-SCB was specifically developed for DREAM. It is a comprehensive model of the inner magnetosphere that self-consistently describes particle dynamics in pressure balance with the geomagnetic field and driven

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by observational inputs of particle fluxes at geosynchronous orbit. The heritage, development, and testing of RAM-SCB is described in more detail in section 2.

2.3. Physics Models [15] Physics-based models of the radiation belts come in a variety of forms. Diffusion models are the most common because they capture most of the known dynamics of the radiation belts while still assuming quasi-linear perturbations of conserved quantities. Radiation belt diffusion equations use phase space density (PSD) as the free variable. Phase space density, f, is defined as the flux (j) over the square of the particle momentum (p); i.e., f = j/p2 which, by Liouville’s theorem, is conserved along a dynamic trajectory (unless particles are added or removed from the system). Space instruments do not directly measure phase space density. Rather, they measure particle flux which is a function of the detector energy passbands and viewing direction(s). [16] Using the global geomagnetic field model, we can calculate the magnetic invariants that correspond to a given electron energy and pitch angle at a given time and spatial location. This allows us to calculate phase space density as a function of those magnetic invariants as required by the physics models. The physics-based radiation belt models (represented by the blue box in Figure 1) can include a variety of processes including radial diffusion, particle energization, pitch angle scattering, precipitation into the atmosphere, detrapping and magnetopause loss, etc. Each process may be parameterized in terms of geomagnetic activity indices (such as Kp or Dst) or they may be directly calculated from measured quantities (such as wave spectra). Some processes may be parameterized by characteristics of the solar wind and interplanetary magnetic field (IMF). [17] In some cases, the physical relationship between input parameters and a given process can be directly modeled. Other parameters may be known statistically but not known for a particular time, place or event. One example of a directly modeled process is the loss of particles on “open” drift trajectories that can be calculated for any interval using solar wind observations and magnetic field models. Statistically driven physics models include those that calculate energy and pitch angle diffusion [e.g., Shprits and Thorne, 2004; Li et al., 2007]. This class of models typically assume a fixed set of wave characteristics (frequency, amplitude, etc.) based on statistical observations [e.g., Meredith et al., 2001, 2003, 2004]. Processes that are parameterized by geomagnetic indices include radial diffusion rates. Brautigam and Albert [2000], for example, parameterize radial diffusion using the Kp index. 2.4. Artificial Radiation Belts [18] For national security applications, DREAM can also include a module that calculates the injection and initial trapping of artificial radiation belts from high altitude nuclear explosions (HANE) [Tokar, 2007; Winske et al., 2009]. The most well-known and well-documented HANE

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belt was produced by the 1962 U.S. atmospheric nuclear test known as Starfish. Starfish was an 1.4 MT device detonated at an altitude of 400 km, 31 km southwest of Johnston Island. HANE electrons are produced by the beta decay of radioactive debris from the nuclear explosion. After the initial diamagnetic cloud collapses, the motion of both the debris ions and the beta electrons are controlled by the geomagnetic field. Once injected, those electrons with pitch angles that mirror above the atmospheric drift loss cone will be trapped for long periods of time. The trapped HANE electrons form a new, artificial radiation belt. But electrons are electrons, regardless of their source, and the intermediate and long-term evolution of “natural” and HANE electrons are both subject to the same physical processes: acceleration, transport, scattering, and losses. DREAM allows a HANE source to be incorporated along with the assimilated observations and physics model to quantitatively model a wide variety of hypothetical scenarios and the “space weather” risks associated with each.

2.5. Data Assimilation [19] The data assimilation engine combines the observations with the physics model by optimizing a solution taking into account both model and observational uncertainties. One of the first attempts at radiation belt data assimilation was by Bourdarie et al. [2005] who used the “direct data insertion” method. This method replaces the model value with satellite observations in those simulation bins for which observations are available (e.g., a given L-shell bin at a given time) which improves the overall solution but can introduce discontinuities in one or more simulation dimensions (space, time, energy, etc.). The results of Bourdarie et al. [2005] did, however, show the power of data assimilation as a tool for radiation belt modeling [O’Brien, 2005]. They also pointed out the importance of understanding the quality, fidelity, and calibration of the data that are assimilated. Producing a well calibrated (and inter-calibrated) set of observations prior to assimilation is as important, and often as difficult, as implementing the actual assimilation model [Friedel et al., 2005]. [20] After the choices of assimilation method and data sets comes the choice of physics model. In choosing a global dynamic model of the radiation belts the level of complexity in the model must be balanced against the ability of the observations to drive and constrain the model. One approach is to include all the physical processes that are thought to play a role in radiation belt dynamics and parameterize them to the best of our ability. Such an approach is intellectually satisfying and pushes the state of the art in physics model development. However, when the physics model relies on more assumptions and free parameters than can be constrained with the available observations the accuracy of the result is largely unknown and could be worse. [21] We have chosen to implement a very simple physical model to begin with and add in physical processes only 4 of 25

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Figure 2. A schematic illustration of the data assimilation process.

when their effects can be tested against observations and their uncertainties can be calculated. Most of the results that we present here use simple models in order to test and validate the assimilative process. Even simple models can produce highly realistic results, though, and can often compensate for missing physical processes - as we show in a later section. It is important to note, though, that DREAM does not depend on any particular constraint on model complexity. It can be used with any model that can be specified as a function of a well-defined state vector. [22] The most simple radiation belt physics model is a 1-dimensional (1D) radial diffusion model. Other radiation belt data assimilation models (e.g., the UCLA 1D VERB model) start with the 1D case before adding further sophistication and complexity [e.g., Shprits and Thorne, 2004; Shprits et al., 2007]. One advantage of using a 1D model is that one can use a simple state vector, namely phase space density as a function of L-shell which is binned into an array of 90 elements of 0.1 RE. At each time step the state vector is compared against available observations for those elements of the array that have new observations. The state vector is “adjusted” according to the observations and the uncertainties assigned to both the observations and the model. [23] DREAM typically uses one of two variations on the standard Kalman Filter. An ensemble Kalman Filter advances the state vector for a single time step using random statistical variation in the starting state vector or in the model. A Kalman Filter can also be run with an augmented state vector which refers to a technique that includes additional parameters as part of the state vector. One

example is to use phase space density at the boundary of the model as a free parameter [e.g., Daae et al., 2011]. Other examples are leaving the radial diffusion rate or electron loss lifetimes [e.g., Kondrashov et al., 2007] as components of an extended state vector. [24] We illustrate how a simple ensemble Kalman Filter (enKF) operates in Figure 2. We start the example with a specified state (x, y) at time Ti (green). The model advances the system to time Ti+1. The enKF perturbs the previous state and/or the model parameters to produce an ensemble of model states. The distribution of those states gives the model uncertainty as represented in blue (x, y)M. The observations at ti+1 are represented in red (x, y)O. They also have an uncertainty which, in this case is assumed to be much smaller than the model uncertainty. The Kalman Filter “picks” a new state (x, y)i+1, that is intermediate between the model and observed states and the process is repeated. The relative distribution of uncertainties is shown schematically, for the Y dimension, on the right hand side of the plot. In this hypothetical example the state of the system is highly constrained by the observations. In more typical cases the observations cover only a portion of the state-space – for example only a limited set of L-shell bins. [25] DREAM actually does a larger number of assimilations simultaneously in order to model the radiation belts. Radial diffusion in the outer electron belt is dominated by magnetic diffusion which is independent of particle energy and pitch angle and therefore independent of the magnetic invariants, m and K (described more below). DREAM selects a finite set of (m, K) pairs and applies

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the assimilation to each. The resolution in m and K and the number of assimilations that needs to be done is dependent on the application but can range from tens to hundreds (or more if needed). The final step is to convert back from phase space density, f as a function of (m, K, L*) to flux as a function of energy, pitch angle and position, j(E, a, R). [26] The data assimilation cycle, represented by the green circle in Figure 1, is at the heart of DREAM. Later sections of this paper will illustrate in more detail how the assimilation is applied to radiation belt modeling and what we can learn from it about radiation belt physics.

