Short time augmented extended Kalman filter for soil analysis: a feasibility study

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1 ATMOSPHERIC SCIENCE LETTERS Atmos. Sci. Let. 13: (212 Published online 13 July 212 in Wiley Online Library (wileyonlinelibrary.com DOI: 1.12/asl.394 Short time augmented extended Kalman filter for soil analysis: a feasibility study Alberto Carrassi, 1,2 * Rafiq Hamdi, 1 Piet Termonia 1,3 and Stéphane Vannitsem 1 1 Institut Royal Météorologique de Belgique, Bruxelles, Belgique 2 Institut Catalá deciéncies del Clima (IC3, Barcelona, Spain 3 Department of Physics and Astronomy, Ghent University, Ghent, Belgium *Correspondence to: A. Carrassi, Institut Catalá de Ciéncies del Clima, Carrer Doctor Trueta 23, 85 Barcelona, Spain. acarrassi@ic3.cat Received: 4 August 211 Revised: 17 May 212 Accepted: 24 May 212 Abstract This paper presents a soil analysis scheme based on an extended Kalman filter (EKF, the short time augmented extended Kalman filter (STAEKF, where the model parameters are estimated along with the system state. We use an off-line version of the interaction soilbiosphere-atmosphere model for the assimilation of screen-level temperature and relative humidity. The leaf area index, the albedo, and the minimum stomatal resistance are estimated with the STAEKF. Results show that the STAEKF gives encouraging results and indicate the benefit of the simultaneous estimation of the state and parameters in soil data assimilation. Copyright 212 Royal Meteorological Society Keywords: data assimilation; parameter estimation; soil analysis 1. Introduction Current soil moisture analysis schemes in most numerical weather prediction (NWP centers are based on analyzed or observed screen-level variables. An optimal interpolation technique (OI developed by Mahfouf (1991 on the basis of the short-range forecasts of 2 m temperature (T 2m and relative humidity (RH 2m is operational in a number of centers: Météo-France (Giard and Bazile, 2, Environment Canada (Bélair et al., 23, and the European Centre for Medium-Range Weather Forecasts (Douville et al., 2 among others. However, some fundamental limitations are inherent with the use of the OI (Balsamo et al., 24; Mahfouf et al., 29, among these the fact that it does not allow for the use of nonlinear observation operator (note that the physical link between near-surface observations and soil variables is nonlinear and that the OI coefficients are usually derived using very simple assumptions and/or model configurations. A particularly promising approach to address the above shortcomings is the use of an extended Kalman filter (EKF. The EKF and a simplified version (SEKF, in which a static forecast error covariance matrix is adopted, have been successfully applied to the assimilation of direct and/or indirect soil observations (Draper et al., 29; Drusch et al., 29; Mahfouf et al., 29; Albergel et al., 21; Mahfouf and Bliznak, 211. Recently, Carrassi and Vannitsem (211a introduced an alternative formulation of the EKF where the uncertain model parameters are estimated along with the system state variables. The algorithm, referred to as short time augmented extended Kalman filter (STAEKF, uses a deterministic formulation for the model error dynamics (Nicolis, 23; an approach already used for the treatment of the error arising from the unresolved scales (Carrassi and Vannitsem, 211b and in the context of variational assimilation (Carrassi and Vannitsem, 21. With the aim of contributing to the current discussion on soil data assimilation strategies, we undertake here a set of numerical twin experiments with observations designed to test the STAEKF for the assimilation of screen-level observations using an offline version of the land surface model, Interactions between surface, biosphere, and atmosphere (ISBA (Noilhan and Mahfouf, 1996 where errors in three land surface parameters, the leaf area index (LAI, the albedo, and the minimum stomatal resistance (RS min are introduced. 2. The land surface model The two-layer version of the land surface model ISBA describes the evolution of soil temperature and moisture contents based on the force-restore method. The model is available within a surface externalized platform (SLDAS; Mahfouf, 27. The equations can be formally written as a dynamical system, dx dt = g(x, λ. The state vector, x = (T s, T 2, w g, w 2, contains the surface and deep soil temperatures T s and T 2 and the corresponding water contents w g and w 2. The vector λ is taken to represent the set of Copyright 212 Royal Meteorological Society

