Advances in Meteorology

Volume 2018 (2018), Article ID 7931964, 17 pages

https://doi.org/10.1155/2018/7931964

## Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale and Multistep Variational Data Assimilation

^{1}NOAA/National Severe Storms Laboratory, Norman, OK, USA^{2}Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, OK, USA

Correspondence should be addressed to Qin Xu; vog.aaon@ux.niq

Received 24 April 2017; Revised 24 November 2017; Accepted 3 December 2017; Published 7 February 2018

Academic Editor: Shaoqing Zhang

Copyright © 2018 Qin Xu and Li Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

When the coarse-resolution observations used in the first step of multiscale and multistep variational data assimilation become increasingly nonuniform and/or sparse, the error variance of the first-step analysis tends to have increasingly large spatial variations. However, the analysis error variance computed from the previously developed spectral formulations is constant and thus limited to represent only the spatially averaged error variance. To overcome this limitation, analytic formulations are constructed to efficiently estimate the spatial variation of analysis error variance and associated spatial variation in analysis error covariance. First, a suite of formulations is constructed to efficiently estimate the error variance reduction produced by analyzing the coarse-resolution observations in one- and two-dimensional spaces with increased complexity and generality (from uniformly distributed observations with periodic extension to nonuniformly distributed observations without periodic extension). Then, three different formulations are constructed for using the estimated analysis error variance to modify the analysis error covariance computed from the spectral formulations. The successively improved accuracies of these three formulations and their increasingly positive impacts on the two-step variational analysis (or multistep variational analysis in first two steps) are demonstrated by idealized experiments.

#### 1. Introduction

Multiple Gaussians with different decorrelation length scales have been used at NCEP to model the background error covariance in variational data assimilation (Wu et al. [1], Purser et al. [2]), but mesoscale features are still poorly resolved in the analyzed incremental fields even in areas covered by remotely sensed high-resolution observations, such as those from operational weather radars (Liu et al. [3]). This problem is common for the widely adopted single-step approach in operational variational data assimilation, especially when patchy high-resolution observations, such as those remotely sensed from radars and satellites, are assimilated together with coarse-resolution observations into a high-resolution model. To solve this problem, multiscale and multistep approaches were explored and proposed by several authors (Xie et al. [4], Gao et al. [5], Li et al. [6], and Xu et al. [7, 8]). For a two-step approach (or the first two steps of a multistep approach) in which broadly distributed coarse-resolution observations are analyzed first and then locally distributed high-resolution observations are analyzed in the second step, an important issue is how to objectively estimate or efficiently compute the analysis error covariance for the analyzed field that is obtained in the first step and used to update the background field in the second step. To address this issue, spectral formulations were derived by Xu et al. [8] for estimating the analysis error covariance. As shown in Xu et al. [8], the analysis error covariance can be computed very efficiently from the spectral formulations with very (or fairly) good approximations for uniformly (or nonuniformly) distributed coarse-resolution observations and, by using the approximately computed analysis error covariance, the two-step analysis can outperform the single-step analysis under the same computational constraint (that mimics the operational situation).

The analysis error covariance functions computed from the spectral formulations in Xu et al. [8] are spatially homogeneous, so their associated error variances are spatially constant. Although such a constant error variance can represent the spatial averaged value of the true analysis error variance, it cannot capture the spatial variation in the true analysis error variance. The true analysis error variance can have significant spatial variations, especially when the coarse-resolution observations become increasingly nonuniform and/or sparse. In this case, the spatial variation of analysis error variance and associated spatial variation in analysis error covariance need to be estimated based on the spatial distribution of the coarse-resolution observations in order to further improve the two-step analysis. This paper aims to explore and address this issue beyond the preliminary study reported in Xu and Wei [9]. In particular, as will be shown in this paper, analytic formulations for efficiently estimating the spatial variation of analysis error variance can be constructed by properly combining the error variance reduction produced by analyzing each and every coarse-resolution observation as a single observation, and the estimated analysis error variance can be used to further estimate the related variation in analysis error covariance. The detailed formulations are presented for one-dimensional cases first in the next section and then extended to two-dimensional cases in Section 3. Idealized numerical experiments are performed for one-dimensional cases in Section 4 and for two-dimensional cases in Section 5 to show the effectiveness of these formulations for improving the two-step analysis. Conclusions follow in Section 6.

