## 1. Introduction

Doppler radar can detect the wind field with high spatiotemporal resolution, but only along the radial direction. Armijo (1969) first demonstrated that it is mathematically feasible to use data collected by two or three Doppler radars to uniquely determine the wind velocity and terminal fall speed of the raindrops. Ray et al. (1975) provided the first three-dimensional wind fields in a tornadic storm using observations from two Doppler radars. Over the past few decades, numerous studies about dual-Doppler or multiple-Doppler wind synthesis techniques have been conducted, allowing the estimation of the error distribution of the derived wind field (Doviak et al. 1976), improved treatment of the top or bottom boundary conditions for the vertical velocity (O’Brien 1970), development of an iterative approach to find the solution for the wind field (Brandes 1977), implementation of variational adjustment (Ray et al. 1980; Ziegler et al. 1983), and so on. New analytical schemes using multiple-Doppler radar measurements have also been proposed by Scialom and Lemaitre (1990), Protat and Zawadzki (1999), and Shapiro and Mewes (1999), among others.

However, a Doppler weather radar is not able to directly observe the thermodynamic parameters such as temperature and pressure. These variables, especially their three-dimensional distribution, are useful for various applications, such as studies of the evolution and triggering mechanisms of mesoscale or storm-scale weather systems, and improvement of numerical model forecast skill through data assimilation. This has made the so-called thermodynamic retrieval technique, a procedure by which one can derive three-dimensional pressure and temperature fields from multiple-Doppler radars synthesized winds, an important research topic in the radar meteorology community. The method developed first by Gal-Chen (1978) was particularly applicable, because the boundary condition for solving a Poisson equation to obtain the pressure perturbation could naturally be determined by the wind fields synthesized by Doppler radars. This method was later widely adopted to study the structure of various weather systems such as a deep moisture convection (Hane et al. 1981), a dry boundary layer (Gal-Chen and Kropfli 1984), a tornadic thunderstorm (Hane and Ray 1985), convection embedded in a squall line (Lin et al. 1986), and a frontal rainband (Parsons et al. 1987). To investigate the scale interactions, Protat et al. (1998) also introduced a new procedure for recovering the thermodynamic fields from Doppler radar data.

However, it should be pointed out that the products retrieved using the aforementioned algorithms only represent the deviation of the pressure and temperature perturbations (defined as the difference with respect to a base state) from their horizontal average at each layer. Fortunately, this limitation does not affect the structure of the thermodynamic fields over a horizontal plane. On the other hand, a correct interpretation of the vertical distribution of the pressure and temperature perturbations is still possible if the weather system is well restricted inside the computational domain and the forcing is sufficiently insignificant along the boundaries (e.g., Brandes 1984). However, when this condition is not satisfied and the vertical structure of the thermodynamic field is needed, as pointed out by Gal-Chen (1978), it is necessary to have at least one independent field measurement of the temperature and pressure for each altitude, which may not always be available.

Roux (1985, 1988) proposed a new approach in which a unique solution of the pressure and temperature perturbations could be achieved up to a volume-wide constant. This constant can be deduced from one independent pressure and temperature measurement at a single point in the domain. Sun and Roux (1988) applied the results obtained by Roux (1985, 1988) to investigate the trailing anvil clouds of squall lines over a vertical cross section. Roux and Sun (1990) and Roux et al. (1993) made further improvements of the Roux (1988) scheme by including a thermodynamic equation along with the equations of motion in their retrieval scheme, so that the temperature gradient could be provided in any direction. Sun and Crook (1996) compared the 4D-Var adjoint technique and the traditional Gal-Chen (1978) scheme. The advantage of the 4D-Var formulation was that one could deduce the three-dimensional thermodynamic fields without extra and independent measurements.

However, all of the retrieval schemes mentioned above are only applicable to recovering the thermodynamic fields over a flat surface. It is known that complex terrain will significantly complicate the computation. For example, in Gal-Chen (1978), a Poisson equation is solved for each horizontal domain to obtain the pressure perturbation. If complex topography exists, the horizontal domain that intersects the mountains will contain hollows with irregular boundaries. Solving the Poission equation over such a domain becomes very complicated (Maury 2001). In recent years, the first author of this manuscript developed a new multiple-Doppler radar wind synthesis algorithm, named the Wind Synthesis System using Doppler Measurements (WISSDOM; see Liou et al. 2012 and Liou et al. 2014). By employing the so-called Immersed Boundary Method to compute the forcing exerted by the terrain on the fluid, WISSDOM was able to synthesize the three-dimensional wind fields over complex terrain using multiple-Doppler radar observations. To fully take advantage of WISSDOM, in this current work a new method, named Terrain-Permitting Thermodynamic Retrieval Scheme (TPTRS), is developed whereby one can use the wind fields synthesized by WISSDOM to derive the three-dimensional thermodynamic fields immediately over complex terrain.

The rest of the manuscript is organized as follows. Section 2 gives an introduction to WISSDOM and TPTRS. The performance of TPTRS is tested in section 3. An example of the application of this scheme to recover the thermodynamic fields for a real case is presented in section 4, followed by some conclusions in section 5.