2 STAEKF for Soil Analysis 269 model parameters. A detailed description of ISBA can be found in the study by Noilhan and Mahfouf ( Surface assimilation schemes 3.1. Extended Kalman filter (EKF Let us write the model equations as a mapping from time t k to t k+1, x f k+1 = M(xa k, where xf and x a are the forecast and analysis state respectively, of dimension I,andMis the nonlinear model forward operator. A set of M observations of the true system, stored as the components of an M -dimensional observation vector y o, is supposed available at the regularly spaced discrete times t k = t + kτ, k = 1, 2..., with τ being the assimilation interval, y o k = H(y k + ɛ k, with ɛ being the observation error, assumed to be Gaussian with known covariance matrix R and uncorrelated in time, and y is taken to represent the unknown true state; H is the (possibly nonlinear observation operator mapping model variables to observation space. The EKF analysis state update equation reads x a = [I KH]x f + Ky o. The analysis error covariance, P a, is updated through: P a = [I KH]P f.thei M gain matrix K is given by K = P f H T [HP f H T + R] 1 with P f being the I I forecast error covariance matrix and H the linearized observation operator. The matrix P f is obtained by linearizing the model around its trajectory between two successive analysis times. Model error is assumed to be a random uncorrelated noise whose effect is modeled by adding an error covariance matrix, P m, at the forecast step so that, P f = MP a M T + P m, with M being the tangent linear model. Note that while P f is propagated in time and is therefore flow dependent, P m is defined once for all and it is then kept constant Short time augmented extended Kalman filter (STAEKF The forecast model, is augmented with P model parameters: z f = [ x f λ f ] = F(z a = [ M(x a ] F λ (λ a, (1 z = (x, λ is the augmented state vector. The augmented dynamical system F includes the dynamical model for the system s state, M, and a dynamical model for the parameters F λ. In the absence of additional information, a persistence model for F λ is often assumed so that F λ = I and λ f t k+1 = λ a t k ; the same choice has been adopted here. The forecast/analysis error covariance matrix, P f,a z, for the augmented system reads P f,a z = ( P f,a x P f,a T xλ P f,a xλ P f,a λ (2 where the I I matrix P f,a x is the error covariance of the state estimate x f,a, P f,a λ is the P P parametric error covariance and P f,a xλ the I P error correlation matrix between the state vector, x, and the vector of parameters λ. These correlations are essential for the estimation of the parameters. In general, one does not have access to a direct measurement of the parameters, and information are only obtained through observations of the system s state. In the controlled experiments given in the next sections the knowledge of the true parameters is used to diagnose the algorithms performance. The forecast error propagation in the STAEKF is given by P f z = CPa z CT, with C being the STAEKF forward operator defined as ( g C = M λ λ aτ (3 I P where I P is the P P identity matrix. Equation (3 embeds the key feature of the STAEKF; the presence of the term g λ λaτ allows for accounting for the contribution of the parametric error to the forecast error as well as to the error correlation between model state and parameters. An augmented observation operator is introduced, H z = [H ] with H as for the standard EKF. Its linearization, H z is now a M (I + P matrix in which the last P columns contain zeros. The augmented state and covariance update complete the algorithm and are equivalent to those of the EKF except that they refer now to the augmented system, and the gain matrix has dimension (I + P M (see Carrassi and Vannitsem, 211a, for more details. 4. Experimental setup We follow here a standard procedure in data assimilation studies, known as observation system simulation experiments (OSSE, in which a trajectory, solution of the model equations, is taken to represent the true unknown evolution which is then sampled at discrete times to obtain simulated observations. The estimated model trajectory is given again by a model solution but using a different set of parameters and initial conditions. The forcing data are the same for the truth and the assimilation solutions. They consist of 1 hourly air temperature, specific humidity, atmospheric pressure, incoming global radiation, incoming long-wave radiation, precipitation rate, and wind speed relative to the ten summers in the decade extract from ECMWF Re-analysis ERA4 and then dynamically downscaled to 1 km horizontal resolution over Belgium (Hamdi et al., 211. The fields are then temporally interpolated to get data consistent with the time resolution of the integration scheme of ISBA (3 s. In this study, ISBA is run in one off-line single column Copyright 212 Royal Meteorological Society Atmos. Sci. Let. 13: (212

