Advances in Meteorology

Volume 2015 (2015), Article ID 530764, 16 pages

http://dx.doi.org/10.1155/2015/530764

## A Study of Coupling Parameter Estimation Implemented by 4D-Var and EnKF with a Simple Coupled System

^{1}Key Laboratory of Marine Environmental Information Technology, State Oceanic Administration, National Marine Data and Information Service, Tianjin 300171, China^{2}Geophysical Fluid Dynamics Laboratory, National Oceanic and Atmospheric Administration, Princeton University, Princeton, NJ 08542, USA^{3}Center for Climate Research and Department of Atmospheric and Oceanic Sciences, University of Wisconsin-Madison, Madison, WI 53706, USA^{4}Laboratory of Ocean-Atmosphere Studies, Peking University, Beijing 100871, China^{5}Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA

Received 24 October 2014; Revised 5 January 2015; Accepted 5 January 2015

Academic Editor: Hiroyuki Hashiguchi

Copyright © 2015 Guijun Han 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

Coupling parameter estimation (CPE) that uses observations to estimate the parameters in a coupled model through error covariance between variables residing in different media may increase the consistency of estimated parameters in an air-sea coupled system. However, it is very challenging to accurately evaluate the error covariance between such variables due to the different characteristic time scales at which flows vary in different media. With a simple Lorenz-atmosphere and slab ocean coupled system that characterizes the interaction of two-timescale media in a coupled “climate” system, this study explores feasibility of the CPE with four-dimensional variational analysis and ensemble Kalman filter within a perfect observing system simulation experiment framework. It is found that both algorithms can improve the representation of air-sea coupling processes through CPE compared to state estimation only. These simple model studies provide some insights when parameter estimation is implemented with a coupled general circulation model for improving climate estimation and prediction initialization.

#### 1. Introduction

Due to its potential to reduce initial shocks between different media in a coupled climate system, coupled data assimilation (CDA) that uses coupled model dynamics to extract observational information in one or more media is emerging as an important topic in the climate community ([1–6] the related discussion at the Sixth World Meteorological Organization Data Assimilation Symposium (http://das6.cscamm.umd.edu/)). Based on Bayes’ rule, two main data assimilation algorithms, four-dimensional variational analysis (4D-Var) [7–9], and ensemble Kalman filter (EnKF) [10] have been used to develop CDA systems [3, 4]. Many efforts have been made to compare the performances of 4D-Var and EnKF either under CDA or not [11–16]. It has been found that 4D-Var and EnKF have a comparable performance when the former uses an appropriate minimization time window (MTW) and the latter adopts a suitable variance inflation scheme [17]. While 4D-Var requires a shorter spin-up time in the weather timescale (like days), EnKF can produce a better forecast skill at a long lead time [18–20] in the season timescale (like months).

To reduce model errors and improve coupled model predictability, coupled model parameter estimation (also referred to as parameter optimization in the literature) has been introduced into CDA [21–27]. However, coupling parameter estimation (CPE) that uses observations in one medium to estimate the model parameters in other media has not yet been studied in 4D-Var and EnKF. In a climate system that has multiple characteristic time scales, model parameters have remarkable impacts on all model variables. Thus, it is expected that CPE may further enhance the consistency of estimated parameters and model states in a coupled system. Although data assimilation in a multiple space and time scale system has been explored [2, 28], the CPE in such a system has not been fully investigated. Both 4D-Var and EnKF can be used to implement CPE. It is well known that although they originated from the same information estimation theory (Bayes’ rule), different numerical implementations make different performances of 4D-Var and EnKF data assimilation [17]. On one hand, by minimizing a cost function that measures the distance between observations and model states within a specific MTW, 4D-Var CPE seeks a posterior maximum likelihood solution of model parameters in different media in terms of the best fitting of modeling trajectory to observations. On the other hand, EnKF CPE uses flow-dependent coupling error covariance (i.e., error covariance between a model variable in the observational medium and a parameter in another medium) to project observational information onto the parameter being estimated, thereby implementing CPE in a sequential manner.

Since evaluating the error covariance of variables residing in different media with a finite ensemble is difficult [6] and a specific MTW is difficult to capture multiple time scales in minimization, CPE is therefore challenging in both EnKF and 4D-Var. A fundamental issue is can 4D-Var or EnKF CPE improve representation of the model air-sea coupling process? Based on a conceptual coupled model [25] which couples a chaotic atmosphere [29] with a slab ocean, we set up an observing system simulation experiment (OSSE) [30] which takes a Nature Run as the truth of model states to answer this question. Within this framework, in both 4D-Var CPE and EnKF CPE, “observations” drawn from a “truth” model that uses the default parameter values are assimilated into the assimilation model which uses erroneously set parameter values for optimizing the coupling parameters.

The paper is organized as follows. Section 2 describes the methodology, starting from introducing the simple coupled model, followed by the OSSE setup as well as implementation of the 4D-Var CPE and EnKF CPE. This section also presents the scale analysis of perturbed terms in the model equations and the sensitivity study of model parameters. Sections 3 and 4 examine the results of 4D-Var CPE and EnKF CPE, respectively. Different performances of EnKF CPE and 4D-Var CPE are investigated in Section 5. Summary and discussions are given in Section 6.

#### 2. Methodology

##### 2.1. The Simple Coupled Model

To study coupling parameter estimation (CPE) implemented by 4D-Var and EnKF so as to answer the question (i.e., can 4D-Var or EnKF CPE improve representation of the model air-sea coupling process?) posed in Section 1, we employ a simple “climate” model [25] which consists of a chaotic atmosphere model [29] coupled to a slab ocean model to simulate the interaction of the fast atmosphere and slow ocean:where an overdot denotes time tendency; , , and (hereafter denoted by if they are presented together) are the high-frequency variables of the atmosphere and represents the slab ocean. (, ) define the oceanic time scale, where is the ratio of heat capacity between the slab ocean and the chaotic atmosphere while is the damping coefficient of the temperature (°C) of the slab ocean. is a dimensionless parameter. The unit of is , where represents the dimensionless time step. The “atmosphere” and the “ocean” interact with each other through the coupling coefficients and . Here, the unit of is while is a dimensionless parameter. The external solar forcing is represented by , where represents the annual mean temperature (°C) of the slab ocean and (in °C) and (in ) represent the amplitude and period of the model seasonal cycle.

While the detailed description of the model construction can be found in [25], here, we only comment on the setting of model parameter values. The “atmospheric” parameters , , and take their standard values of 9.95, 29, and 8/3. For maintaining the chaotic nature of the “atmosphere” and the stability of the system, the values of and are chosen as 0.1 and 1. and are set to 10 and 1, which defines the oceanic time scale as 10 times of the atmospheric time scale. The parameters and are set as 10°C and 1°C, respectively. The is chosen as 10 Time Units (TU, 1 TU = 100 time steps = 100) so that the period of the forcing is comparable with the oceanic time scale, defining the time scale of the model seasonal cycle. Given the value of , the model calendar year is defined as 10 TUs. If we assume that one year has 360 model days, is equivalent to 0.36 model days.

Using a leap-frog time stepping scheme with a Robert-Asselin time filter [31, 32] and starting from the initial conditions (note that since the leap-frog time stepping scheme is applied, initial conditions at both time 0 and time 1 are set to .), the model is first freely run for 10^{4} TUs by setting the parameters (, , , , , , , , , , and ), where is the time filtering coefficient, to be the default values as prescribed in Table 1.