SPACE WEATHER, VOL. 10, S08001, doi:10.1029/2012SW000767, 2012
Localized adaptive inflation in ensemble data assimilation for a radiation belt model H. C. Godinez1 and J. Koller2 Received 9 January 2012; revised 8 June 2012; accepted 11 June 2012; published 1 August 2012.
[1] In this work a one-dimensional radial diffusion model for phase space density, together with observational satellite data, is used in an ensemble data assimilation with the purpose of accurately estimating Earth’s radiation belt particle distribution. A particular concern in data assimilation for radiation belt models are model deficiencies, which can adversely impact the solution of the assimilation. To adequately address these deficiencies, a localized adaptive covariance inflation technique is implemented in the data assimilation to account for model uncertainty. Numerical results from identical-twin experiments, where data is generated from the same model, as well as the assimilation of real observational data, are presented. The results show improvement in the predictive skill of the model solution due to the proper inclusion of model errors in the data assimilation. Citation: Godinez, H. C., and J. Koller (2012), Localized adaptive inflation in ensemble data assimilation for a radiation belt model, Space Weather, 10, S08001, doi:10.1029/2012SW000767.
1. Introduction [2] The Earth’s radiation belts, discovered by Van Allen et al. [1958], experience significant and abrupt changes due to acceleration, loss, and transport processes [Reeves et al., 2003] of the trapped energetic electrons in Earth’s magnetic field. Since high-energy particles can potentially damage space infrastructure, understanding and predicting the dynamics of the radiation belts is of particular importance. Nevertheless, their structure remains poorly described since satellite observations are typically restricted to single point measurements, resulting in a very sparse and incomplete observational field [Chen et al., 2006]. [3] Data assimilation comprises methodologies that combine information from a first-principle physics model with observational data and include relevant error statistics to provide an improved state of the system. Among the most popular assimilation methods are the classical Kalman filter (KF) [Kalman, 1960], extended Kalman filter (EKF) [Jazwinski, 1970], and ensemble Kalman filter (EnKF) [Evensen, 1994]. These methods are now widely accepted in a large variety of different fields ranging from control application to terrestrial weather prediction. 1 Applied Mathematics and Plasma Physics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA. 2 Space Science and Applications, Intelligence and Space Research Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA.
Corresponding author: H. C. Godinez, Applied Mathematics and Plasma Physics, Theoretical Division, Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. (
[email protected])
©2012. American Geophysical Union. All Rights Reserved.
[4] Recently, data assimilation has become popular in space physics and space weather modeling because of the following reasons: (i) Given that observations in the radiation belts are very sparse, assimilation methods are able to fill the gaps using a first-principles physics-based model while maintaining consistency between model dynamics and observations. (ii) Data assimilation can provide an efficient method to provide a best possible estimate of the current state of the system. The current state can be used as an accurate and global initial condition for forecasting models and thus improving their prediction skill. These underlying benefits are specifically addressing the need of today’s space physics and space weather modeling efforts. For example, (a) radiation belt models and data assimilation can be used to develop better specifications models of the space environment and replacing empirical models like AE8 [Vette, 1991]. An additional benefit of data assimilation can be provided to (b) spacecraft operators who may want to know the radiation levels for spacecraft orbit where there is no monitoring of radiation. Since assimilation methods can effectively fill data gaps, the combined model-data output can be used to (c) determine whether any particular satellite failure has been caused by an increase in radiation or not. These benefits of data assimilation methods are just examples but have long been recognized in the terrestrial weather forecasting community who makes effort to deal with related issues. [5] Today, data assimilation has been proven to be an effective method for optimally combining radiation belt models with observational data. One of the first adopters were Naehr and Toffoletto [2005], who implemented the EKF