3 27 A. Carrassi et al. T s T 2.5 EKF STAEKF W g W [m 2 /m 2 ].3 Leaf Area Index.25.2 [m 2 /m 2 ] 2 1 P a LAI.5 Figure 1. RMS estimation error in the four state variables for the EKF (red and the STAEKF (blue. The RMS error in the estimate of LAI obtained with the STAEKF is shown in the bottom-left panel, while the corresponding estimated error variance, P a LAI, in the bottom-right panel. mode for a 9 period, and the forcing parameters are those relative to the grid point closest to Brussels. An one-point soil model has been also used by Orescanin et al. (29, for parameter estimation using an ensemble-based assimilation algorithm. The simulated observations are T 2m and RH 2m, interpolated between the forcing level ( 2 m and the surface with the Geleyn s interpolation scheme (1988, at, 6, 12, and 18 UTC. The assimilation interval is τ = 6 h, while the observational noise is drawn from a Gaussian, N(, R, with zero-mean and covariance given by the diagonal matrix R with elements: diag(r = (σ 2 T 2m,σ 2 w 2m = (1K 2, 1 2. As explained in Mahfouf et al. (29, the observation operator H, relating the state vector to the observation includes the model integration. The initial P a and P m required by the EKF are set as diagonal with elements diag(p a = (σ 2 T s,σ 2 T 2,σ 2 w g,σ 2 w 2 = (1 K 2, 1 K 2, 1 2, and diag(p m = (σ 2 T s, σ 2 T 2,σ 2 w g,σ 2 w 2 = ( K 2, K 2, 4 1 4, (Mahfouf, 27. Parametric errors are introduced by perturbing either alternatively or simultaneously the LAI, the albedo and the RS min. These parameters strongly influence the surface energy balance budget and partitioning, which in turn regulate the circulation patterns and modify the hydrological processes. According to the STAEKF setup, we compute (not shown the functional dependence of the model equations on the 1 The values of σ wg and σ w2 are expressed as soil wetness index SWI = (w w wilt /(w fc w wilt where w fc is the volumetric field capacity and w wilt is the wilting point. uncertain parameters, the term g λ λ, required in Equation (3, relative to the ISBA model for the LAI, the albedo, and the RS min. For each summer in the period , a reference trajectory is generated by integrating the model with LAI = 1 m 2 /m 2, albedo =.2 and RS min = 94 s/m. Around each of these trajectories, Gaussian samples of 1 initial conditions and uncertain parameters are used to initialize the assimilation cycles. The initial conditions are sampled from a distribution with standard deviation (σ Ts,σ T2,σ wg,σ w2 = (5 K, 5 K, 1, 1, whereas LAI, the albedo, and the RS min, are sampled with standard deviations, σ LAI =.5 m 2 /m 2, σ albedo =.5 and σ RSmin = 5 s/m respectively (Ghilain et al., 211. The initial P a λ in the STAEKF read P a LAI = 1 (m2 /m 2 2, P a albedo = 1 4 and P a RS min = 5 (s/m 2, P a x is taken as in the EKF while P a x,λ is initially set to zero. 5. Results In the first set of experiments, only LAI is uncertain. Figure 1 displays the time series of the RMS estimation error in the four state variables for the EKF (red and the STAEKF (blue. The root-mean-square (RMS in the estimate of LAI obtained with the STAEKF is shown in the bottom-left panel, while the corresponding estimated error variance, P a LAI, in the bottom-right panel. The error is averaged over the ensemble of 1 experiments and over the ten summers. Note that in view of the rapid fluctuations of T s and T 2, to improve Copyright 212 Royal Meteorological Society Atmos. Sci. Let. 13: (212