2.6. User Requirements [27] When the assimilation step is complete the result is a global representation of the radiation belts. The resulting global model is multidimensional and therefore contains much more information than can be intuitively understood. At this stage it is essential to incorporate user requirements (the orange box in Figure 1) to produce information tailored to specific users and applications. [28] The most common application that we envision is to use DREAM to specify the environment at a particular satellite that does not have its own environmental measurements. DREAM is built to use satellite ephemeris files or the two line elements (TLEs) from the NORAD space catalog which are then used to extract flux (or dose) as a function of time along the orbit of the satellite of interest. For longer intervals we can calculate orbit-averaged values. An even more useful application would compare current conditions against the historical probability distributions derived from long-term reanalysis products (e. g., radiation belt climatology). [29] One important feature of DREAM is that it has been designed with flexibility in mind. It can produce a variety of space weather products to meet a variety of user needs without changes to other parts of the code. Likewise, it is not designed to use any specific set of satellite observations. The same codes will run in the same way whether there are data from ten satellites in ten different orbits or just one. It is robust to heterogeneous data from different satellites providing data over different time spans. It also robustly accommodates data gaps with no observations and can even accommodate a mixture of real-time and latent (i.e., old) data. [30] Different geomagnetic field models and different physics-based radiation belt models can be configured together or independently. Because the components work together in the same way regardless of configuration the results from different configurations can be compared quantitatively against one another. For example, the effects of including electron energization can be directly compared against runs with no explicit energization. Similarly, the quantitative improvement in accuracy from the addition of new satellite data can be assessed and validated. As we describe the details of each step in the process and the configuration of the different components of DREAM, we

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will present different examples that illustrate how DREAM has been used to date.

3. Observations, Phase Space Densities, and Magnetic Invariants [31] While the data-assimilative process can rightfully be said to have more than one starting point, one useful way to think about the process is to start with the transformation of satellite observations into phase space densities as a function of the magnetic invariants. 3.1. Magnetic Invariants [32] Electrons trapped in the geomagnetic field undergo three periodic motions (Figure 3). The first is gyration around a magnetic field line. The second is “bounce” motion back and forth along a magnetic field line between magnetic mirror points. The third is azimuthal (longitudinal) motion around the Earth along the magnetic drift shell. Each of these periodic motions can be described in terms of a magnetic (or adiabatic) invariant. Changes in the geomagnetic field that are slow compared to the period of each motion will conserve the magnetic invariants of radiation belt electrons in the inner magnetosphere and deviations from pure adiabaticity can be formulated as perturbations to those motions. [33] The gyro, bounce-, and drift invariants are designated m, J (or K), and F (or L*) [Schulz and Lanzerotti, 1974]. The adiabatic invariants are a function of the magnetic field configuration as well as the electron’s energy, E, and pitch angle, a. Throughout this paper, unless otherwise noted, the pitch angle distributions are given in terms of their equatorial values. Particles with 90 equatorial pitch angle mirror exactly on the magnetic equator. [34] The first invariant, m, is proportional to p2?/B. Since p? is the momentum component perpendicular to the local magnetic field, B, m is therefore a function of an electron’s pitch angle and energy. Since B refers to the local magnetic field, m can be determined directly from observations. The required measurements are the magnetic field vector and particle flux over a range of energies and look directions. [35] NASA’s Polar satellite provides all the measurements needed for very accurate direct calculation of m, but many data sources provide only partial information. The NOAA GOES geosynchronous satellites, for example, have excellent magnetometer data but only a single look direction with a wide field of view. Therefore we cannot measure the pitch angle distribution and have to use assumed distributions in order to calculate m. Similarly, data are typically available from only two integral energy channels which requires certain assumptions about the energy spectrum. Conversely, the LANL SOPA instruments, which are also at geosynchronous orbit, have excellent energy resolution and good look direction coverage over the full unit sphere, but the satellites have no magnetometer. Through analysis, the symmetry axis of the

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Figure 3. Motion of charged particles in the geomagnetic field. (a) Electrons (and ions) will gyrate around the magnetic field and will bounce along the field between magnetic mirror points which are determined by the local magnetic field strength and the particle’s pitch angle, a. Therefore it is convenient to label any point on a magnetic field line with the same value of “L” which, in a dipole field, is defined as the radial distance to the magnetic equator normalized to units of Earth radii, RE: e.g., geosynchronous orbit is at L = 6.6. (b) Charged particles drift azimuthally and the motion around the Earth and between the mirror points defines a surface of motion called a “drift shell.” The magnetic field and drift shells change and are distorted by solar and geomagnetic activity. distribution can be determined and look directions converted to pitch angle [Thomsen et al., 1996] but the strength of the magnetic field must be estimated using models or other means. The GPS satellites have neither angularresolved measurements or a magnetometer. [36] While the first invariant, m, can be calculated using measurements alone, the second and third invariants, J and F, cannot. Both require a model of the magnetic field. The second invariant, J, is an integral along the magnetic line connected between the mirror points. Often it is convenient to use another form of the invariant, K which is only a function of electron pitch angle and the magnetic field and is independent of particle energy. In a dipole magnetic field, K can be calculated analytically whereas, in a realistic, non-dipole field, K must be evaluated by numerical integration along the field line. [37] The third invariant, F, is defined by an integral along a surface defined by the “drift shell” traced out by the azimuthal drift of electrons around the earth and their bounce between the magnetic mirror points (see Figure 3b). A more convenient parameter, however, is the parameter L* which is proportional to 1/F. In a dipole field L is exactly equal to the distance (in units of Earth radii, RE) from the center of the Earth to the point where the drift shell crosses the magnetic equator (Figure 3a). [38] In a dipole field L* = L but as the field becomes less and less like a dipole (i.e., more distorted by the external geomagnetic current systems) L* adiabatically deviates from the dipole L value. Similarly, in a dipole field, L is independent of the electron pitch angle but in a realistic

geomagnetic field L* is a function of pitch angle and therefore a function of m and of K which requires separate calculation of L* for each (m, K) pair used in the data assimilation and a unique transformation from phase space density back to flux for each position in space (R). [39] Accurate calculation of L* (or F) is critical for radiation belt modeling but is also one of the most computationally intensive steps and can be prohibitive for real-time applications or for processing of very large data sets. Therefore, as part of the DREAM project we have developed a very fast, very accurate, neural-net based algorithm for the calculation of L* which we call LANL* [Koller et al., 2009; Koller and Zaharia, 2011]. Either direct numerical integration or the neural net can be used depending on the particular application of DREAM.