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for state estimation in a physics-based radiation belt model using synthetic observational data. Kondrashov et al. [2007] used the EKF with observations from the Combined Release and Radiation Effects Satellite (CRRES) to estimate the electron phase space density (PSD) along with model empirical parameters that characterize the lifetime of relativistic electrons in Earth’s radiation belts. [6] Koller et al. [2007] and Shprits et al. [2007] were first to perform data assimilation to estimate the localized acceleration term and validated their results by performing simulations with synthetic data. Shprits et al. [2007] performed data assimilation using Kalman filter and applied it to 50days of CRRES data, while Koller et al. [2007] applied the ensemble Kalman filter to multispacecraft observations for a storm in October of 2003. [7] Furthermore, data assimilation has been validated by comparing reanalysis produced from assimilating observational data from two different satellites [Ni et al., 2009a], and it has been shown that data assimilation can produce reanalysis with errors that are smaller than errors of PSD reconstructed from individual satellites [Ni et al., 2009b]. In a more recent paper Daae et al. [2011] demonstrated that reanalysis is relatively insensitive to the assumed boundary conditions when sufficient data is available at all considered L-shell for assimilation. [8] More recently, Kondrashov et al. [2011] have developed a Kalman filter in log-normal space that better captures the gaussianity of both the model and observations. These studies show that data assimilation can be successfully applied to radiation belt models, and will be an important element to specify the radiation belt environment, as well as estimate the current state of the system and thus improve the prediction skill or radiation belt models. [9] One of the most efficient assimilation methods is the ensemble Kalman filter (EnKF), which was introduced by Evensen [1994]. The EnKF has gained wide acceptance due to its ease of implementation and robustness, where the first-principle physics models can be used as forecast models in the assimilation without linearization of nonlinear dynamics. Of particular concern in EnKF, and all assimilation methods in general, is the presence of model errors or uncertainty. In this work we extend the results of Koller et al. [2007] by accounting for the effects of model errors in the EnKF with a localized adaptive inflation scheme for the model forecast covariance matrix. [10] There are various sources of errors in radiation belt models, such as approximate parameterization of physical processes, incomplete physics, numerical discretizations, and the such. In many data assimilation applications, including radiation belt models, it is assumed that model errors are relatively small compared with errors in initial conditions and parameters [Zupanski, 1997; Dee and Da Silva, 1998; Gillijns and De Moor, 2007; Trémolet, 2007]. On the contrary, for one-dimensional radial diffusion models, the errors in the model itself are often larger than errors in the initial conditions. As a result, the model
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forecast covariance matrix for one-dimensional diffusion models will be underestimated. The underestimation of the forecast covariance can lead to filter divergence, where the observations are completely ignored in the assimilation. To adequately include the effects of model errors in the EnKF, approaches include inflating the forecast error covariance matrix with multiplicative inflation [Anderson and Anderson, 1999]. This technique involves multiplying the covariance matrix by an inflation factor, which is a scalar, to increase the uncertainty of the model. Wang and Bishop [2003] presented an adaptive technique to compute the inflation factor which satisfies certain statistical properties. [11] The effectiveness of the adaptive inflation technique depends on the observation distribution in time and space. Since most of the derived PSD observations experience a large variability, that can be of several orders of magnitude, the use of a single inflation factor for the entire model spatial domain can cause an over-inflation of the covariance matrix. The over-inflation can lead to states that are unphysical and unrealistic. To overcome this difficulty, we implemented a localized adaptive inflation technique, where the inflation is localized to small regions where observational data is available. The advantage of the localized inflation is that the covariance matrix is not overinflated over regions where there is no observational data, and model errors are more accurately represented. Additionally, the computation of the inflation factors is performed adaptively by using the statistical principle in Wang and Bishop [2003], which ensures an appropriate local inflation factor. To test the adaptive inflation and localized adaptive inflation techniques, we performed twin-experiments, where synthetic data is taken from a reference run of the model. [12] The paper is organized as follows. Section 2 presents the one-dimensional radial diffusion model for PSD, as well as the derived PSD observations from satellite data. Section 3 provides a brief review of the EnKF, along with the adaptive inflation technique and the localized adaptive inflation technique implemented in this paper. The assimilation results from a twin-experiment, as well as with real satellite data, are discussed in section 4. Conclusions and future work are presented in section 5.
