AE9, AP9 and SPM: New Models for Specifying the Trapped Energetic Particle and Space Plasma Environment

Abstract

The radiation belts and plasma in the Earth’s magnetosphere pose hazards to satellite systems which restrict design and orbit options with a resultant impact on mission performance and cost. For decades the standard space environment specification used for spacecraft design has been provided by the NASA AE8 and AP8 trapped radiation belt models. There are well-known limitations on their performance, however, and the need for a new trapped radiation and plasma model has been recognized by the engineering community for some time. To address this challenge a new set of models, denoted AE9/AP9/SPM, for energetic electrons, energetic protons and space plasma has been developed. The new models offer significant improvements including more detailed spatial resolution and the quantification of uncertainty due to both space weather and instrument errors. Fundamental to the model design, construction and operation are a number of new data sets and a novel statistical approach which captures first order temporal and spatial correlations allowing for the Monte-Carlo estimation of flux thresholds for user-specified percentile levels (e.g., 50th and 95th) over the course of the mission. An overview of the model architecture, data reduction methods, statistics algorithms, user application and initial validation is presented in this paper.

Appendices

Appendix A: Acronyms

ACE:

Advanced Composition Explorer (satellite)

BDDII:

Burst Detector Dosimeter II

CAMMICE:

Charge and Mass Magnetospheric Ion Composition Experiment

CEASE:

Compact Environment Anomaly Sensor

CPME:

Charged Particle Measurement Experiment

CRRES:

Combined Radiation and Release Experiment (satellite)

DD:

Displacement Damage

DEMETER:

Detection of Electro-Magnetic Emissions Transmitted from Earthquake Regions (satellite)

Dos:

Dosimeter

EPAM:

Electron, Proton, and Alpha Monitor

GEO:

Geosynchronous Orbit

GPS:

Global Positioning System (satellite)

HEEF:

High Energy Electron Fluxmeter

HEO-F1:

Highly Elliptical Orbit—Flight 1 (satellite)

HEO-F3:

Highly Elliptical Orbit—Flight 3 (satellite)

HISTe:

High Sensitivity Telescope—electrons

HISTp:

High Sensitivity Telescope—protons

HYDRA:

Hot Plasma Analyzer

ICO:

Intermediate Circular Orbit (satellite)

IGE:

International Geostationary Electron (model)

IDP:

Instrument for Particle Detection

IPS:

Imaging Proton Spectrometer

LANL-GEO:

Los Alamos National Laboratory-Geosynchronous Orbit satellite

LEO:

Low-Earth Orbit

MEA:

Medium Energy Analyzer

MEO:

Medium-Earth Orbit

MICS:

Magnetospheric Ion Composition Sensor

MPA:

Magnetospheric Plasma Analyzer

PET:

Proton/Electron Telescope

PROTEL:

Proton Telescope

SAMPEX:

Solar Anomalous and Magnetospheric Particle Explorer (satellite)

SCATHA:

Spacecraft Charging at High Altitudes (satellite)

SC3:

High Energy Particle Spectrometer

SEE:

Single event effects

SEM:

Space Environment Monitor

SOPA:

Synchronous Orbit Particle Analyzer

Tel:

Telescope

TIROS:

Television Infrared Observation Satellite (satellite)

TSX5:

Tri-Services Experiment-5 (satellite)

Appendix B: Construction of the Flux Maps, Principal Components and Time Evolution Matrices

Described in this appendix are the methods used to build the flux maps and components of the auto-regression scheme (Eq. (7)) introduced in Sect. 5 and illustrated in Fig. 5. Much of the theory underlying the V1.0 architecture can be found in O’Brien (2005), O’Brien and Guild (2010) and Johnston et al. (2013). A good deal of statistical analysis is needed to build the autoregressive model and only a cursory overview is given here. Interested readers are referred to Wilks (2006) for information on the basic techniques and O’Brien (2012a) for the application to radiation belt models.

As mentioned in Sect. 5 the statistical quantities tracked in the flux maps are the 50th and the 95th percentile unidirectional flux values m50 and m95, respectively. Actually, the variable m95′=m95−m50 is used instead of m95 so the restriction that m95>m50 imposed on the analysis takes the simple form m95′>0. Hereinafter the ′ will be dropped. From these two quantities the entire particle distribution can be determined by assuming a two-parameter functional form for the distribution function.