3.2. Phase Space Density [40] Phase space density is conserved along a particle’s trajectory and the motion of electrons in the Earth’s field is specified by the three invariants. Therefore, if the invariants are conserved then phase space density (as a function of m, K, and L*) is conserved whereas electron flux (as a function of R, E, and a) is not conserved. This feature is very convenient for radiation belt modeling. For a given pair of (m, K) phase space density will be constant along a drift shell (L*). If observations are converted to phase space density as a function of m, K, and L* (or equivalently m, J, and F) then all calculations can be done in magnetic invariant space and the azimuthal dimension can be ignored. 7 of 25

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Figure 4. Apparent motion in L*. A geosynchronous satellite orbits at a fixed radial distance of R ≈ 6.6 RE but measures different populations of electrons as it orbits around the Earth. The distortion of the geomagnetic field maps different drift shells (defined by L*) to geosynchronous points along geosynchronous orbit. The exact mapping depends on the geomagnetic field, the orbit (longitude and latitude) of the satellite, and the pitch angle of the electrons of interest. This apparent motion in “magnetic space” can be used to test magnetic field models and to reveal gradients of phase space density [Chen et al., 2005, 2007a]. [41] Figure 4 shows observations from two geosynchronous satellites converted to phase space density (PSD) as a function of L* and time. For the values of m and K shown here, geosynchronous satellites sample higher L* on the dayside where the field is compressed and lower L* on the night side where the field is stretched. Although the satellites are always separated by the same azimuthal angle (longitude) there are points on the orbit where they measure the same L*-shell and therefore the same population of electrons. If the field model used to calculate the invariants is accurate, then the phase space densities at those points should be equal. Figure 4 shows that, for this interval, when the field is quiet, the phase space densities match very closely. [42] The availability of data from multiple geosynchronous satellites with nearly identical instrumentation is a great advantage for data assimilative modeling. When the satellites are separated in L* they provide a direct measure of the phase space density (PSD) gradients [Onsager et al., 2004; Chen et al., 2005] which is an important constraint on physics models. On the other hand, when the satellites are measuring the same L* the comparison of PSD provides an estimate of the observational uncertainty. The uncertainty in the original flux values can be quite small but it is the uncertainty in the PSD, f(m, K), that is important for the assimilation and that is driven primarily by uncertainties in the calculation of m and K. We use this

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technique to provide an uncertainty to the “observations” that is a function of both the original measurements and the global magnetic field model [Chen et al., 2005]. We have also used this technique to select among different magnetic field models in order to find the one that best matches the actual geomagnetic conditions [Chen et al., 2007a]. [43] During quiet times the choice of magnetic field model may not greatly affect the results of the assimilation. Ni et al. [2009a] presented an interesting analysis of the sensitivity of the UCLA 1D Versatile Electron Radiation Belt (VERB) data assimilation model to the choice 4 magnetic field models. They found that, while the actual values of the invariants for a particular observation point could be sensitive to the choice of model, the overall results of the assimilation were relatively insensitive to the choice of model. However, as with Chen et al. [2007a], they found that the greatest sensitivities were found during storm times when the field distortions are most dependent on the choice of model.

3.3. Magnetic Field Models [44] As we have noted, the geomagnetic field is dynamic and therefore the transformation from physical position (R) to L* is a function of time and geomagnetic conditions. In Figure 4 the mapping between position and L* is approximately the same for the two orbits because the geomagnetic field was very quiet. During storm times the magnetic field can be highly distorted. One of the major causes is the build up of the storm time ring current which is measured by the Dst index. As the ring current builds it produces a magnetic field that opposes the Earth’s magnetic field and the field “inflates.” Therefore the mapping of L* changes. Figure 5 illustrates this effect schematically. In the quiet time field geosynchronous orbit lies just outside L* ≈ 6 and the heart of the radiation belts are at L* ≈ 4.

Figure 5. Other apparent motions in L*. The stormtime ring current “inflates” the Earth’s magnetic field which changes the drift orbits of radiation belt electrons and, in turn, changes the population of electrons that can be measured by a particular satellite. In magnetic coordinates this real motion of electrons becomes an “apparent motion” of the spacecraft in L*. 8 of 25

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Figure 6. The phase space density (PSD) (color coded) as a function of L* for three geosynchronous satellites during a geomagnetic storm in 2001. As the ring current intensifies the drift shells “inflate” which produces an “apparent motion” of the satellites to very low L*. As the field inflates due to the storm time ring current the new magnetic mapping can put L* ≈ 4 very close to geosynchronous orbit. In other words, geosynchronous satellites are now measuring electrons that used to be orbiting much closer to the Earth. It is, of course, the electrons that move but in L* space there is an apparent motion of the satellite to lower L*. Figure 5 also illustrates another important effect. Notice that L* ≈ 6 now intersects the magnetopause (white arc). Electrons whose drift orbit hits the magnetopause are quickly lost to interplanetary space. [45] The apparent motion of a satellite in L* is critical for accurate radiation belt modeling. The assumption that a geosynchronous satellite continuously measures the same population of radiation belt electrons is strongly violated during geomagnetic storms. Furthermore, most of the acceleration, transport, and loss processes that we want to model are also strongest during geomagnetic storms. Figure 6 shows L* and phase space density (PSD) for three

geosynchronous satellites during a storm in October 2001. Prior to the storm Dst was slightly negative and the geosynchronous satellites sampled L* ≈ 5.7. During the peak of the storm, however, the geosynchronous satellites all measured L* ≈ 4.2 – a completely different population of electrons. [46] There are now many geomagnetic field models to chose from and we have included many of them in the DREAM computational framework including T89c [Tsyganenko, 1989], OP77 [Olson and Pfitzer, 1977], OP88 [Pfitzer et al., 1988], T96 [Tsyganenko and Stern, 1996], MO97 [Ostapenko and Maltsev, 1997], T01 [Tsyganenko, 2002a, 2002b], and T01s [Tsyganenko et al., 2003]. All these models account, in one way or another, for the distortion of the geomagnetic field by solar wind dynamic pressure, storm time ring current and (to a varying degree) other geomagnetic currents. The choice of model is driven by a number of factors including model complexity, availability 9 of 25

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of input parameters to drive the models, and perceived accuracy. We say “perceived” accuracy because accuracy can be measured in a variety of ways and there is no widely accepted way to measure the accuracy for radiation belt applications. Even PSD matching is generally only available in limited orbits. [47] There are several reasons to have multiple magnetic field models available in DREAM. Some users may simply have a “favorite” model. In other cases we wish to compare against previous calculations that used specific models. Some applications need a very fast computation of L*. Some applications need a model that is sensitive to multiple geomagnetic and solar wind conditions. Ideally, multiple, parallel runs that use different magnetic field models but assimilate the same observations could provide an “ensemble” estimate of the dependence on choice of field model.

3.4. RAM-SCB [48] The dynamics of plasma and fields in the inner magnetosphere has much broader implications for radiation belt dynamics than just providing the “coordinate system” for the motion of energetic particles. Electrons with energies of hundreds of keV (or lower) provide the “seed” population that can get energized to radiation belt energies. Plasma anisotropies develop as ions and electrons of different energies convect through the inner magnetosphere. When those anisotropies reach instability thresholds they generate the electromagnetic waves that control many radiation belt processes including acceleration and precipitation. Ideally the energetic particle, plasma, and fields in the inner magnetosphere should be modeled as the coupled system that they are in reality. Therefore, since its inception, DREAM has included the development of a coupled inner magnetosphere model based on the Ring current-Atmosphere interactions Model, RAM [Jordanova et al., 1994, 2006]. [49] Initially, RAM (like similar ring current models) was not well-suited to coupling to radiation belt models. RAM is a kinetic model, meaning that it solves the equations of motion for individual electrons and ions (of several species) for each discrete energy and pitch angle. A great simplification of the numerical solution can be achieved by assuming that the particles move in a dipole magnetic field where many of the integrals have closed form solutions and do not need to be evaluated numerically. The result, however, is that no matter how sophisticated the simulation of ring current processes, the field is always dipole and does not include the self-consistent effects of the build up of plasma pressure and the diamagnetic inflation of the field. [50] In a self-consistent treatment the global electric and magnetic fields prescribe the motion of charged particles. The global distribution of charged particles then modifies the global electric and magnetic fields which, in turn, modifies the motion of those particles. In a fully selfconsistent treatment, the plasma distributions, magnetic fields, and electric fields are all mutually consistent. In the