2. Radiation Belt Model and Observations 2.1. 1-D Radial Diffusion Model [13] The time evolution of the distribution of electrons contained in the radiation belts is described by their phase space density f(m, K, L*, t), calculated as a function of the three adiabatic invariants. Here m, K, and L* are the adiabatic invariants for periodic gyration, bounce motion, and drift motion of an electron in the geomagnetic field at time t. A radial diffusion model for the distribution of electrons is derived using the Fokker-Plank equation with constant adiabatic invariants m and K. This simplified 1-D model [Schulz and Lanzerotti, 1974] only describes the radial
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evolution of electron phase space density, as a function of L*, and is given by ∂f ∂ DLL ∂f ¼ L2 þ Q; ∂t ∂L L2 ∂L
ð1Þ
where L was used for abbreviation of L*, DLL is a diffusion coefficient and Q is a forcing term representing radiation belt losses, sources, energy diffusion and pitch angle diffusion including mixed diffusion. The unit for PSD variable c 3 , where c is the speed of light, cm are cenf are cmMeV timeters, and MeV are megaelectron volts. The diffusion coefficient [Brautigam and Albert, 2000; Lanzerotti and Morgan, 1973; Lanzerotti et al., 1978] is a given function of magnetic activity DLL ¼ 100:506Kp9:325 L10 ;
ð2Þ
where Kp is a geomagnetic activity index. The forcing term Q is unknown here, and the omission of this term may introduce significant model uncertainty. One of the main objective of this work is to compensate for this forcing term by appropriately including the effects of model uncertainty into an ensemble data assimilation method. The methods developed for this simple 1-D diffusion model can also be applied for multidimensional diffusion models [e.g., Subbotin et al., 2010; Varotsou et al., 2008]. For this purpose we implement a localized adaptive inflation technique, within the ensemble assimilation method, to reduce the effects of model error and produce an improved model state. This particular technique can also be applied to other radiation belt models with higher dimensions. [14] We use the SpacePy package for the numerical simulation of the 1-D radial diffusion model [Morley et al., 2010]. The radial diffusion equation (1) is discretized with a modified Crank-Nicolson implicit finite difference solver which is second-order accurate in space and time [Crank and Nicolson, 1947]. The computational grid is of dimension n = 90 for the model domain 1 < L* < 10. The CrankNicolson scheme is an implicit, unconditionally stable numerical method, which converges with any reasonable time step Dt [Welling et al., 2011]. The inner boundary condition at L* = 1 is fixed at zero, and the outer boundary condition is modeled with a strong loss term (timescale of a drift period) for L* > Lmax, the last closed drift shell.
2.2. Data [15] Data from three Los Alamos National Laboratory Geosynchronous (LANL-GEO) satellites (LANL-97a, 1991– 080, and 1990–095), as well as from Polar and GPS-ns41, are assimilated into our model. The data set targets a highspeed solar wind stream storm on 25 October 2002, and observation are considered over the course of 12 days. [16] Since the data is originally in units of flux, a transformation is applied to obtain phase space densities for the data set [Chen et al., 2006]. This step is necessary to bring
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the data set into the same coordinate system as the model [Koller et al., 2007].
3. The Ensemble Kalman Filter and Adaptive Inflation [17] There are a number of data assimilation techniques used throughout a wide range of disciplines [Jazwinski, 1970; Daley, 1991; Kalnay, 2003]. One type of assimilation technique that is widely accepted and implemented is the Kalman filter method and its variations such as the Kalman filter (KF) [Kalman, 1960], extended Kalman filter (EKF) [Jazwinski, 1970] and, ensemble Kalman filter (EnKF) [Evensen, 1994] to name a few. In our current work we use the EnKF data assimilation for radiation belt models described in Koller et al. [2007]. A brief overview of the EnKF is presented in the following section. 3.1. Ensemble Kalman Filter [18] The EnKF is a Monte Carlo approximation to Kalman filtering for non-linear models, and has gained wide acceptance in data assimilation applications. The EnKF is a suboptimal method that approximates the results of the classical Kalman filter. The EnKF has the advantage of propagating the model covariance matrix using an ensemble of model simulations, which can be more computationally efficient than the KF for multidimensional problems. The EnKF is a sequential data assimilation method that uses an ensemble of model integrations or forecasts to calculate the forecast mean and error covariance matrix needed for the analysis. The ensemble is updated with every analysis to reflect information provided by the observations, and is evolved using the forecast model between analysis. By providing a flow-dependent estimate of the forecast error covariance matrix, the EnKF can optimally adjust the forecast to newly available observations. In doing so, it may be able to produce analyses and model forecasts that are more accurate than other data assimilation schemes which assume a constant covariance matrix. [19] Let Mtk →tkþ1 be a numerical model that advances an n-dimensional discrete state vector x from time tk to tk+1 , xðtkþ1 Þ ¼ Mtk →tkþ1 ðxðtk ÞÞ
ð3Þ
For a vector of m measurements yo 2 Rm and an ensemble of N state forecast vectors xi 2 R n, i = 1, …, N the EnKF analysis equation is given by: xai ¼ xi þ K yoi Hxi ;
i ¼ 1; …; N
1 K ¼ PHT HPHT þ R ;
ð4Þ ð5Þ
where xai 2 Rn is the analysis, Pf 2 Rnn is the forecast covariance matrix, R 2 Rmm is the observation covariance matrix, H 2 Rmn is the linear observation operator. The matrix K 2 Rnm is referred to as the Kalman gain matrix.
