Efficient implementation of inverse variance-covariance matrices in variational data assimilation systems

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1 ERAD THE SENVENTH EUROPEAN CONFERENCE ON RADAR IN METEOROLOGY AND HYDROLOGY Efficient implementation of inverse variance-covariance matrices in variational data assimilation systems Dominik Jacques, Isztar Zawadzki J. S. Marshall Radar Observatory, McGill University 805 Sherbrooke St. W., Montreal, Quebec, Canada, H3A 2K6 dominik.jacques@mail.mcgill.ca May 17, Introduction We are investigating the correlation of model and observation errors in the context of radar data assimilation at the mesoscale. When radar measurements are performed, various factors such as beam averaging and varying drop size distributions will lead to errors that are correlated in time and space. In the framework of data assimilation, error correlations are represented by the non-diagonal terms in the variance-covariance (vc) matrices appearing in the cost function. In this research, we are demonstrating a method to efficiently consider non-diagonal inverse vc matrices. In the most standard form of a data-assimilation cost function, J(x) = [x x b ] T B 1 [x x b ] + [h(x) y] T R 1 [h(x) y] (1) we require the final analysis x to be a compromise between the background term from the model x b and the observations y. The operator mapping model variables to observation is symbolized by h( ). B and R are the vc matrices representing model and observation errors respectively. The relative magnitude of elements composing these matrices will determine which of the model or observations will have the most impact on the analysis. Off diagonal elements of these matrices represent the covariance of errors between neighboring data points. Historically, error correlations have been neglected (Sun and Crook 1997; Xu et al. 1994, among others) and vc matrices were assumed diagonal. This simplification was made for two main reasons. First, the true structure of error correlations is unknown. Second, the numerical cost of using non-diagonal vc matrices was (and still is) prohibitive. One of the most tangible consequence of considering error correlations is the generation of smooth analyses. In atmospheric retrievals, where number of parameters to estimate is very large, smoothness is a severe constraint that greatly helps convergence to a plausible solution. It is interesting to observe that the developers of the earliest retrieval systems (where error correlation were neglected) acknowledged the need smooth analyses. They achieved this by including smoothness constraints to their cost functions, an empirical method to simulate the use of an isotropic and homogeneous background vc matrix. In more recent years, the recursive filters (Purser et al. 2003) have been proposed to perform the same task. This method allows an efficient implementation of background error correlations that is consistent with the cost function formulation. In this aspect, recursive filters improved over smoothness constraints. In its function, the exponential correlations we are proposing are comparable to the recursive filters. which are presented in section 3 of the present document. The demonstration of exponential covariances is given in section 4. Assimilation metrics demonstrating the usefulness of cyclical insertion of observations are then presented in sections 5. Before we explain different methods for representing non-diagonal terms in assimilation, it is worthwhile to demonstrate the comparative effect of multiplying a vector by a vc matrix and its inverse. This will give a more intuitive notion of the role of these matrices in data assimilation. 2 Multiplication of a vector by a variance-covariance matrix For this example, we let z be a 1D error vector. In this section, we illustrate the multiplication of z by an vc matrix and its inverse as it is the case in eq. 1. Lets consider the correlation between elements of the error vector z. It seems reasonable to assume that the closer two data points are, the more correlated their error should be. This assumption is the basis for modeling the correlation of errors by functions decreasing with increasing distance between points. We here define δ ij as the spatial distance