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RAM Self-Consistent magnetic field model (RAM-SCB) the electric field is prescribed externally but the plasma distributions and magnetic fields are solved selfconsistently [Zaharia et al., 2005, 2006; Zaharia, 2008]. [51] RAM-SCB uses boundary conditions from plasma and medium-energy particle measurements from the LANL Magnetospheric Plasma Analyzer (MPA) and Synchronous Orbit Particle Analyzer (SOPA) instruments at geosynchronous orbit. Therefore the inner magnetosphere model (including global magnetic field) are data-driven and are uniquely calculated for each event (as opposed to calculated statistically from solar wind parameters and geomagnetic indices). We can then use RAM-SCB to calculate the magnetic invariants (m, K, L*) and to convert fluxes to and from phase space density using exactly the same code structure as for any other magnetic field model. [52] A current limitation of RAM-SCB for radiation belt data assimilation is that it is too computationally intensive to be run for long validation intervals (e.g., years). Therefore, most of the DREAM radiation belt simulations to date have been run using the T89 or T01s models. For the future we are pursuing a two-pronged approach for ring current modeling. RAM-SCB will continue to be the testbed for developing new capabilities and testing new physics models. At the same time we are developing an operational version, the DREAM Ring Current (DREAM-RC) model, that can be run continuously in real time or for many years of observations. [53] RAM-SCB, and DREAM-RC also calculate the unstable plasma distributions that lead to whistler, electromagnetic ion cyclotron (EMIC), and magnetosonic waves [Jordanova et al., 2007; Blum et al., 2009; Chen et al., 2010; Jordanova et al., 2010]. These waves resonantly interact with radiation belt electrons and are now thought to play a dominant role in radiation belt acceleration [e.g., Summers et al., 1998; Reeves et al., 2009] and loss [e.g., Millan and Thorne, 2007; Summers et al., 2007; Jordanova et al., 2008]. We envision that future development of DREAM will include both measurements and models to specify the spatial and temporal distribution of waves that drive energy and pitch angle diffusion.

4. DREAM Results 4.1. Identical Twins Experiments [54] The initial results of the DREAM model focused primarily on understanding how Kalman Filter based data assimilation could best be applied to radiation belt dynamics (rather than on understanding those dynamics per se). Koller et al. [2006] described the data assimilation technique and some of those initial results. One of the most important tests is the so-called “identical twins” test [Naehr and Toffoletto, 2005]. For this test we produce a set of initial conditions that are known exactly. This is known as the “true state” of the model. We then sample the true state using a sparse sampling similar to the limited times and locations of observations in the radiation belts. The final step is to apply random errors to the sampled 10 of 25

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Figure 7. The results of an “identical twins experiment” used to determine the interplay between the spatial and temporal distribution of observations and the complexity of the physics model. DREAM uses this kind of test to determine how well various physical processes can be constrained by the available observations. “observations” and assimilate them into a radiation belt model. In conducting the identical twins test the model used to produce the true state can be the same model used in the assimilation or it can be different. This enables us to test how well available satellite measurements could (or could not) constrain the physics model and therefore how complex a physics model is justified given measurement limitations. [55] Figure 7 shows the results from one identical twins experiment similar to that of Naehr and Toffoletto [2005]. The true state (initial conditions) are produced by solving the Fokker-Planck equation for 1D radial diffusion ∂f DLL ∂f 2 ∂ ¼L ∂d ∂L L2 ∂L

ð1Þ

where f is the phase space density, L is the drift shell (actually L*), and DLL is the radial diffusion coefficient.

Here we use DLL = 2.1 103(L/4)11.7, as derived by Selesnick et al. [1997] using data from the Polar satellite. Here we use a true state (xtrue) with constant boundary conditions and a constant rate of radial diffusion. We then add an artificial source region followed by an artificial loss (Figure 7, top left plot). We sample the true state every 6 h with five artificial satellites at constant L*-shells between 3 < L * < 8 and randomly perturb these observations to simulate observational uncertainty. These observations are then fed into the Kalman Filter algorithm for assimilation with a physics model that contains only radial diffusion but no source or loss processes. [56] The result of the assimilation is shown in the bottom left plot of Figure 7 while the plots at right show the log of the ratio of the true state to the assimilated state on top and the relative residual on the bottom [Koller et al., 2007a]. It is clear that, even with very sparse sampling and no explicit source or loss terms, the Kalman filter can very accurately reproduce the global state of the system 11 of 25

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Figure 8. DREAM assimilation of a storm in October 2002. (a) The satellite observations converted to phase space density. We selected observations very close to the magnetic equator (K = 0.1 G1/2) from one GPS satellite, three LANL geosynchronous satellites, and Polar. (b) The assimilated state that fills in the entire state space. White lines show the location of the plasmapause and magnetopause. White dots show where observations were available. (c) The innovation vector. This shows where and by how much the state vector is modified by the assimilated observations. and “mimic” the effects of missing physical processes by adjusting the solution to match the observations - in this case adding phase space density where the residual is positive and removing it where the residual is negative. [57] This rather simple example does not exhibit the full power of identical twins experiments but it does illustrate the technique. Other interesting cases include: creating a true state using one diffusion parameter but doing the data assimilation with a physics model that assumes a different diffusion parameter; or, quantitatively testing how the addition (or subtraction) of a particular data set constrains the physics in our models [Reeves, 2010]. We illustrate cases like this using real observations below.

4.2. Assimilation With Satellite Observations [58] We turn now to an example of DREAM results using satellite observations. Koller et al. [2007a] studied a moderate storm in late October 2002. That study used data from 5 satellites: LANL-GEO (LANL-97A, 1991–080, and 1990–095), GPS-NS41, and Polar. This example shows one (m, K) pair, specifically m = 2083 MeV/G which corresponds to several MeV at geosynchronous orbit and K = 0.1 G1/2RE which corresponds to equatorial pitch angles near 90 . Figure 8a shows the phase space density as a function of L* and time with the Dst profile below. Polar was in a 9 RE elliptical orbit which, in 2002, crossed the magnetic equator several RE outside geosynchronous orbit. GPS is in a 4.2 RE circular, inclined orbit so it crosses the magnetic equator near L* = 4.2. Geosynchronous orbit is at 6.6 RE in radius and for this moderate storm that puts it generally at L* > 5. These satellites were chosen to constrain the phase space density near geosynchronous, near the plasmapause, and near the magnetopause. Note that, at any given time, the three geosynchronous satellites are at different L* which provides a “spread” in the observations. When two satellites are at nearly the same