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The main advantage of the EnKF is that the forecast covariance matrix Pf is approximated using the ensemble of model state vectors xi. This matrix is obtained using the relation Pf ¼
N 1 X ðxi x Þðxi x ÞT ; N 1 i¼1
ð6Þ
where x is the average of the forecast ensemble members. The vector yoi 2 Rm is a perturbed observations vector defined as yoi ¼ yo þ ɛi ;
i ¼ 1; …; N;
ð7Þ
where ɛi 2 Rm is a random vector from a normal distribution with zero mean and a specified standard deviation so. This standard deviation is usually specified to be equal to the observational error. In the standard formulation of the EnKF, so is equal to the observational error, and this error is assumed to be known. This rarely happens in practice and usually some simple approximations are made. In the current study the observational error is set to 30% of the observed PSD variance based on a conjunction study of geosynchronous satellites observations in phase space coordinate L* [Koller et al., 2007]. The observation covariance matrix R can be constructed from the observations perturbations as N 1 X R¼ ɛi ɛTi : N 1 i¼1
ð8Þ
[20] For a detailed description of the EnKF algorithm see for example Evensen [2003] and Houtekamer and Mitchell [1998]. We adopted EnKF with a singular value decomposition algorithm similar to Evensen [2003]. 3.2. Accounting for Model Uncertainty in EnKF [21] In many realistic physical applications, model errors may be significant enough to disrupt the EnKF data assimilation. The forecast covariance matrix can be underestimated if the increase in uncertainty, due to model errors, is not included in the calculation of the covariance matrix. This underestimation will cause filter divergence, where the data assimilation completely ignores the observations and fails to correct the forecast. To mitigate the impact of model errors in EnKF, the current work presents a method to adjust the forecast covariance matrix in order to cope with the increase in uncertainty due to unaccounted model errors. 3.3. Adaptive Covariance Inflation [22] In the ensemble Kalman filter, the forecast covariance matrix is approximated using the ensemble, as given in equation (6). This approximation tends to underestimate the true covariance matrix due to the sampling error associated with the small number of ensemble members,
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and unaccounted model errors in the ensemble. As a result, the filter gives too much confidence to the model. This results in the underestimation of the forecast errors in the next assimilation cycle, which can eventually lead to filter divergence (the assimilation ignores the observations). [23] Covariance inflation techniques are typically used to adjust the forecast covariance matrix in order to account for increased uncertainties. A popular approach for covariance inflation was introduced by Anderson and Anderson [1999], which is a multiplicative inflation technique. This technique increases the forecast covariance by multiplying the prior covariance matrix by an inflation factor a as, ~ f ¼ aP f ; P
a > 1:0:
ð9Þ
The main drawback in this method is that the inflation factor a has to be tuned at each assimilation cycle, until a satisfactory analysis is obtained. Understandably, the tuning process can become computationally expensive and time consuming. Furthermore, one should not expect the inflation factor to be constant in space and time. [24] To address this issue, Wang and Bishop [2003] proposed an adaptive inflation methodology, where the inflation factor is adaptively estimated at each assimilation cycle. The forecast covariance matrix P f and observation covariance matrix R should satisfy the relationship zzT ¼ HP f HT þ R;
ð10Þ
where the vector z ¼ yo Hx is the innovation vector of observations minus forecast [Houtekamer et al., 2005]. The over-bar in equation (10) represents the average over many cases or statistical expectation. Following the multiplicative covariance inflation and using equation (10) the inflation factor can be estimated as a¼
zT z TrðRÞ ; Tr HP f HT
ð11Þ
where Tr denotes the trace of a matrix. It must be noted that this inflation factor is estimated in observation space and not in model space. In the case where observational data are sparse, such as in radiation belts, an appropriate inflation factor that applies to the whole domain may be difficult to attain.
