Advances in Meteorology

Volume 2016 (2016), Article ID 4305204, 11 pages

http://dx.doi.org/10.1155/2016/4305204

## Development of the Nonstationary Incremental Analysis Update Algorithm for Sequential Data Assimilation System

^{1}Faculty of Earth Systems and Environmental Sciences, Chonnam National University, Gwangju, Republic of Korea^{2}Korea Institute of Atmospheric Prediction Systems, Seoul, Republic of Korea^{3}Seoul National University, Seoul, Republic of Korea

Received 21 March 2016; Revised 22 August 2016; Accepted 10 October 2016

Academic Editor: Takashi Mochizuki

Copyright © 2016 Yoo-Geun Ham et al. 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

This study introduces a modified version of the incremental analysis updates (IAU), called the nonstationary IAU (NIAU) method, to improve the assimilation accuracy of the IAU while keeping the continuity of the analysis. Similar to the IAU, the NIAU is designed to add analysis increments at every model time step to improve the continuity in the intermittent data assimilation. However, unlike the IAU, the NIAU procedure uses time-evolved forcing using the forward operator as corrections to the model. The solution of the NIAU is superior to that of the forward IAU, of which analysis is performed at the beginning of the time window for adding the IAU forcing, in terms of the accuracy of the analysis field. It is because, in the linear systems, the NIAU solution equals that in an intermittent data assimilation method at the end of the assimilation interval. To have the filtering property in the NIAU, a forward operator to propagate the increment is reconstructed with only dominant singular vectors. An illustration of those advantages of the NIAU is given using the simple 40-variable Lorenz model.

#### 1. Introduction

Data assimilation is a mathematically rigorous procedure of combining forecast models with relatively sparse, discontinuous observations to obtain an optimal estimation of the underlying system state. The theory and practice of data assimilation are continuously evolving with several issues still needing to be fully addressed. One of these issues is related to the intermittence of certain assimilation schemes. The initialization problem relates to the discontinuities introduced in the model integration right after the model is restarted from the assimilated fields, which can destroy the intrinsic balances of the model and cause spurious high-frequency oscillations [1, 2].

To prevent these problems, Bloom et al. [2] introduced the incremental analysis updates (IAU). The IAU distributes the analysis increments over a fixed assimilation time window. It allows one to smoothly determine the influence of the observations without the use of an explicit initialization procedure, such as the use of nonlinear normal modes [3]. The IAU behaves like a low-pass filter and dampens high-frequency oscillations [4]. This property of filtering spurious oscillations is important in significantly improving operational forecasts, because the high-frequency waves dominate most of the initial forecast errors [5]. Therefore, the IAU method has been widely used in many atmospheric and oceanic data assimilation systems [6–8]. Even though most of the atmospheric assimilation system adopted original Bloom’s algorithm that the analysis is performed at the middle of the time windows for adding the IAU forcing, this type of the IAU (i.e., centered IAU) has a deficiency to be applied in the complex ocean-atmospheric coupled GCM as the computational time is increased 1.5 times than that in the intermittent method. That is, first half of the time window for adding the IAU forcing should be computed twice to calculate the observational increment and to add the increment. To avoid this time overlapping, several operational data assimilation products implemented the IAU that the analysis is performed at the beginning of the time window for the IAU (i.e., forward IAU) [6, 9, 10].

While there is an advantage in terms of this filtering property, the IAU produces assimilated fields that are less accurate than those of the intermittent assimilation procedure. It represents a suboptimal solution to the assimilation problem (as pointed out by [2]). For example, Ourmières et al. [11] have shown the root-mean-square (RMS) error using the assimilating fields updated by IAU is worse than that obtained by using an intermittent assimilation scheme.

To overcome the deficiency of the IAU, a weighting function, which is a function of time, is multiplied by the constant IAU forcing. For example, Bergemann and Reich [12] applied a hat function to decrease the IAU forcing as the difference between the increment calculation time and the increment-addition time increases. Similarly, an IAU with a weighting function based on a sinusoidal function is comparable to a digital filter that can filter out high-frequency noise without excessive dampening of the slower waves [4]. Even though there is improvement in applying weighting functions to IAU forcing, the selection of the weighting function is arbitrary and problematic, and the inconsistency between the timing for calculating the increment and that of adding the increment still exists.

With this issue in mind, this study introduces a modified version of the IAU, referred to as the nonstationary incremental analysis updates (NIAU) method as a potential alternative to the forward IAU. In NIAU, the forcing term is evolved over time using a forecasting model-derived formulation (see Section 2 for the details). Section 2 introduces the properties of the NIAU. Section 3 illustrates the experimental design, and Section 4 shows the performance of the NIAU in a Lorenz-96 model consisting of 40 variables [13]. Section 5 presents a brief summary.

#### 2. Description and Characteristics of the Nonstationary IAU (NIAU) Procedure

##### 2.1. The IAU and the Nonstationary IAU (NIAU)

The analysis equation for the intermittent assimilation has the form , where . The terms , , and are the analysis increment, the analysis, and the background fields, respectively; is the observation; and , , and are the background error covariance at time 0, the observational error covariance, and the observation operator, respectively.

The time differencing scheme to integrate the forecasting model is applied times between two consecutive analysis times. Using the observational increment at time 0, the IAU is applied between two analysis times (i.e., between and , where is the model time step). Note that this slightly differs from that introduced in Bloom et al. [2], which performs the analysis at the middle of the time windows to add the increment. This type of the IAU is utilized in several operational data assimilation products [6, 9, 10] to avoid an overlapping period before the analysis time as mentioned in Introduction.

Then, the IAU is in the form, where is the state-independent forcing of the IAU. The subscript is defined for times 0 to . The operator corresponds to the nonlinear dynamics integrating from time to time , and is the background vector at time . Equation (1) implies that constant forcing is added at every time step.

Alternatively, the proposed NIAU procedure obtains the state of time as where the matrix is the linear operator of the nonlinear model linearized about the state vectors (i.e., Tangent Linear Operator, TLM). The main difference between NIAU and IAU is that the NIAU forcing evolves over time by using the TLM per each time step.

The solution of the NIAU at the end of each assimilation cycle is expected to be quite similar to that of the intermittent method. In an intermittent method, the state at the end of the assimilation cycle is This can be approximated using a first-order approximation as follows:where is the TLM linearized around and . Under the perfect model assumption, is the optimal solution with all observations before the th time step taken into account.

In the NIAU case where the analysis state is denoted with the hat function, the state at the end of the assimilation cycle is obtained from the iterative procedure where for . Hence, the NIAU value at the end of the assimilation cycle is nearly equal toComparing this equation with (4), the NIAU state propagates in a similar way as the first-order approximation of the propagation of the intermittent analysis.

Analogously, the first two iterations of IAU in which the analysis state is denoted with a tilde give where . At the end of the assimilation cycle, the analyzed state is of the formand therefore it is quite different from the solution of the intermittent assimilation.

Figure 1 is a schematic diagram that shows that the solution of the NIAU is equivalent to that of the intermittent method at the end of the assimilation cycle. In the schematics, a simple linear system is assumed that the values are doubled at each time step. Because the IAU forcing is constant, the analysis value of the IAU is far from that of the intermittent method. Conversely, the NIAU forcing is evolved to the same degree as the state vector; therefore, the solution of the NIAU is equivalent to that of the intermittent method at the end of the assimilation cycle.