L* the phase space densities generally agree closely which gives us confidence in the magnetic field model for this period. For these (m, K) values and this storm the PSD gradient is generally negative indicating a source inside geosynchronous orbit [Koller et al., 2007a]. [59] Figure 8b shows the DREAM results from the assimilation, illustrating how the assimilation can produce global solutions from sparse observations. Figure 8b also plots the locations (in L*) of the original observations are shown by white dots. The white lines show two magnetospheric boundaries that are important physically (and in other DREAM runs) but that are not actually used in this assimilation. The outer boundary is the last closed drift shell specified by its L* value. We calculate the last closed drift shell as the L* value that intersects the magnetopause. We also plot the statistical location of the plasmapause. While those boundaries are not used in this assimilation, other instances can set loss rates or other processes according to their relationship to these boundaries. [60] In Kalman filtering the physics model is called the “forecast model” because it projects the state vector forward in time (see Figure 2). Here we again use a 1D radial diffusion model (Fokker-Planck equation) but this time we use the radial diffusion coefficient DLL given by Brautigam and Albert [2000], DLL = 10(0.506Kp-9.325)L10 (where L is actually L*). The PSD at the outer boundary (L* = 10) is a free parameter and there are no source or loss terms in the model. [61] There are a number of interesting features in the assimilation results. We first notice that the assimilated state is strongly controlled by observations. Even though the physics model is based on radial diffusion from an outer boundary the assimilation produces strong, persistent peaks in phase space density near or inside geosynchronous orbit. The strong dropout at the storm main phase is well-represented. In fact it is the low PSD values 12 of 25

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during the main phase that take the moderate PSD values at the boundary and reduce them to be consistent with the observations. It is somewhat surprising that the last closed drift shell follows the contour of very low PSD near the outer boundary because it was not used in the physics model at all. Rather, the low PSD at high L* and the negative outward gradient of PSD is driven nearly entirely by the low PSD values obtained from the Polar observations consistent with Chen et al. [2007b]. [62] There are also likely artifacts in the assimilation. The early time conditions before the storm main phase are influenced by the choice of the initial state vector f(L*) used to initialize the run. This is particularly true because Kp is small and diffusion is relatively slow. Following the minimum Dst when phase space density is beginning to increase at both GEO and GPS we see another likely artifact. Our physics model has no source term so all increases in PSD that do not come from the outer boundary are driven by the observations. Here both GEO and GPS observations produce local peaks in PSD. This is physically unrealistic. A more realistic interpretation is that there was some acceleration process (source of PSD) at L-shells between GEO and GPS that is not captured by the model but is responsible for the increases seen at both satellites. Later when radial diffusion has mixed the two populations the peak in PSD does move to that region. [63] The effect of the observations on the assimilation results is shown quantitatively in Figure 8c. Here we show the magnitude of the “innovation vector” as a function of L* and time. Figure 2 illustrates schematically how the “forecast” (given by the physics model) can deviate from the observations. The separation between the forecast and measured state indicates where the model and the observations disagree. The observations influence more than the single bin where the satellite was located because, in this case, radial diffusion creates strong correlations in the vicinity of the original data. Where the innovation is large the forecast produced by the physics model is not doing a good job of reproducing the actual dynamics of the radiation belts and the data are “pulling” the global solution back closer to reality.

4.3. Are High L-shells a Source or Sink? [64] DREAM was developed not only to accurately represent the state of the magnetosphere but also to test the physical processes that are in the forecast model. One of our early sets of studies was to test the nature of the outer boundary. This is an interesting question because the outer boundary could be either a source or a sink. We know that the magnetopause is a sink of electrons because the electrons that hit this boundary are lost from the radiation belts into interplanetary space. On the other hand the plasma sheet is a source of electrons that get convected into the inner magnetosphere or injected by substorms. DREAM does not have a local time dependence and therefore processes on the day side (magnetopause loss) and night side (plasma sheet source) are intermingled. We would expect that at high m (high energy)

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there are few plasma sheet electrons and the net effect is loss through the outer boundary while at low m (low energy) the plasma sheet is a source [see, e.g., Shprits et al., 2006; Chen et al., 2007a]. [65] The quantity L* is only properly defined and can only be calculated for electrons with closed drift orbits that do not intersect the magnetopause. We call this L*max. We cannot assimilate data from satellites measuring electrons on open drift orbits because we cannot calculate L* and therefore cannot place the observations in the assimilation. The forecast model does not have the same limitation. It simply solves the Fokker-Planck equation and does not use any information about the geomagnetic field. Therefore we can calculate diffusion both inside and outside L*max. [66] Figure 9 shows a comparison between three assumptions for electron behavior at high L*. All three plots are for m = 2083 MeV/G and K = 0.03 G1/2RE. Three sets of curves are over-plotted in each. Solid lines show the plasmapause and L*max. Dotted lines show the position of satellites used in the assimilation. The top plot allows the phase space density at the outer boundary (L* = 10) to be a free parameter optimized by the assimilation. The middle plot sets an arbitrary finite lifetime of t = 1 h for particles with L* > L*max, and the bottom plot sets the lifetime for those particles at t = 1 min. [67] We can immediately see that, for these values of m and K, the outer boundary is more of a sink than a source of electrons during this interval. When the DREAM allows the high L* phase space density to be a free parameter it does not set the boundary condition high. This is primarily because the observations from Polar near the last closed drift shell (L*max) have low phase space density. Setting the loss lifetime to very low values (1 min) produces phase space densities that are too low at L* < L*max which is confirmed by comparison with the Polar data (not shown). On the other hand setting the loss lifetime to 1 h outside the last closed drift shell produces results that agree quite well with Polar and differ only slightly from the free outer boundary case. This is quite reasonable because electrons outside L*max are on open drift orbits and are lost to the magnetopause in less than one drift period. The DREAM results are also consistent with the results from other models studying radial transport [e.g., Shprits et al., 2006]. An additional practical conclusion, though, is that magnetopause loss in radiation belt models can be very accurately captured with the very simple assumption of finite lifetimes outside the last closed drift shell – much like the finite lifetimes assumed for inside and outside the plasmapause.

4.4. Including a Source for Local Acceleration [68] The location, timing, and energy dependence of phase space density peaks is a subject of great interest because the existence of PSD peaks is a predicted consequence of acceleration by gyroresonant wave-particle interactions [e.g., Reeves et al., 2009]. The simple 1D radial 13 of 25

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Figure 9. DREAM assimilation of a storm in October 2002 with three different assumptions for the outer boundary. Top plot shows an assimilation that leaves the phase space density at the outer boundary (L* = 10) as a free parameter. Middle plot sets a finite electron lifetime, t = 1 h, outside the last closed drift shell (L*max). Bottom plot is similar to the middle plot but with t = 1 min. All three plots are for m = 2083 MeV/G and K = 0.03 G1/2RE, and for these values a model with a lossy outer boundary at L*max with a lifetime of t = 1 h provides a good physical description of the electron dynamics. diffusion model does not include a physical process for producing local phase space density peaks so the peaks are produced by the assimilated data. As seen in Figure 8 this can sometimes produce a peak near the GEO observations, another peak near the GPS observations, and a minimum in between (i.e., L* ≈ 5). The PSD minimum between GEO and GPS does not have any likely physical explanation. Rather, it is more likely that there is a process that accelerates electrons and increases phase space density in the location where we have no observations. If we postulate the existence of a local acceleration mechanism in the vicinity of L* = 5 it would increase PSD in that region. Electrons would then diffuse to both higher and lower L* and match the satellite observations in those regions in a more realistic way. [69] We can test the effects of a localized acceleration process using a simple modification of the Fokker-Planck equation by adding a “source” term, S(L*, A). For a single value of m, a process that accelerates particles from lower energies to higher energies will appear as a source of new phase space density for that m (assuming a spectrum with

higher flux at lower energy). A more complete treatment of local acceleration would have to self-consistently diffuse electrons from lower to higher energy. But, in our 1D model we calculate phase space density for each m independently and can treat energization as a source of PSD for that particular m. [70] Figure 10 illustrates the concept schematically. We chose the functional form of the source term and assume it is centered on L* = 5 with a Gaussian width of L* = 1 and a time-dependent amplitude, A(t), that is a free parameter optimized by the assimilation process. (We have tried leaving the center location and width as free parameters but find that all three parameters are not well constrained for a single m.) [71] Figure 11 shows the results of the assimilation and the parameter estimation. The top plot shows the assimilation results with our standard 1D radial diffusion model (with a lossy outer boundary). The middle plot shows the effect of adding a PSD source term with an amplitude, A(t), as shown in the bottom plot. As expected, the addition of a PSD source term reduces or eliminates 14 of 25

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Figure 10. We show, schematically, the addition of a PSD source term to the assimilation. The source, S(L*, A), has a spatial extent given by a Gaussian in L* centered on L* = 5.6 with a width of 1. The amplitude, A(t), is a free parameter in the assimilation that is chosen to best match the observations.