3.4. Localized Adaptive Inflation [25] The effectiveness of the multiplicative adaptive inflation depends on the sparsity of observations and model error characteristics. When only sparse and irregular observations are available, the ensemble can be overinflated in regions where no observations are available. To address this issue Anderson [2009] developed an adaptive inflation technique that varies with the location and time of individual variables of the state vector. Following the work by Anderson [2009] and Li et al. [2009] we develop and implement a localized adaptive inflation technique, which inflates the ensemble locally where observations 4 of 11
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observations that fall within the local region in observation space, and let z‘i;j be the local innovation vector for ensemble member i, that is ‘ ‘ z‘i; j ¼ yo‘ j Hj xi; j ;
ð12Þ
where H‘j is the local observation operator that maps the local state to observation space. Following the adaptive inflation technique, given by equation (11), a localized inflation coefficient is defined as T z‘j z‘j Tr R‘j a‘j ¼ T f‘ Tr H‘j Pj H‘j
ð13Þ
where z‘j is the average of the local innovation vector over
Figure 1. Localization example in L* for the 1D radial diffusion equation. The figure shows the model PSD (blue lines with dots), and synthetic observations (green triangles) at a particular time instance over an L*-grid for illustrative purposes. For our paper, the local region of influence is a interval in L* composed of three grid points. As an example of the localization, a grid point is chosen at L* = 6.5 (red star) and shown with the adjacent variable grid points (black stars) that fall within the local region of influence. The corresponding observations are chosen that fall within this interval in L* for the particular grid point in question. The local model grid point, as well as the observations, are shown in the inset plot. are available, for a radiation belt model. The inflation is adaptive locally, that is, it satisfies the correlation relationship from equation (10) for a local region of influence. [26] For each grid point j = 1, …, n in the model spatial domain, we define a local region of influence. The local region of influence is a spatial region defined in such a way that it includes model grid point variables that are strongly correlated in space. This can be found by studying the correlations between the values of the state vector on the mesh. For our 1D radial diffusion model, the diffusion coefficient will determine how PSD values are propagated or diffused throughout the L*-shells, which can be based on the study of Brautigam and Albert [2000]. Using this information, we defined our local region of influence for each grid point in the L*-grid to be an interval consisting of three grid points, one central grid point and one at each side (see Figure 1 for a simple example of a local region). More generally, the definition of the local region of influence will depend on the particular model being used [see Houtekamer and Mitchell, 2001; Hamill et al., 2001]. Prior knowledge of how the model variables are correlated will help define an appropriate local region for each grid point in the model. [27] Let xi; j denote the value of the state vector at grid point j for ensemble member i. Let yjo‘ denote the
the ensemble, R‘j denotes the local observation covariance f‘
matrix, and Pj is the local forecast covariance matrix. Using this localized adaptive inflation factor a new state vector ~ x is defined, where the jth grid point of ensemble i is inflated locally, in the following way: ~ x i; j ¼ xj þ
qffiffiffiffiffi a‘j xi; j xj ;
ð14Þ
where xj is the jth grid point in the ensemble average. This localization will inflate the ensemble members only where observations are available. In case no observations are available that correspond to the local area of influence of the particular grid point, the inflation factor is set to 1. In this way, the ensemble is not over-inflated in regions without observations and the model covariance matrix, defined by equation (6) using ~ x i , is only inflated in the appropriate locations. [28] In summary, the localized adaptive inflation technique in the EnKF proceeds as follows. [29] 1. The ensemble state variables xi are advanced to the time of the next set of observation using the model equation (3), [30] 2. For each grid point j in the state variable xi;j , define a local region of influence in the state-space and do the following: [31] (i) define the local observation operator H‘j , [32] (ii) identify which observation fall within the local region of influence to determine yo‘ j , f‘ [33] (iii) define the local forecast covariance matrix Pj and the local observation covariance matrix R‘j , [34] (iv) compute the local innovation vector z‘i;j , [35] (v) compute a local inflation factor a‘j using equation (13), [36] (vi) for each ensemble member i, inflate the ensemble at grid point j using equation (14) to obtain ~ x i;j , [37] 3. Compute the analysis using equations (4) and (5) with the inflated ensemble members ~ xi. f‘ [38] For step (iii), the local covariance matrix Pj can either be computed using the local state vectors x‘i;j or 5 of 11
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Figure 2. (top) A model forecast simulation without an artificial forcing term Q, where the x-axis is the day of the year for 2002, y-axis is the L* shell, the color pffiffiffifficontours are the phase space density in log space, for fixed m = 2083 MeV/G and K ¼ 0:03 GRE, and the white line indicates the last closed drift shell Lmax. (bottom) Derived Dst (red line) and Kp (black dotted line) index for the model simulation. Clearly, without the additional forcing term Q, a one dimensional radiation belt model would not provide a very accurate forecast when compared to the observations (see Figure 5). The model diffuses the initial condition until a zero PSD solution is reached. Data assimilation, together with an inflation technique, can provide a correction to the model solution and better approximate the observed radiation belt environment. extracted from the global covariance matrix P f. For our f‘ Pj
case, we computed using the local state vectors x‘i;j . This cycle will be repeated each time there are observations available. [39] For our particular application, the observations and state vector are expressed in the same physical variables, where the observations are interpolated to model grid locations. In this case, the observation operator H is a matrix with 0 or 1 entries, indicating where observations are available for the state vector. In other cases where the observation operator H is a non-linear function, a linearization of H must be used for the analysis equations (4) and (5) and for the computation of the inflation factor a.
