Advances in Meteorology

Volume 2016 (2016), Article ID 2170746, 10 pages

http://dx.doi.org/10.1155/2016/2170746

## Reconstruction of Typhoon Structure Using 3-Dimensional Doppler Radar Radial Velocity Data with the Multigrid Analysis: A Case Study in an Idealized Simulation Context

^{1}Key Laboratory of State Oceanic Administration for Marine Environmental Information Technology, National Marine Data and Information Service, State Oceanic Administration, Tianjin 300171, China^{2}NOAA Earth System Research Laboratory, Boulder, CO 80305-3328, USA^{3}NOAA Geophysical Fluid Dynamics Laboratory, Princeton University, Princeton, NJ 08540-6649, USA

Received 25 March 2016; Revised 2 June 2016; Accepted 6 June 2016

Academic Editor: Mario M. Miglietta

Copyright © 2016 Hongli Fu 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

Extracting multiple-scale observational information is critical for accurately reconstructing the structure of mesoscale circulation systems such as typhoon. The Space and Time Mesoscale Analysis System (STMAS) with multigrid data assimilation developed in Earth System Research Laboratory (ESRL) in National Oceanic and Atmospheric Administration (NOAA) has addressed this issue. Previous studies have shown the capability of STMAS to retrieve multiscale information in 2-dimensional Doppler radar radial velocity observations. This study explores the application of 3-dimensional (3D) Doppler radar radial velocities with STMAS for reconstructing a 3D typhoon structure. As for the first step, here, we use an idealized simulation framework. A two-scale simulated “typhoon” field is constructed and referred to as “truth,” from which randomly distributed conventional wind data and 3D Doppler radar radial wind data are generated. These data are used to reconstruct the synthetic 3D “typhoon” structure by the STMAS and the traditional 3D variational (3D-Var) analysis. The degree by which the “truth” 3D typhoon structure is recovered is an assessment of the impact of the data type or analysis scheme being evaluated. We also examine the effects of weak constraint and strong constraint on STMAS analyses. Results show that while the STMAS is superior to the traditional 3D-Var for reconstructing the 3D typhoon structure, the strong constraint STMAS can produce better analyses on both horizontal and vertical velocities.

#### 1. Introduction

Doppler radar has long been a valuable observational tool in meteorology. Three-dimensional (3D) Doppler radar radial velocity data can provide an opportunity to estimate both horizontal and vertical velocities. Therefore, in recent years, Doppler radar data assimilation for short-term numerical weather forecasting or called nowcasting has become a focal point of research [1–4]. Lots of techniques have been developed to retrieve wind field from Doppler radar radial velocity observations [2–24].

In Doppler radar radial velocity data assimilation used in the above literatures, in a three-dimensional variational (3D-Var) framework, a background error covariance matrix is always needed to determine the spatial spreading of observational information. It is well known that an analysis field at different locations may have different correlation scales [25], which are difficult to be estimated. Unfortunately, the traditional 3D-Var always employs an empirical and static background error covariance matrix and therefore usually can only correct single-scale wavelength error. However, the errors in short wavelength scales cannot be sufficiently corrected until the long waves are corrected [25, 26].

To minimize the errors of long and short waves in turn, a sequential 3D-Var approach has been proposed by Xie et al. [25, 26], implemented by either a recursive filter [27] or a multigrid technique [28] at Global Systems Division (GSD) of Earth System Research Laboratory (ESRL) in National Oceanic and Atmospheric Administration (NOAA) for a Federal Aviation Agency (FAA) project joined by the research team from the Lincoln Laboratory in Massachusetts Institute of Technology (MIT). Since this system also uses the temporal observation information, it is called a Space and Time Mesoscale Analysis System (STMAS, thereafter; see Xie et al. [26]). The STMAS has been applied to assimilating 2-dimensional (2D) Doppler radar radial velocity data to improve the wind field analyses [29].