Figure 11. Here we show the same storm from October 2002 with and without a source term. The source term represents acceleration and is selected as a fixed Gaussian in L* with an amplitude, A(t), that is optimized by the assimilation. Top plot, with no source, shows peaks in PSD near the observation locations (L* ≈ 4 and L* ≈ 6) with a minimum in between. Adding a PSD source centered at L* = 5 fits the same observations better and generally removes the physically implausible minimum. 15 of 25

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Figure 12. Approximately 6 months of PSD values from GEO, GPS, and Polar satellites for m = 462 MeV/G and K = 0.005 to 0.3 G1/2RE. See section 4.5 for a description of the plot. the physically implausible minimum between the GEO and GPS observation locations. Interestingly, it also produces higher absolute PSD values than the assimilation without a source term. This result further illustrates (1) the importance of a realistic physical forecast model and (2) the importance of having sufficient observations to constrain the free parameters in that model. [72] Clearly these results are only the first steps. A Gaussian distribution with fixed width and location is certainly not physically realistic. Observations that fill a larger volume of L* space will likely provide enough information to allow amplitude, location, and span to all free parameters but an even better physical model would use those parameters to constrain a true energy diffusion term rather than a source of PSD at fixed m.

4.5. Data Assimilation as a Function of m and K [73] Radial diffusion does not change the first two adiabatic invariants, m, K. This is what allows us to treat each individual (m, K) pair as an independent assimilation. There is, however, a great deal of information available by comparing the assimilation results for different m and K values. [74] Even more importantly, when we want to validate the assimilation results against independent observations, we must transform back from phase space density as a function of magnetic invariants to flux (or dose) as a function of absolute spatial position, electron energy, and pitch angle. Even to calculate flux along a particular spacecraft orbit at a single energy and pitch angle it is necessary to calculate a large number of (m, K) pairs.

[75] A simplifying assumption that is often used is to assume a fixed functional form for the pitch angle and spectral distributions that is independent of location. Adopting this assumption allows one to do an assimilation for a single (m, K) and to extrapolate to other energies and pitch angles using the assumed functional forms. This greatly simplifies and speeds up the calculations but, as we will see, introduces unphysical artifacts in the assimilation outputs and reduces the validity for real-world space weather applications. [76] DREAM was initially tested using 25 individual assimilation runs (5 m and 5 K values). Currently we run 324 (18 18) or 1296 (36 36) pairs. At any given position in space we can calculate flux at a fixed energy and pitch angle by interpolation. [77] In Figures 12 and 13 we illustrate some of the features of the DREAM assimilation as a function of K. Here we use a relatively low value of m = 462 MeV/G and K = 0.005 to 0.3 G1/2RE. Figures 12 and 13 now show approximately the last 6 months of 2002. Figure 12 shows the observations from GEO, GPS, and Polar that were used in the assimilation and Figure 13 shows the results. [78] One obvious feature of Figure 12 is that the density of observations is a strong function of K. Very small values of K correspond to equatorial pitch angles very close to 90 which can only be directly measured when the satellite is very close to the magnetic equator. For K = 0.005 we clearly see where the satellites sampled near 90 equatorial pitch angles. GEO is nearly equatorial and has nearly continuous data for all values of K. The range of equatorial pitch angles (and K) that GPS and Polar 16 of 25

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Figure 13. Approximately 6 months of PSD values calculated from the DREAM model for m = 462 MeV/G and K = 0.005 to 0.3 G1/2RE. See section 4.5 for a description of the plot. can measure is a strong function of L*, or, equivalently, the range of L* that can be measured is a strong function of K. GPS orbit measures low K only where it crosses the magnetic equator near L* = 4 but for larger and larger K, GPS can provide PSD over larger and larger ranges of L*. The same is true of Polar. For K = 0.005 the Polar data are quite sparse while for K = 0.3 much of the outer radiation belt is directly sampled. [79] In Figure 12 we have kept the color scale for PSD the same for each K. The other striking feature is the strong dependence of PSD on K at a given point in space. This is, of course, a reflection of the highly anisotropic pitch angle distributions in the original data. Generally the radiation belts exhibit “pancake” distributions with higher fluxes near 90 and lower fluxes in the fieldaligned directions (0 and 180 ). If we extrapolate the measured pitch angle distributions to a presumed equatorial distribution we could fill in much more of the observational volume. This is equivalent to extrapolating from large K to small K. But, as Figure 12 shows, this can be a very dangerous assumption that is very sensitive to the shape of the distribution. Note, for example, how little variation there is at K = 0.1 compared to K = 0.005. [80] Figure 13 shows the results from the DREAM model for the same time and same set of K values. Here we have not included any losses at the magnetopause but, rather, allow the phase space density at L* = 10 to be a free parameter. A significant difference between this plot and Figure 9 is the behavior at high L*. Here the assimilation can frequently fit the observations by setting the appropriate PSD value at L* = 10 and allowing it to radially

diffuse. This is particularly true at low values of K and is consistent with the high PSD values from Polar seen in Figure 12. This can be understood in terms of the relative roles of source or sink at high L* as a function of energy. Figure 9 shows m = 2083 MeV/G while Figures 12 and 13 show m = 462 MeV/G. As we have noted, for lower m (lower energy) the radial gradient of PSD can be nearly flat or positive outward which allows the outer boundary to be a source. [81] It is also true, however, that for the low K values, the observations for L* > 7 are quite sparse which relaxes constraints on the assimilation. Notice that, in Figure 13, as K increases the outer boundary acts more and more like a sink of particles. This is a function of the negative radial gradients seen in Figure 12 for those K. There are at least two possible explanations that need to be investigated further: (1) the boundary values set by DREAM at low K may be an artifact of the assimilation or (2) convection from the plasma sheet could produce the observed gradients at low K and variations in pitch angle diffusion could produce different pitch angle distributions as a function of L*, energy, and time. In any case, the DREAM output provides a wealth of information to test against theory and observations.