4. Assimilation Experiments and Results 4.1. Twin Experiments [40] We verified the performance of the ensemble data assimilation with the proposed localized adaptive covariance inflation, described in section 3.4, by performing an identical twin experiment, where a reference solution (“truth”) is produced by solving equation (1) with a nonzero forcing term Q. The model forecast, or control run, is produced by solving the same equation but with a zero forcing term, that is Q = 0 in equation (1). Therefore, the objective of the data assimilation is to obtain an analysis that reflects the “truth” by assimilating synthetic data, sampled from the “truth” solution, using the control
solution with its forcing set to zero. This type of experiments will test the ability of the proposed inflation technique to consider the effects of model errors in EnKF, as well as to increase the ability of the EnKF to approximate the true state of the system. [41] For the assimilation twin experiment, both the reference and forecast are simulated for a period of 12 days. All simulations were p performed for the fixed ffiffiffiffi m = 2083 MeV/G and K ¼ 0:03 GRE. The observations are sampled from the reference solution along the track of three LANL geosynchronous satellites, as well as Polar, and GPS satellites (described in section 2.2) for 23 October to 4 November 2002. The observations are averaged over a time window of 30 min to avoid frequent assimilation cycles due to the high time frequency but sparse spatial distribution of the recorded satellite position for the given dates. Observations are taken only inside the outer boundary where PSD is actually defined, which is the last closed drift shell Lmax. The initial condition for both reference and control simulations is taken by spatially interpolating the PSD data given at the satellite locations to the model grid using a Gaussian function. [42] Figure 2 shows the solution of the radiation belt model with Q = 0, and as can be seen, the lack of a nonzero forcing term will lead to a diffusion of the initial condition, and ultimately, a zero PSD solution. [43] An artificial forcing term was added to the numerical model for the reference run, which is constant in time.
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Figure 3. (a) Reference run of radiation belt model with an artificial forcing term, generated c 3 with a Gaussian function centered at L0 = 5.0 with s = 0.5 and A ¼ 1:0 108 cmMeV . (b) Observations taken from the reference run along the track of the Polar, GPS, and three LANL satellites for October 23 to November 4 2002, where the black line indicates the outer boundary defined by last closed drift shell Lmax. (c) Assimilated results with EnKF using a global adaptive inflation with the control run (no forcing term), where the assimilation was not able to capture the effect of the forcing term. (d) Assimilated results with EnKF using localized adaptive inflation, where even though the forecast was lacking the forcing term, the assimilation is able to approximate the “true” state of the system around the observations. Figures 3c and 3d clearly show how the different inflation scheme perform in the presence of model error, which in this case is simulated by the lacking of the forcing term in the control runs. Axis and contours are the same as in Figure 2.