4 STAEKF for Soil Analysis 271 Table I. RMS estimation error in the four state variables averaged over the 9 s of experiment for the Open Loop, EKF and STAEKF. T s (K T 2 (K w g (m 3 /m 3 w 2 (m 3 /m 3 LAI Open Loop EKF STAEKF Albedo Open Loop EKF STAEKF RS min Open Loop EKF STAEKF LAI and Albedo Open Loop EKF STAEKF visualization the corresponding error curves are displayed every 15 time-steps (equivalent to 12.5 h in all the figures. For surface and deep temperature, T s and T 2,the EKF and STAEKF give similar results for the first month with a slight advantage for the STAEKF at the error peaks, while a larger improvement is visible afterward. For the water contents, w g and w s, the STAEKF takes a longer time to exhibit a benefit over the EKF, but after around 2 s the STAEKF estimation error attains a lower level. After a sufficiently long time the estimation error essentially comes from the parametric error and the STAEKF becomes advantageous. The estimation of LAI with the STAEKF is also satisfactory as depicted by the error reduction observed in the bottom-left panel of Figure 1. The reduction of the actual error is tracked by the associated estimated variance, as it can be observed in the bottom-right panel. A complete perspective on the relative performance of the assimilation algorithms is given in the upper-most part of Table I which shows the RMS estimation error averaged in time (over the 9 s of experiment; results relative to an experiment without assimilation (referred to as open loop are also shown for completeness. By looking at the error level for the open loop experiment, the benefit of implementing a data assimilation technique is evident; then the slight advantage of the STAEKF over the EKF comes from its additional parameter estimation feature. After around 2 s the component of the prediction error coming from the growth of the previous analysis error, becomes progressively smaller and the error mainly originates from the misspecification of the parameters. Figure 2 is the same as Figure 1, but for the experiments with parametric error in the albedo. Similar to the previous case, the STAEKF shows better results after around 2 s, but now the improvements in the analyzed temperatures are a bit more rapid. A smooth error reduction is observed in the estimate of the albedo and is accompanied by a corresponding reduction in the associated error variance. In the second block of Table I the total RMS error relative to the experiment with error in the albedo is provided. Comparing the skill of the open loop with that obtained when LAI was uncertain, it is possible to 1.5 EKF T s T 2 STAEKF W g W Albedo x P a Albedo Figure 2. The same as in Figure 1, but for an error in the albedo. Copyright 212 Royal Meteorological Society Atmos. Sci. Let. 13: (212

5 272 A. Carrassi et al. deduce the strong impact of the albedo on the estimates of temperatures. This is also reflected by the performance of the EKF and STAEKF: the estimation errors are now larger for the temperatures and lower for the water contents, compared with those obtained in the previous case. A further indication of the different roles played by these two parameters in soil modeling is given by observing their relative performances. When LAI is uncertain the STAEKF improves over the EKF of approximately 14 and 9% in T s and T 2 but the improvement raises to 25 and 24% for w g and w 2 respectively. The situation is reversed when the albedo is uncertain: the improvement is approximately 27 and 24% in T s and T 2 in contrast to 14 and 9% for w g and w 2. In practice, reducing the parametric error leads to improving the estimate of those variables which are more sensitive to the uncertain parameters. In analogy to Figures 1 and 2, Figure 3 refers to experiments with error in RS min. In this case the overall performance of the STAEKF is less satisfactory. The estimation of RS min appears more difficult (see bottom-left panel and although a trend of error reduction is visible, it is now lower than in the two previous cases, with a parametric error reduction of about 4 and 35% in the LAI and albedo, respectively, and a reduction of 14% for RS min now. The error is also subject to significant fluctuations. As a consequence, the STAEKF is not able to reach the skill of the EKF, although it improves over the open loop (Table I. The different behavior of the STAEKF when dealing with different parameters was already observed in the study by Carrassi and Vannitsem (211a in the context of a low-order atmospheric model, and it was found connected to the sensitivity of the model dynamics to the uncertain parameters. In the STAEKF, the effect of parametric error is accounted for on the basis of a first-order approximation, a truncation which turns out to be poorly accurate in certain cases, such as the RS min here, affected by very large errors. Finally, Figure 4 describes the results relative to the experiments with the simultaneous presence of error in the LAI and the albedo. Note that now the bottom panels display the parametric error in LAI and albedo respectively. Although similar conclusions to the