2 separating any two points with index i and j in z. Gaussian and exponential functions are common models for the correlation or errors. These functions can be used to construct a generic vc matrix L using the following formulae Gaussian ) L ij = exp ( (δij)2 σ 2 Exponential ( L ij = exp δij σ ) (2) where the parameter σ controls the dispersion (the spread) of both the Gaussian and the exponential correlation function. Figure 1 illustrates the effect of multiplying a correlated error vector z by the corresponding inverse vc matrix L 1. We can observe that L 1 transforms a correlated vector into a uncorrelated one. As such the inverse of a vc matrix can be thought of as a roughing operator. Conversely, an uncorrelated vector will be smoothed when multiplied by L. Counter intuitively, it is the inclusion of L 1, a roughing operator, in the cost function that imposes a smooth analysis field. When we are using the inverse of exponential vc matrices, we are parametrizing this roughing operator. On the other hand, the recursive filters emulate the multiplication of a vector by a vc matrix itself. This explains the necessity to rewrite eq. 1 in order for it to be a function of The vc matrix B rather than its inverse B 1. It is worthwhile to write down the key steps of this procedure as this will highlight some of the limitations of this technique. Figure 1: Applying a variance-covariance matrix L on a vector y consisting of 100 independent samples from a Normal distribution produces the correlated field z. The original vector y can be recovered by applying the inverse of the covariance matrix L 1 on the correlated field z. 3 Simulating background error correlation: The recursive filter The Recursive filter is a numerically efficient method for simulating the convolution with a Gaussian kernel. For example, to simulate Ly in the previous figure, information from neighboring points would be recursively mixed from left to right y i = αy i 1 + (1 α)y i for i = 1,..., n (3) and then from right to left. y i = αy i+1 + (1 α)y i for i = n,..., 1 (4) The parameter α determines in which proportion the information from two neighboring points will be mixed. Purser et al. (2003), describe the relation between α and the dispersion σ of a Gaussian function. Several passes (the successive application of eq. 3 and 4) of the recursive filter converge to a convolution with a Gaussian kernel. In order to take advantage of the recursive filters, the cost function must then be rewritten so that the inverse of the background vc matrix is no more required. Only the main steps are reproduced here. Purser et al. (2003) and Lewis et al. (2066) can be consulted for full derivation of the cost function in this form. First, the analysis vector x is defined as an increment δx to the background x b x = x b + δx (5) and the observation operator h( ) is replaced by a linearization of itself H about the background. h(x b + δx) h(x b ) + Hδx. (6) We also define a variable d representing the distance between the observations y and the background x b. d = y h(x b ) (7)

3 The explicit computation of B 1 is avoided through a variable change. with δx = Cχ (8) B = CC T (9) By including eqns. 5-9 into eq. 1 we obtain the cost function in its incremental and preconditioned form. J(χ) = χ T χ + [HCχ d] T R 1 [HCχ d] (10) In this form, the computation of B 1 becomes unnecessary and the recursive filters are used to efficiently emulate multiplication by the matrix C. Using the analogy between matrix multiplication and convolutions, Oliver (1995) shows that square root of a Gaussian convolution kernel is also Gaussian. Using his results, we find that C is a Gaussian with a spread σ C = σ B / 2. The fact that the square root of a Gaussian B is also a Gaussian make possible the use of recursive filter to represent a Gaussian covariance matrix B. Analytical consistence throughout the derivations gives an advantage to the recursive filters with respect to the smoothness constraint. Additionally, the smoothing effect of including background error correlations is very intuitive. However, this formulation does not allow to take observation errors R 1 into account. Also, the recursive filter is an intrinsically serial algorithm that would be very difficult to implement in parallel to speed-up computations on large domains. We are proposing the use of exponential covariances as an answers to these limitations. 4 Exponential covariances and the direct application of inverse vc matrices The following clues allow us to conjecture that atmospheric errors errors may decay exponentially rather than following a Gaussian function. 1.0 Correlation LWC(DSD) - LWC(Reflectivity) dbz dbz time [min] Figure 2: The temporal auto-correlation of errors made by estimating Liquid Water Content (LWC) through a single Z-R relationship during a stratiform precipitation event. These errors were computed using disdrometric measurements allowing estimation of reflectivity and true LWC (Zawadzki et al in this conference). Dash lines are exponential functions fitted to observations. First is an argument based on the general structure of atmospheric fields. Energy cascade leads to atmospheric fields with auto-correlations decaying exponentially in space and time. This was demonstrated for dynamical fields in general (Kolmogorov 1941) and precipitation fields (Fabry 1996) in particular. One can suppose that the exponential decay is also a valid model for the observation errors of such fields. Another clue comes from looking at maps of differences between radar observations and model output at the continental scales. These differences combine both model and observation errors and cannot easily be interpreted. However, these fields display exponential correlation. Could both these errors decay exponentially? Yet another argument for exponential covariances, comes from a study on the natural variability of drop size distributions (DSD) by Zawadzki et al. (2012 see presentation in this conference). In fig. 2, we show that the error induced by