The constant forcing term is given by a Gaussian function in space 2 ! L* L0 ; Q L* ¼ A exp 2s2
ð15Þ
where A specifies the magnitude of the constant forcing term, L0 the central position in L*, and s the width of the localized acceleration. For the reference run in our numerical experiments the forcing term was placed in the middle of the L* domain, that is L0 = 5.0, with s = 0.5 and c 3 . This forcing term is representing A ¼ 1:0e 8 cmMeV localized energizations which is added every time step and diffused by the 1-D radiation belt equation (1). The
reference run is simulated for a total of 12 days with a time step of 30 min. [44] An ensemble of 50 members was used for the EnKF data assimilation experiments. The initial condition of each ensemble member is generated by introducing a relative perturbation to the control initial condition as xi ðt0 Þ ¼ ð1:0 þ ɛÞxc ðt0 Þ;
ð16Þ
where xi ðt0 Þ and xc ðt0 Þ are the initial conditions for ensemble i and the control run, respectively, and ɛ is a random number drawn from a normal distribution with mean zero and a standard deviation of 0.35. [45] A pair of data assimilation experiments was performed: the first assimilation experiment used the global 7 of 11
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inflating the ensemble globally instead of locally, which can lead to over-inflation. [47] For this experiment, the main cause for over-inflation is the large discrepancy between observations and model forecast at lower L*-shells. Given that the computation of the inflation parameter a is directly linked to the innovation vector (see equation (11)), this large discrepancy will directly affect a, causing it to attain a larger value than needed for the whole spatial domain. [48] The over-inflation is resolved by using the localized adaptive inflation technique in the EnKF, as shown in Figure 3d. From the plots in Figure 3, it is possible to note that the adaptive inflation produces a slightly overestimated PSD field (Figure 3b), while the localized adaptive inflation produces an underestimated PSD field (Figure 3d). From the EnKF-LAI plot we see that, even though the forecast does not contain the artificial forcing term, the assimilation is able to approximate the “truth” around the observation locations better than EnKF-AI. Due to the nature of the localized adaptive inflation technique, only regions where there are observations available will be inflated. [49] To measure the performance of the assimilation a normalized root mean square (rms) error is computed over the whole spatial domain, where the RMS is defined as rmsðxÞ ¼
rffiffiffiffiffiffiffiffiffiffiffi 1 T x x; n
ð17Þ
where x 2 Rn. The RMS error is normalized by the RMS of the reference solution, that is, the normalized error is
Figure 4. (a) Normalized root mean square (rms) error measure for EnKF-AI (red line) and EnKF-LAI (blue line). (b) The observation variance in time (black line), and the prediction skill for EnKF-AI (red line) and EnKF-LAI (blue line). The prediction skill for EnKF-AI between L* = 3.7 and 4.2 goes up to 2.2 1012, which is out of the relevant range of the plot within this L*-shell interval. The forecast performance of the data assimilation with a localized adaptive inflation scheme is much better than with a global adaptive inflation scheme.
adaptive inflation technique (equation (11)) and is referred to as EnKF-AI. The second experiment used the localized adaptive inflation technique (equation (13)) presented in section 3.4, and is referred to as EnKF-LAI. Both experiments use the same ensemble model setup and synthetic data. [46] The solution of the EnKF-AI assimilation, shown in Figure 3b, exhibits pulses which correspond to the location of observations that appear in lower L* shell values (Figure 3c). These pulses are a result of indiscriminately
a Þ rmsðxt x ; rmsðxt Þ
ð18Þ
a the where xt denotes the reference model state and x analysis ensemble average. [50] Another useful measure for assessing the effectiveness of an assimilation scheme is to analyze the innovation vector sequence f
zk ≡ yok Hxk ;
ð19Þ
where k is a time index, and the forecast is evolved from the previous analysis, that is xk ¼ Mtk1 →tk xak1 . The innovation vector is the difference between the observations and the model forecast from a previous analysis state. Hence, it provides the improvement to the forecast due to the assimilation of observations at a previous time. In the context of the radiation belts, Koller et al. [2007] and Shprits et al. [2007] used zk to identify missing physics in radiation belt models. Using the innovation vector, a prediction skill measure (PS) [Fukumori, 2006] is defined as
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Figure 5. (a) Phase space density processed from count rates for the dates from October 23, 2002 to November 11 2002. Satellite observations were gathered from three LANL-GEO satellites (LANL-97a, 1991–080, and 1990–095), and from Polar and GPS-ns41.The observations pffiffiffiffi were converted to phase space densities at constant m = 2083 MeV/G and K ¼ 0:03 GRE using the same procedure as Koller et al. [2007], and is used in the ensemble Kalman filter assimilation experiments. (b) Assimilation results for EnKF-LAI experiment with satellite observational data. The solution is smoother and more accurate than with the global adaptive inflation.
the variance of the time distributed sequence zk as a function of L*, that is PS L* ≡
Nt 2 1 X zk L* z L* ; Nt 1 k¼1
ð20Þ
where Nt is the total number of time instances where observations are available, and z is the average over time. The predictive skill measures the discrepancy between the observations and the model forecast. A large value in the prediction skill indicates a large disagreement between the forecast and observations, and a low value indicate low disagreement.
[51] Figure 4a shows the normalized root mean square (rms) error for EnKF-AI (red line) and for EnKF-LAI (blue line). The RMS error for the assimilation using adaptive inflation (EnKF-AI) shows again large oscillatory pulses, which correspond to the observations in lower L* shells, where the assimilation tries to consider this new set of data in the global inflation factor estimation. Since no localization is performed, the complete state is inflated. This unrealistic pulse-like behavior has been removed with the localized adaptive inflation (EnKF-LAI) where the error remains relatively low. Figure 4b shows the time variance of the observations (black line), as well as the prediction skill for EnKF-AI (red line) and EnKF-LAI (blue line), as
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Figure 6. Time variance of the satellite observations (black line), as well as the prediction skill of the EnKF-AI (red line), and EnKF-LAI (blue line) showing that the EnKF-LAI method provides a solution with higher accuracy. As with the twin experiment, the prediction skill for EnKF-AI goes up to 1012 between L* = 3.7 and 4.2, which is out of the plot.