Here, we study the analysis of 3D Doppler radar radial velocities using the STMAS to reconstruct the 3D wind structure. As for the first step, this study is performed in a twin experiment framework. In the next section, we first briefly review the theory of the multigrid 3D-Var data assimilation scheme in the STMAS. Some important aspects of the STMAS techniques such as smoothing, constraint, and Doppler radar radial wind operators used in the cost function of the STMAS multigrid 3D-Var are described. Section 3 first introduces the twin experiment framework for 3D Doppler radar radial wind data assimilation with the STMAS and then gives the evaluation by comparing it to the traditional 3D-Var. Section 3 also examines the performance of the STMAS in weak and strong constraints for 3D Doppler radar radial velocity analysis. Conclusions and discussions are given in Section 4.

#### 2. Smoothing, Constraint, and Radar Radial Wind Operators in STMAS

In this study, the STMAS implemented by the multigrid 3D-Var is applied to the analysis of 3D Doppler radar radial velocities. This method can extract long and short wavelength information in turn efficiently from observations and provide objective and accurate analysis. The basic idea of this multigrid implementation can be referred to Li et al. [28–30].

To assimilate 3D Doppler radar radial velocities, with the control variables being , where and represent zonal and meridional components of wind vector, the cost functional for the th level grid iswhere the subscript denotes the background term and the smooth term, the conventional observation data term, and the radar radial wind observation data term. The smooth matrixes and in the smooth term are derived from the Laplacian of control variables and , respectively, at grid points. Let represent the vertical component of wind vector. The details of the conventional observation data term and the 3D Doppler radar radial wind observation data term are as follows:where is the amount of radar radial wind observations, is the azimuth angle of the radar beam relative to north with positive clockwise, and is elevation angle of the radar beam. Of course, since radar scans at nonzero elevation angles, the fall speed of precipitation particles should be taken into account, and the algorithm of Sun and Crook [6] can be used to calculate terminal velocity. But for this study, we just neglect this terminal velocity, which does not lose its generality. The matrix is an error covariance matrix for radar radial wind observation; its superscript stands for the reverse matrix, and its subscript represents the radar radial wind observation.

During the procedure of sequential multiscale analyses, the operators , , , and remain the same when the full observation dataset is used through all multigrid levels; therefore, the superscript is omitted from these operators.

To make a strong constraint on these three components of wind vector, incompressible continuity equation is employed and discretized to calculate vertical velocity from the other two horizontal components. The discretized incompressible continuity equation is as follows:The adjoint codes are recursively developed for represented by and .

#### 3. Simulation Methodology

##### 3.1. Synthetic Typhoon Structure

The study domain covers a square region with 10 km thickness. The Doppler radar locates at the center (250 km, 250 km) of the study domain. A simulated typhoon field can be constructed by using the following function which consists of two subsection functions:This formula allows that the 2-order derivatives of this function exist. Let and , where km^{2}* *s^{−1}, km, km, km^{2}* *s^{−1}, km, and km; then a stream function of a two-scale typhoon field can be constructed. The amplitude of one large-scale ( km) is scaled by and the amplitude of one small-scale ( km) is scaled by . The locations of the maximum wind horizontal velocity of these two scales are km for large-scale and km for small-scale, respectively, from the typhoon center.

The typhoon center is set at 300 km and 150 km. Then, the stream function and velocity potential function can be constructed as follows:where , km, and km. The horizontal components can be expressed in terms of and :In this study, incompressibility is assumed. And the true vertical velocity field can be obtained by integrating the continuity equation . That is,where the bottom boundary condition is . The wind speed field in this simulated typhoon field contains two different scale information. The first one is about 35 km and the other is about 10 km. The radial wind, component, and component pattern of middle level of this simulated typhoon field and a section wind field across the center of this typhoon are shown in Figure 1. This typhoon pattern is located at the southeast part of the study domain, so only the southeast square part is shown for the detailed structure. This simulated typhoon wind field is referred to as the “truth” typhoon field in this twin experiment.