4 choosing a single Z-R relationship in a environment where DSD s are naturally variable lead to exponentially correlated errors in time. If we assume that fields are stationary over such timescales, spatial errors would also by exponentially correlated. The true error structure of both the background and the observations are flow dependent and vary in time and space. We here make the assumption that representing these errors with a isotropic and homogeneous exponential decay is a good first-order approximation. 4.1 Parametrizing R 1 for exponential correlations The inverse of a vc matrix built using exponentially decaying covariances (eq. 2) is very sparse. This is illustrated in fig. 3. In the case of a 1D field, R 1 is tridiagonal. In 2D the inverse is slightly more complicated but nevertheless mostly empty. This holds true for 3D fields. 1D field 2D field Figure 3: Exponential variance-covariance matrix Land its inverse L 1 for 1D and 2D error fields. Matrices are color coded with warmer colors indicating positive values and black indicating null entries. A cross-section of each matrix is plotted on the right hand side to illustrate the shape in the matrix. By taking advantage of the sparse nature of exponential inverse vc matrices, the numerical cost of using them in the cost function is no longer prohibitive. This avoids the need to rewrite the cost function in incremental form and allows to specify error correlations of both the background and observations. Here is how we parametrize R 1 by defining a stencil that can be applied to the desired error vector. First we choose the dispersion σ required to built R using eq. 2. The dispersion should be set according to experiments such as depicted in fig. 2. Int the experiments presented here, σ = 3km was chosen in every direction. We then build a L for a small domain with the same grid spacing as the assimilation domain. The small domain make numerical inversion of L possible. Note that inversion needs to be performed only once as the significant entries of L 1 are saved in a lookup table. 5 Results and Discussion Inverse exponential covariance functions have been implemented in McGill s assimilation system. The main conclusions of this inclusion are briefly discussed here. First, convergence to the final analysis is slower with the exponential covariance than with the recursive filters. Because of this exponential covariances are numerically more expensive. This extra cost might however be beneficial. MAS uses model equations residuals as weak constraints to complement ambiguous radar measurements, (Chung et al. 2009). It turns out that these equations can be better minimized using exponential covariance than with the recursive filters and the smoothness constraints. Second, no mater the method for introducing covariances in assimilation, assimilation of radar data will fail if the background instability cannot lead to a thunderstorm. This turns out to be the greatest of all problems for mesoscale assimilation. Unless the background s instability is sufficient for a thunderstorm to be triggered at the location of observations, the assimilation will fail to seed a thunderstorm in the model. For this purpose, we use a crude version of latent heat nudging.

5 This consists in increasing the moisture content of the atmosphere where precipitation is observed at the beginning of the assimilation. This method works well. So well in fact that one could wonder why we should perform an expensive assimilation at all. The following figure is included to demonstrate that assimilation leads to better analyses than simple latent heat nudging. The two top panels of this figure depict the observational terms of the cost function as a function of time. We can observe that errors with respect to Doppler velocity and Reflectivity observations are lower when the assimilation is performed. The discontinuities in these lines occurs every five minutes when assimilation is performed. The bottom two panels show the magnitude of two model residual terms. If both latent heat nudging and full assimilation start out with similar values, full assimilation achieves lower values towards the end of the run. This indicates that the assimilation succeeds in seeding a thunderstorm with minimal generation of spurious gravity waves. Full assimilation Convective buble Doppler velocity errors Rain water errors u-momentum eqn residuals cloud water equation residuals :04 16:08 16:13 16:17 16:22 16:26 16:31 16:35 16:40 time Figure 4: A full assimilation run using exponential covariances is compared to a convective bubble seeded using latent heat nudging. The assimilation run displays less errors with respect to observations (two top panels) and lower model equation residuals (two bottom panels). Lower equation residuals imply the generation of a lower level of (sometimes) spurious gravity waves in the vicinity of the thunderstorm. 6 Conclusion Visual inspection of atmospheric analyses and cost function terms such as fig. 4 demonstrate the direct inclusion of inverse exponential vc matrices as a valid option for including error covariance in assimilation systems. The idea of using the simple structure of inverse exponential vc matrices has been suggested in various sources (Oliver 1998; Sun and Crook 2001; Tarantola 2005; Stewart 2009) but it is often overlooked in assimilation systems. We here promote its use since this method has two main advantages over the recursive filters. First, it is much simpler to implement. Second, it can be applied to both the background (B 1 ) and observation (R 1 ) errors. Third, this method is better suited for parallel implementation. Successful assimilation requires specification of moisture fields not constrained by radar observations. This is where will be directed our future work. References Chung, K.-S., I. Zawadzki, M. K. Yau, and L. Fillion, 2009: Short-term forecasting of a midlatitude convective storm by the assimilation of single doppler radar observations. Monthly Weather Review. Fabry, F., 1996: On the determination of scales ranges for precipitation fields. Journal of Geophysical research, 101,

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