For each satellite data set the unidirectional flux measurements are sorted into a set of time sequential maps where the spatial bins are defined by the coordinate grid and the time bin for each map is one day for electrons and 7 days for protons. Note that the term “spatial” is used in a general sense to denote all the non-temporal coordinates including energy. For each satellite pass through a spatial bin during the time bin a value for the flux and variance is computed as a weighted average of the j and dlnj measurements during the pass. Weights are determined by the relative values of dlnj which themselves are computed from a cross-calibration procedure discussed in Sect. 4.3. Bin pass average values are then averaged for each time bin. These preliminary maps can be spatially sparse as only coordinate bins through which the satellite passes will contain values. With the tracked percentile values defined as the vector $\underline{\theta}= (m50, m95)$ and their deviations about the average $\underline{\bar{\theta}}$ as $\delta\underline{\theta} = \underline{\theta} - \bar{\underline{\theta}}$, the average value and the covariance matrix $\mathrm{cov}(\delta\underline{\theta})$ are computed for each bin by using a bootstrap technique over the set of time averaged values. With the bootstrap a random selection of time binned values is chosen, with replacement, to equal the original set size. Each selected value is perturbed randomly in a manner constrained by its standard deviation (in a lognormal sense) and the resultant set sorted to obtain a value for $\underline{\bar{\theta}}_{i}$ in each spatial bin i (hereafter the subscript i will be dropped to avoid notational overload). Repeating this process 200 times yields a distribution of $\underline{\bar{\theta}}$ estimates that are used to compute an average $\underline{\bar{\theta}}$ and a 2×2 local $\mathrm{cov}(\delta\underline{\theta})$. This process is performed for each spatial bin for each sensor data set.

The filling-in procedure using the templates is as follows. A realization of $\underline{\bar{\theta}}$ on the sparse grid described above is constructed by randomly perturbing the original $\underline{\bar{\theta}}$ values consistent with a normal distribution characterized by $\mathrm{cov}(\delta\underline{\theta})$ in each bin. From this realization the quantity $\Delta\underline{\theta} = \underline{\bar{\theta}} - \underline{\theta}^{(0)}$, where $\underline{\theta}^{(0)}$ is the template estimate, is computed. The $\Delta\underline{\theta}$ grid is filled in first by using energy interpolation and extrapolation, and then applying nearest-neighbor averaged and smoothed before being added to the original sparse $\underline{\bar{\theta}}$ grid to produce an estimate for the full $\underline{\bar{\theta}}$ grid. This process is repeated 10 times for each template and the distribution of $\underline{\bar{\theta}}$ obtained is used to compute a new best estimate of the $\underline{\bar{\theta}}$ and $\mathrm{cov}(\delta\underline{\theta})$ over the entire grid for each satellite.

To compute the final $\underline{\bar{\theta}}$ map, denoted as the generalized vector $\bar{\underline{\boldsymbol{\theta}}}$ with a single index covering all the (E,K,Φ or h _min ) grid, the individual satellite $\underline{\bar{\theta}}$ maps are averaged with weighting by the standard deviations computed from $\mathrm{cov}(\delta\underline{\theta})$ in each bin. The maps are then smoothed. The final covariance $\mathrm{cov}(\delta\underline{\boldsymbol{\theta}})$ is then captured by computing the “anomaly matrix” S where the number of rows in S are the number of grid points (i.e. equal to the number of values in $\bar{\underline{\boldsymbol{\theta}}}$) and each column of S is a normalized bootstrap realization of $\underline{\bar{\theta}}$ on the grid obtained by selecting a set of random sensor groups, randomly perturbing $\underline{\bar{\theta}}$ in each bin assuming a normal distribution of $\underline{\bar{\theta}}$ characterized by $\mathrm{cov}(\delta\underline{\theta})$ and averaging the result. This layer of bootstrapping captures the uncertainties of measurement errors, spatial interpolation and extrapolation and the temporal coverage limitations of a finite set of sensors. By construction, $\mathrm{cov}(\delta\mathbf{\underline{\theta}})=\mathbf{SS}^{T}$where T represents the transpose operation. Nominally, $\mathrm{cov}(\delta\underline{\boldsymbol{\theta}})$ would be a very large matrix of size N×N, where N is twice the number of grid points (∼50,000 for AE9/AP9). By constructing S of 50 bootstrap realizations, a number found to be sufficient, only an N×50 matrix need be computed and stored. Singular value decomposition of S keeping only the number of dimensions needed to re-compute 90 % of the total variance further reduces the stored matrix size to N×10.

The end result of the process is a flux map $\bar{\underline{\boldsymbol{\theta}}}$ of the 50th and 95th percentile unidirectional flux values with the anomaly matrix S allowing for computation of the spatial error covariance across the entire grid. With the assumed Weibull or lognormal distribution functions the mean or any percentile level flux can be computed from $\bar{\boldsymbol{\theta}}$. Uncertainties in these values can be calculated from $\mathrm{cov} (\delta\mathbf{\underline{\theta}}) = \mathbf{SS}^{T}$ and represent estimates of the combined uncertainty imposed by imprecise measurements, lack of spatial and temporal coverage, and the templates used for interpolation and extrapolation.