#### 2. Analysis Error Variance Formulations for One-Dimensional Cases

##### 2.1. Error Variance Reduction Produced by a Single Observation

When observations are optimally analyzed in terms of the Bayesian estimation (see chapter 7 of Jazwinski [10]), the background state vector** b** is updated to the analysis state vector** a** with the following analysis increment: and the background error covariance matrix** B** is updated to the analysis error covariance matrix** A **according towhere** R** is the observation error covariance matrix, is the innovation vector (observation minus background in the observation space), is the observation vector, denotes the observation operator, and is the linearized . For a single observation, say, at in the one-dimensional space of , the inverse matrix reduces to , so the th diagonal element of** A** in (1b) is simply given by where , (or ) is the background (or observation) error variance, is the background error correlation function, denotes the th point in the discretized analysis space , and is the number of grid points over the analysis domain. The length of the analysis domain is , where is the analysis grid spacing and is assumed to be much larger than the background error decorrelation length scale .

Note that is a continuous function of , so (2) can be written into also as a continuous function of , whereis the error variance reduction produced by analyzing a single observation at . The error variance reduction in (3) decreases rapidly as increases, and it becomes much smaller than it peak value of at as increases to . This implies that the error variance reduction produced by analyzing sparsely distributed coarse-resolution observations can be estimated by properly combining the error variance reduction computed by (3) for each coarse-resolution observation as a single observation. This idea is explored in the following three subsections for one-dimensional cases with successively increased complexity and generality: from uniformly distributed coarse-resolution observations with periodic extension to nonuniformly distributed coarse-resolution observations without periodic extension.

##### 2.2. Uniform Coarse-Resolution Observations with Periodic Extension

Consider that there are coarse-resolution observations uniformly distributed in the above analysis domain of length with periodic extension, so their resolution is . In this case, the error variance reduction produced by each observation can be considered as an additional reduction to the reduction produced by its neighboring observations, and this additional reduction is always smaller than the reduction produced by the same observation but treated as a single observation. This implies that the error variance reduction produced by analyzing the coarse-resolution observations, denoted by , is bounded above by ; that is, where denotes the summation over for the observations. The equality in (4) is for the limiting case of only. The inequality in (4) implies that the domain-averaged value of is larger than the true averaged reduction estimated by , where is the domain-averaged analysis error variance estimated by the spectral formulation for one-dimensional cases in Section 2.2 of Xu et al. [8].

The domain-averaged value of can be computed by where denotes the integration over the analysis domain, denotes the summation over for the grid points, and is used in the last step. By extending with the analysis domain periodically, can be also estimated analytically as follows:where denotes the integration over the infinite space of , = = = is used in the second to last step, and is used with in the last step. For the double-Gaussian form of used in (5) of Xu et al. [8], we have . The analytically derived value in (5b) is very close to (slightly larger than) the numerically computed value from (5a). With the domain-averaged value of adjusted from to , can be estimated byThe analysis error variance, , is then estimated by

As shown by the example in Figure 1 (in which = 110.4 km and = 10 so km is close to km), the estimated in (7) has nearly the same spatial variation as the benchmark that is computed precisely from (1b), although the amplitude of spatial variation of , defined by , is slightly smaller than that of the true , defined by . As shown in Figure 2, the amplitude of spatial variation of benchmark decreases rapidly to virtually zero and then exactly zero (or increases monotonically toward its asymptotic upper limit of m^{2} s^{−2}) as decreases to 0.5 and then to (or increases toward ∞), and this decrease (or increase) of the amplitude of spatial variation of with is closely captured by the amplitude of spatial variation of the estimated as a function of .