previous cases can be drawn, some differences are worth mentioning: the STAEKF shows now a higher skill along the entire duration of the experiment and the parametric error in LAI and albedo is successfully reduced. As it can be further deduced by looking at Table I, the simultaneous presence of errors in two parameters enhances the benefit of the parameter estimation feature of the STAEKF. Overall, a main concern revealed by these results is the slow rate of error convergence of the STAEKF. This rate is notably related to the initial parametric error variance, P a λ, which in turn affects the correlation between observables and parameters as well as the effect of parametric uncertainty on the state estimation error. After a sufficient long time the STAEKF (as all Kalman filter-type algorithms loses memory about the initialization, and the error covariances are modulated by the dynamics. In this study, P a λ has been chosen as proportional to the actual initial parametric error variance; the value used in the experiment is then the outcome of a short tuning. A more extensive tuning would have led to an acceleration of the rate.5 T s T.5 2 EKF STAEKF.1.5 W g W Minimum Stomatal Resistance (RSmin 5 P a RSmin [s/m] 25 2 [s 2 /m 2 ] Figure 3. The same as in Figure 1, but for an error in the RS min. Copyright 212 Royal Meteorological Society Atmos. Sci. Let. 13: (212

6 STAEKF for Soil Analysis 273 T s T 2.8 EKF STAEKF W g.15 W [m 2 /m 2 ].32 Leaf Area Index Albedo Figure 4. The same as in Figure 1, but for error in both the LAI and the albedo. of error convergence, but is out of the scope of this study. 6. Conclusions This paper describes a first application of the STAEKF to soil analysis, for the assimilation of screen-level observations into an off-line version of the land surface model, ISBA. A twin experiment approach has been adopted for the summers for the decade Model error is simulated by perturbing either independently or simultaneously the LAI, the albedo, and the minimum stomatal resistance, three key parameters in soil modeling, usually affected by significant errors. The effect of model error is implicitly accounted for in the EKF forecast error covariance, while in the STAEKF the uncertain parameters are estimated at the analysis time along with the system s state. Other schemes such as an EKF including a bias correction scheme also exist in the literature (Dee and da Silva, Results indicate that the STAEKF is able to retrieve the true value of the uncertain parameter and progressively reduces the associated estimation error. The accuracy of these estimates is inherently related to the type of parameter to be estimated, given the model sensitivity to the specific parameter and the accuracy of the short-time approximation at the core of the STAEKF. These conditions have to be verified case by case. The present results suggest that the on-line estimation of LAI and/or the Albedo helps in improving the soil analysis skill to a larger extent than the estimate of RS min. The rate of error convergence in the STAEKF is related to the initial parametric error variance. Such a rate could be accelerated, to a certain extent, by an optimal tuning of the initial error covariance, a case-dependent procedure which has not been implemented in this feasibility study. By reducing the parametric error a better guess for the system state can be obtained and this in turn improves the analysis field and again the accuracy of the parameter estimate. Moreover, given that this feature is incorporated using the short-time formulation (Carrassi and Vannitsem, 211a, the additional computational cost with respect to the standard EKF is almost negligible. This proof-of-concept study is a first step in the validation of the STAEKF scheme and encourages its use in a more realistic model and observational scenario, possibly in the context of a full threedimensional NWP model. This research is currently on-going. Acknowledgements The authors are thankful to J.-F. Mahfouf (Météo-France for providing the SLDAS platform and to N. Ghilain for the useful discussion on the estimation of the leaf area index and the albedo. This work has been partially supported by the Belgian Federal Science Policy Program under contract MO/34/17 and MO/34/2. References Albergel C, Calvet J-C, Mahfouf J-F, Rudiger C, Barbu AL, Lafont S, Roujean J-L, Walker JP, Crapeau M, Wigneron J-P. 21. Copyright 212 Royal Meteorological Society Atmos. Sci. Let. 13: (212

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