To determine the quantities involved in auto-regression equation (Eq. (7)) it is first necessary to estimate the spatial (Σ) and spatiotemporal ($\hat{\mathbf{R}}$) covariance matrices for the flux. Although there is certainly considerable error in computing the spatial and spatiotemporal covariance matrices it will be neglected hereinafter because only a low-order model of the dynamics is sought and the uncertainty in the flux values due to measurement and space weather is tracked through flux maps and associated covariance.

The starting point is the set of time-average fluxes in each spatial bin for each satellite. Randomly selecting two bins (possibly from different sensors), we compute a “Gaussian” correlation coefficient at several different time lags. The time-averaged flux values in each bin are transformed to Gaussian-equivalent variables z _i according to the relation,

$$ {\varPsi}(z_{i,k}) = F_{i}(j_{i,k})\approx \frac{k}{1 + N_{i}} $$

(8)

where Ψ (often denoted Φ in the statistical literature) is the cumulative distribution of a standard Gaussian with unit variance and a zero average, F _i is the empirical cumulative distribution within the ith bin, and k is the index of the sort list of k=1 to N _i fluxes j _i,k within the bin. This transformation is independent of the choice of Weibull or lognormal distributions. By transforming to the Gaussian-equivalent variables the formalism of multivariate normal distributions can be used to develop the autoregressive prediction model. In particular, the spatial and temporal covariance matrices are defined as,

(9)

(10)

where z is the vector of z _i values spanning the entire grid, τ _k is the kth time lag and the 〈⋯〉 notation represents the average. Because of limited spatial and temporal coverage, the initial estimates of the covariance matrices is incomplete. They are filled in using a 100-point nearest neighbors average.

To reduce the substantial storage requirements of what is nominally a N×N dimensional matrix a principal component decomposition of Σ is employed, i.e.,

(11)

(12)

where $\mathbf{Q}=[\hat{\mathbf{q}}_{1}, \hat{\mathbf{q}}_{2}, \ldots,\hat{\mathbf{q}}_{N_{q}}]$ is a matrix of the i=1,2,…,N _q principal component eigenvectors $\hat{\mathbf{q}}_{i}$ and q is the state vector of principal component amplitudes representing a particular realization of z. Σ contains many noise factors, which we remove by excluding any principal component that explains less than 1 % of the variance. Using the remaining N _q ∼10 principal components the spatiotemporal covariance can be expressed as,

$$ \hat{\mathbf{R}}_{k} = \mathbf{Q} \bigl\langle\mathbf{q}(t)\mathbf{q}^{T}(t - \tau_{k}) \bigr\rangle\mathbf{Q}^{T} = \mathbf{QR}_{k}\mathbf{Q}^{T}, $$

(13)

where R _k =〈q(t)q ^T(t−τ _k )〉. When k=0 then τ _k =0 by definition and R=I, the identity matrix, so that R ₀=Σ. In summary, the procedure to obtain the spatiotemporal covariance matrices from the data is to (a) compute elements of Σ and each $\hat{\mathbf{R}}$ from time averages in spatial bins (Eqs. (9) and (10)), (b) fill in the missing elements of Σ and $\hat{\mathbf{R}}$ via nearest neighbors averaging, (c) determine the principal components Q of Σ (Eq. (12)), and (d) determine each R _k using Eq. (13).

The autoregressive time-evolution equation (Eq. (7)) of order N _G is used to advance the N _q principal component amplitudes in time. An expression for the expectation value 〈q(t)q ^T(t−τ)〉 can be derived from the time-evolution equation,

(14)

where R _m−k =〈q(t−τ _m )q ^T(t−τ _k )〉 and the CC ^T term arises from 〈η(t)q ^T(t)〉 because η(t) is uncorrelated with all prior q(t). With the R matrices determined from the data, Eq. (14) can be inverted to obtain G and C (O’Brien 2012a).

Statistically realistic flux profiles are generated by choosing at t=0 a scenario-specific random seed which determines the initial principal component amplitudes q(0) and a set of flux conversion parameters, i.e. the $\underline{\boldsymbol{\theta}}$ percentiles characterizing the distribution computed from the flux map $\bar{\underline{\boldsymbol{\theta}}}$ with a random perturbation added consistent with the global spatial error covariance cov(δ θ) encoded in the anomaly matrix S. A time history of the q(t) is generated with Eq. (7), the Gaussian equivalent fluxes z(t) determined from Eq. (11) and the physical flux values j(t) from the left-side of Eq. (8) using the conversion parameters given by $\underline{\boldsymbol{\theta}}$.