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exhibiting unrealistic pulses in PSD. The PSD values from the EnKF-LAI experiment are in better agreement with the actual satellite data than those from EnKF-AI. [55] Figure 6 shows the time variance of the satellite observations (black line), the prediction skill of EnKF-AI (red line) and of EnKF-LAI (blue line). The figure shows that EnKF-LAI provides a more accurate solution than EnKF-AI. Note that the prediction skill for EnKF-AI is greater than the maximum plot range around L* = 6.3 and from L* = 3.7 to 4.2. Particularly, the spike in prediction skill for EnKF-AI in the lower L*-shells can be attributed to the high discrepancy between the forecast and the observations. This is the result of missing or unknown physical properties in the 1-D radial diffusion model. From Figure 6 it is clear that the EnKF with global adaptive inflation is having difficulties in properly assimilating the observations around these L*-shells. These spikes are not present in the prediction skill of EnKF-LAI, which is better approximating the observations around these L*-shells. Hence, the localized adaptive inflation technique EnKFLAI should be the preferred choice of methodologies to account for the effects of model errors in EnKF for radiation belt models.
5. Conclusions measured by the variance of their respective innovation vectors, equation (19). From the plot, it is clear that the prediction skill of the EnKF-LAI is better than EnKF-AI, since the former provides a more accurate prediction of the state of the system.
4.2. Satellite Data Assimilation [52] After the identical twin experiments, we also assimilated actual satellite data using our EnKF algorithm with global and localized adaptive inflation. As mentioned in section 2.2 the data is transformed from flux to phase space pffiffiffiffi density, for the fixed m = 2083 MeV/G and K ¼ 0:03 GRE . Data from three Los Alamos National Laboratory Geosynchronous (LANL-GEO) satellites (LANL-97a, 1991–080, and 1990–095), as well as from Polar and GPS-ns41, is used for assimilation. The data set includes a high-speed solar wind stream storm on 25 October 2002, and observations are considered over the course of 12 days. We employed 50 ensemble members for our assimilation, initialized by introducing a relative perturbation to the initial state as in equation (16). [53] Figure 5a shows the PSD observations derived from flux, for the specified dates, as well as Dst and Kp for this time period. These are the same observations used by Koller et al. [2007], where the difference in magnitude is due to the fact that Koller et al. [2007] normalized the log PSD values in all of their plots. No such normalization is used in our results. [54] As expected, assimilating these observations with the global adaptive inflation produced the unrealistic pulsing solution, but using the localized adaptive inflation (see Figure 5b) provided an improved solution without
[56] Model uncertainties are of particular concern for one-dimensional radial diffusion models for Earth’s radiation belts. These uncertainties can come from various sources, such as parameterization, incomplete physics, and numerical errors to name a few. Accounting for model uncertainty in data assimilation is of particular importance for radiation belt models, since assimilation techniques are increasingly being used to combine model and observational data in order to better understand, predict and specify radiation belts. In this paper we implement a localized adaptive inflation technique to the ensemble Kalman filter for a one-dimensional radial diffusion model, to better account for model error in radiation belt models. Numerical twin experiments show that the localized adaptive inflation technique provides an improved analysis, with a better predictive skill, compared to a standard global adaptive inflation technique. In addition to the twin experiment, we also assimilated real observational data to determine the PSD for radiation belts from three LANL satellites (LANL-97a, 1991–080, and 1990–095), as well as from Polar and GPS-ns41 satellites. The predictive skill of the analysis obtained with the localized inflation shows again a major improvement over a global inflation technique. We conclude that the localized adaptive inflation technique provides an improved assimilation skill for radiation belt models which still may contain large errors due to unknown physical processes. It is crucially important to properly account for the effects of model errors in data assimilation to improve upon the analysis. After all, radiation belt data assimilation will eventually be used to estimate initial conditions for other forecast models just like in terrestrial weather prediction. These initial 10 of 11
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conditions will need to be accompanied by their uncertainties for proper forecasting and forecast uncertainty quantification.
[57] 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. This paper is Los Alamos Unlimited Release LA-UR 12–00003.
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