5. Testing and Validation [82] No model has value for space weather unless it can be tested and validated. Quantitative testing allows the model’s performance to be evaluated against other models or performance can be evaluated as function of time as a 17 of 25


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Figure 14

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given model evolves. Validation of the model enables developers to discover the conditions under which the model performs better or worse and allows users to make judgements about the utility of the model prediction. We have conducted several initial tests of DREAM that are quite encouraging [Reeves et al., 2008] but, at the same time, we realize that systematic, independent testing and validation must still be done. Here we discuss a few of the initial tests. [83] A constant challenge in testing and validation is finding an appropriate data set to use and there are several pitfalls to be avoided. The data must be truly independent. Using geosynchronous data as input and different geosynchronous data (even from a different satellite series) as the validation set only tests the validity in that particular orbit (i.e., as a function of longitude or local time). The validation data must also have known calibrations with an accuracy and fidelity at least as high as what is required from the model. If long time periods of validation are required then instrumental drift and aging must be considered and, often, careful multisatellite inter-calibration is required. [84] To test DREAM we wanted to use data from a type of instrument and orbit that were different from any of the data used as input. Initially we chose data from the Aerospace dosimeters on HEO which is in a Molniya orbit [O’Brien et al., 2008]. Unfortunately, the process of applying assumed pitch angle distributions and fitting a spectrum to integral energy dosimeters resulted in an absolute uncertainty in the validation data that was higher than desired. While more robust processing could greatly improve the validation data set, we chose instead to use Polar observations for validation. This, of course, meant that we could not use Polar as input to DREAM. Therefore, the DREAM results that we have validated to date use only LANL geosynchronous and GPS observations. We use 3 geosynchronous satellites but only a single GPS satellite as input. Polar is excellent for validation of these results because (1) the orbit is very different from either GEO or GPS, (2) it covers a large part of the outer electron radiation belts, and (3) the instruments are extremely well calibrated and have been inter-calibrated against many other observations from many other satellites including CRRES (the current “gold standard”). [85] In validating models it is useful to compare against other models using the same validation data. Currently two radiation belt models must be considered as the standards: AE-8 [Vette, 1991] and CRRESELE [Brautigam and Bell, 1995]. The AE8 model is the accepted international standard model. It represents an average compiled from statistical observations sorted by magnetic coordinates (L and B/B0). AE8 is not a time-dependent model but

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there are slightly different versions for solar maximum or solar minimum. This study uses AE8 min. The CRRESELE model is based on observations taken in 1990–1991 by the joint USAF-NASA Combined Release and Radiation Effects Satellite (CRRES). It is a time-dependent model that is parameterized by geomagnetic activity (Ap15) as well as magnetic coordinates. (Ap15 is a 15-day average of the Ap index which is nearly equivalent to the Kp index.) [86] Figure 14 shows 1 MeV electron flux measured by Polar as a function of L and time along with output from DREAM, AE8 and CRRESELE. The Dst and Kp indices are also plotted for reference. Our validation interval is the year 2005 when Polar’s orbit sampled a broad range of L-shells. [87] The value of L* at a given point in space is also a function of pitch angle. Therefore, to transform from phase space density to flux we cannot ignore the azimuthal dimension (local time or longitude) and the actual satellite position (in 3D space) must be used. While we could do the validation at each point along the orbit we have chosen here to bin both the measurements and model by McIlwain L (not L*) and time. [88] There are a variety of factors that limit where Polar observations are available and other factors that limit where model outputs are available and the model outputs are shown only for times and locations along Polar’s orbit for which a comparison could be done. [89] Visually, it’s clear that DREAM captures much of the spatial structure, intensity, and variability of the Polar observations. AE8 is a static model and, as such, cannot show variability but it is interesting to note that the absolute value of the fluxes is low for many L-shells. CRRESELE is dynamic but is statistically parameterized by Ap15. While there are times when the Ap15 variations are well correlated with the MeV electron variations there are other times when they are not. [90] We now consider the quantitative comparisons. Figure 15 shows the electron flux as a function of L averaged over a full year. All three plots show, in black, the average profile of 1 MeV electron flux as a function of L measured by Polar. The mean of the log(flux) is the solid curve and the standard deviation around the mean is shown with dotted curves. The average Polar fluxes are compared to the average model fluxes as shown. Both the DREAM and CRRESELE averages are within a factor of 2 of the average Polar fluxes and inside one standard deviation across the full range of L. Both show a profile that is very similar to the observations. The DREAM averages match best near L = 4 and 6 and are worst at L < 4 where no data were used in the model. CRRESELE and AE8 match best between L = 4–5 where the peak fluxes are observed but the AE8 model systematically

Figure 14. DREAM model results. The top four plots all show 1 MeV electron flux (color-coded with units of cm2-ssr-keV) as a function of time and L. Data are binned in 1-day, 0.5 L intervals. From top to bottom, the plots show flux measured by the Polar satellite, flux predicted by the DREAM model, the AE8min model, and the CRRESELE model. Bottom plot shows geomagnetic activity measured by Kp (blue) and Dst (black). 19 of 25

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Figure 15. Average flux values predicted by the models. Each plot shows the average and standard deviation of fluxes measured by Polar as a function of L (geocentric radius). The average fluxes predicted by the DREAM, AE8, and CRRESELE models reproduce the average observations reasonably well with the exception of AE8 at high L values. underestimates flux at lower L and overestimates flux at higher L (at least for this time period and this energy). The underestimate at low L is less than a factor of 2 but the overestimate at L = 6 is a factor of 10 and continues to get worse at higher L. [91] As one would hope, all three models are able to reproduce the average conditions with reasonable accuracy but where the DREAM data assimilation really shows its utility is in predicting the variation around the mean. A standard metric for measuring the ability to track variations is the prediction efficiency, PE, which is defined as P ðmi oi Þ PE ¼ 1 P ðo i ho iÞ

where mi is the model output at time i, oi is the observation at time i, and 〈o〉 denotes the average value of the

observations which is the average flux as shown in Figure 15. A perfect prediction is indicated when PE = 1 because, in that case, the model matches the observations at every time i and (mi oi) = 0. If the “model” simply assumes that the flux is equal to the average observed value then mi = 〈o〉 which gives a prediction efficiency of PE = 0. Positive values of PE indicate that the model predicts the variations better than assuming that flux is constant and negative values of PE mean that the model is worse. [92] Figure 16 shows the prediction efficiency for DREAM, AE8, and CRRESELE. Clearly the DREAM model excels at predicting the variation of electron flux observed by Polar. For all values of L > 4, the prediction efficiency is positive and, as we noted above, this study did not include data at L < 4 because it only used data from GEO (L = 6.6) and GPS (L > 4.2) orbits. The CRRESELE model has a variable output that is scaled by geomagnetic activity but the fact that the prediction efficiency is

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Figure 16. Prediction efficiency. The prediction efficiency, PE (defined in section 5), tests the ability of the model to predict the variation of fluxes around the mean. A prediction efficiency of 1 is perfect agreement at all times. Prediction efficiencies less than or equal to zero do not provide useful predictions of the time variation of the observations.

negative at all L indicates that the variations are at the wrong times or are of the wrong magnitude. The AE8 model has no time variation so it may seem surprising that the prediction efficiency is not zero for all L. The reason it is negative for most values of L is that the AE8 value is not equal to the average observations. Where the AE8 and Polar averages are equal the prediction efficiency is zero. By applying a multiplicative factor to AE8 one could make the model match the observations at any given L but because the curves have different shapes they will only cross at one L and the prediction efficiency will still be less than or equal to zero for any satellite orbit. [93] One might expect the DREAM prediction efficiency to be close to 1 for L* ≈ 4 and L* ≈ 6 because the assimilation should be well-constrained by data in those regions. There are several reasons why this might not be the case. One is that the data from Polar, GEO and GPS might have inadequate inter-calibrations which would introduce systematic differences. The other contributing factor is that Polar samples a different part of the field line (and therefore different pitch angles) than GEO or GPS. Since we have done careful inter-calibration of the three data sources we suspect the pitch angle distributions are the primary cause of lower prediction efficiencies. This is particularly true for GPS data for which we have to use an assumed, statistical pitch angle distribution. If this suspicion is confirmed it is somewhat comforting because we can hope that more accurate pitch angle assumptions, other data sets, and/or physics models with pitch angle diffusion could all potentially improve the results.

6. Event Studies: January 1997 [94] Often deeper physical understanding comes from detailed analysis of specific geophysical events. It is becoming more and more common to use models and data together to help overcome the limitations in each. Data assimilation models such as DREAM provide a new tool for event studies.

[95] The January 1997 relativistic electron event was pivotal for changing our understanding of radiation belt dynamics and for raising awareness of space weather effects. On January 10, a magnetic cloud from a coronal mass ejection (CME) hit the Earth’s magnetosphere producing series of dramatic geophysical changes. The event also generated considerable public interest because of its association with the catastrophic failure of the Telstar 401 satellite (see Washington Post, 23 January 1997, p. 1). [96] The event was exceptionally well studied with authors examining the storm/substorm response [Kamide et al., 1998; Li et al., 1998; Lyons et al., 2000], radiation belt electron dynamics [Buhler et al., 1998; Reeves et al., 1998a, 1998b; Selesnick and Blake, 1998], and the overall geoeffectiveness [Baker et al., 1998]. It was also extensively modeled using both MHD models [Goodrich et al., 1998] and inner magnetosphere/ring current models [Jordanova et al., 1998, 1999]. [97] Reeves et al. [1998a] combined data from 5 geosynchronous satellites, 3 GPS satellites, and Polar to understand the spatial and temporal dynamics in higher resolution than would be possible from a single satellite. Here we use DREAM with only 2 spacecraft - a single LANL geosynchronous satellites and a single GPS satellite - to predict relativistic electron fluxes along the Polar satellite orbit and compare against the Polar observations themselves in order to illustrate the strengths (and limitations) of eventoriented data assimilation with minimal observations. [98] Figure 17 shows L-shell sorted fluxes of 700 keV electrons as measured by Polar and as given by DREAM along with the Dst index and solar wind velocity. Both the observations and the assimilation are binned in steps of 0.5 RE and 1-day resolution. The L-shell is given by L* as calculated from the T89 magnetic field model. The geosynchronous data used in the assimilation is from 1994 to 084 and the GPS data is from NS33. To calculate phase space densities we use pitch angle resolved data from GEO but for GPS (with a nearly hemispherical field

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Figure 17

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of view) we assume a pitch angle distribution based on statistical distributions measured by CRRES. [99] As discussed above, DREAM is run for 324 (18 18) m, K pairs. To convert back to flux at a specific energy and pitch angle we first need to calculate the m, K, and L* at each point along the orbit to know which part of the PSD volume to sample. Because PSD is converted to flux at each point along the Polar orbit prior to binning by L* and time, it is possible to be quite specific about the data comparison. Here we have chosen to look at energies of 700 keV and pitch angles of 50 . The pitch angle is the local pitch angle at Polar’s location, not the equatorial pitch angle, so local pitch angles from 0 to 90 can all be compared. Similarly any energy within the range of m values chosen. [100] It is clear from Figure 17 that the DREAM assimilation results capture the general features of the January 1997 event quite well but do not capture every detail. In particular the timing and intensity of the event are very similar in the observations and the assimilation. For this choice of energy and local pitch angle, however, the DREAM assimilation does not show the same radial profile as the observations and do not accurately capture the rate of decay in the quiet period between January 15 and 25. The specifics of where the assimilation and observations differ depend on the particular energy and pitch angle selected but can generally be attributed to one of the following: the use of only two satellites in the assimilation, the limited pitch angle information for GEO and GPS (combined with the high inclination of the Polar orbit), and the limitations of a 1D radial diffusion model for accurately capturing the dynamics of the belts. [101] In future we intend to be able to directly compare energy spectra and pitch angle distributions along the Polar orbit (as opposed to sorted by L*) which will allow highly detailed model-data comparisons. Such detailed model-data comparisons should lead to greater physical understanding and improvements in the models. For space weather applications, however, the true power is to use all available data within the assimilation and to specify the distribution of electrons (spectrum, pitch angle distribution, flux, and fluence) at any point within the radiation belts and, specifically, along the orbits of satellites that do not have instruments to measure the environment.

7. Prospects for DREAM Forecasts [102] We now turn our attention to uses of the DREAM model for real-time applications and for space weather

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forecasts. While retrospective analysis or “near real-time” specifications have many uses for assessing satellite anomalies, state-of-health, or survivability analysis, Space Situational Awareness (SSA) applications also need to answer the question “What is happening at my satellite right now”? or the even more challenging question “What is likely to happen to my satellite in the next few days”? Up to this point we have discussed DREAM results that use data that has already been collected and is available for the model. [103] Real-time operations require real-time observations. We have designed DREAM to be run in real time with whatever data sets are available. DREAM is not dependent on any given data stream. Observations from any satellite will populate (or modify) the predicted environment in any orbit. In our example, if GPS data always lags real time by 12 h then the DREAM output will be less accurate in the region around L = 4 during that 12-h gap but the model will still provide a global real-time specification for all orbits. Currently few radiation belt observations are available in real time. GOES data from geosynchronous orbit is the most widely available. We have implemented a real-time prototype of DREAM using GOES-13 observations [Reeves et al., 2010] and post the results at http://dream.lanl.gov/. While there are many limitations in the current implementation it allows us to exercise the software under real-world conditions and provides a basis for more useful future products. [104] To date, we have spent only limited effort applying DREAM for forecasting. We have based our development on the principle that any forecast is completely dependent on accurate and timely nowcasts. There is no fundamental feature of DREAM that prevents forecasting but at this time we lack confidence in the forecast accuracy of the physics models and/or lack observations that can accurately constrain the physical processes in those models. [105] A bright light on the horizon is the upcoming launch of NASA’s Radiation Belt Storm Probes (RBSP) mission [e.g., Reeves, 2007]. RBSP includes a real-time space weather broadcast that will be collected and disseminated by Johns Hopkins Applied Physics Laboratory. The RBSP orbit and instrumentation makes it ideal for real-time assimilation in DREAM or similar models. RBSP is also designed and instrumented to produce the understanding needed for true radiation belt forecast models which we will have the opportunity to test, validate, intercompare, and ultimately, implement in an operational setting.

Figure 17. The January 1997 radiation belt event. Top two plots show fluxes of electrons with energy 700 keV and a local pitch angle of 50 , and the bottom two plots show Dst and solar wind velocity. Top plot shows measurements from Polar binned in L* with a resolution of 0.5 RE and in time with 1-day resolution. (The Polar orbital period is 18 h) The second plot shows the output from DREAM as sampled along the Polar orbit trajectory for the same energy and local pitch angle and the same binning. Using only two satellites (1 GEO and 1 GPS) the DREAM assimilation captures the overall timing and intensity of the event but differences in time dependence and radial extent can also be seen. Similar calculations over a broad range of energies and pitch angles can be used for highly-detailed reanalysis of events as well as for general space weather applications. 23 of 25

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[106] Acknowledgments. This research was conducted as part of the Dynamic Radiation Environment Assimilation Model (DREAM) project at Los Alamos National Laboratory. We are grateful to the sponsors of DREAM for financial and technical support. We would also like to thank the many people who have contributed to the DREAM project by designing the space environment instruments, by collection and processing of data, by developing software and network infrastructure, and by frequently providing knowledge, expertise, and guidance, particularly: Joachim Birn, Bern Blake, Joe Borovsky, Tom Cayton, Rod Christensen, Dot Delapp, Ted Fritz, Peter Gary, Marc Kippen, Liz MacDonald, Evan Noverovsky, Paul O’Brien, Yuri Shprits, Harlan Spence, Davis Thomsen, Bob Tokar, Athina Varotsou, Dan Welling, and Dan Winske.

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