## 2. Methodology

### a. WISSDOM: A multiple-Doppler radar three-dimensional wind synthesis method over complex terrain

In this study, an advanced algorithm (named WISSDOM) is employed to conduct wind field analysis. As introduced in Liou et al. (2012), the wind fields are solved by variationally minimizing a cost function, which includes a set of weak constraints representing the multiple-radar Doppler radial wind observations, anelastic continuity equation, vertical vorticity equation, background flow field, and spatial smoothness terms. The background wind field is utilized to fill in the data void region, and can be provided by the outputs from a mesoscale numerical model (Liou et al. 2014). Compared with other commonly used traditional approaches, this method has several advantages. For example, it is able to recover the wind field along the radar baseline (Liou and Chang 2009) and immediately above complex terrain (Liou et al. 2012). The latter is accomplished by employing the immersed boundary method (IBM, to be explained below) to handle the bottom boundary conditions, allowing topographic forcing on the fluid to be considered during the wind synthesis. Data from any number of radars can be easily merged (Liou et al. 2016). In addition, the vertical vorticity equation is implemented to constrain the retrieved three-dimensional winds, thus the production of a residual term when performing the vorticity budget analysis can be avoided (Liou et al. 2012). Protat and Zawadzki (2000) also pointed out that a wind field that satisfies the vertical vorticity equation, if applied to thermodynamic retrieval, could yield results with a higher accuracy. Applications of WISSDOM for studying the convection structure or orographic effects can be found in Liou et al. (2013), Lee et al. (2014), Chang et al. (2015), Liou et al. (2016), Lee et al. (2018), and Tsai et al. (2018).

Figure 1 illustrates the configuration of the grid system in IBM. The grid points below the terrain but nearest to the terrain surface are defined as the ghost cells (Tseng and Ferziger 2003). The winds at these so-called ghost cell points are obtained through an interpolation using data at neighboring grid points located in the flow region. The interpolation is designed so that certain boundary conditions for the flow fields can be satisfied along the terrain surface (Liou et al. 2012). As a result, when computing the gradients of the flow fields at these grid points located in the flow region and closest to the terrain surface (marked by a letter T and displayed by blue triangles in Fig. 1), the wind information needed at nearby ghost cell points would be available.

Given that most radar scans are made from a positive elevation angle and Earth’s surface is curved, the atmosphere at lower levels is usually undetectable by a radar. In this study WISSDOM is further improved by merging the wind fields observed by surface stations in order to obtain information about the flow fields near the ground, where the radar data are usually not available.

*A*can be horizontal wind components

*u*or

*υ*at a grid point,

*A*is adjusted, and the weighting coefficient

*α*

_{sfc}is given a value of 500. Note that

*J*

_{sfc}is applied only to the analysis level with available information provided by surface stations. This is accomplished by moving the surface station data vertically to the nearest analysis level using a method described later in this section.

*A*

_{stn}denotes the value of

*A*from surface station observation,

*N*is the total number of applicable surface stations,

*i*is the index of the station, and wt

_{i,new}is a normalized weighting coefficient explained in the following.

*x*,

*y*) and (

*x*

_{s},

*y*

_{s}) are the coordinates of a grid point and a surface station, respectively, and

*σ*is specified to be 5.0 km. A cutoff value of 0.5 is specified, meaning the parameter at a given grid point is adjusted by a surrounding surface station only when the weighting wt from this particular station is greater than 0.5. If a grid point is influenced by more than one nearby surface station, the weighting coefficients from each station need to be normalized by

*i*is the index of the station, and wt

_{i}(or wt

_{j}) and wt

_{i,new}are the weighting coefficients before and after the normalization, respectively.

*A*

_{bgrd}stands for the value of

*A*from the background field, which can be obtained from reanalysis data or the output of a mesoscale model.

*u*and

*υ*at the surface stations are shifted vertically to the nearest analysis level through an empirical power law as suggested by Peterson and Hennessey (1978):

_{stn}and wind

_{WISSDOM}represent the

*u*or

*υ*wind component observed by the station at the altitude

*H*

_{stn}, and the extrapolated results located at the WISSDOM analysis levels

*H*

_{WISSDOM}, respectively. According to Hsu et al. (1994) and Chen et al. (2016),

*P*is specified to be 0.143. Finally, for those grid points located between the lowest radar observational level and the surface, a penalty term, shown in (7), is implemented so as to adjust the wind fields toward the background values that are obtained through a linear interpolation using the wind information below and aloft:

*A*

_{r}and

*A*

_{s}are the

*u*or

*υ*winds at the grid points located on the lowest analysis level (

*Z*=

*Z*

_{r}) with available radar data and the surface level (

*Z*=

*Z*

_{s}), respectively, and

*A*

_{i}is the wind field located at these grid points in between (

*Z*=

*Z*

_{i}). The radar scans in Taiwan are often blocked by the Central Mountain Range (CMR), whose height is roughly 3.0 km. In addition, the applicable range of the linear interpolation should not be too long. Thus, (7) is utilized to fill in the low-level data gap only when the height of the available radar data

*Z*

_{r}is less than 3.0 km. The coefficient

*α*

_{middle}is given a value of 100.

Figure 2 shows the result from an idealized experiment in which the “true” wind field is from a numerical model simulation. It can be seen that by using the procedure introduced in this section, WISSDOM is able to use the surface station observations to provide a better description of the wind field in low levels where the radar observations are usually unavailable.

Since WISSDOM is able to provide three-dimensional wind fields over complex terrain, in the following section TPTRS is introduced so that one can directly use the WISSDOM-produced wind fields to recover the thermodynamic parameters over complex terrain.

### b. A Terrain-Permitting Thermodynamic Retrieval Scheme (TPTRS)

*u, υ*,

*w*) denote the Cartesian wind components,

*f*is the Coriolis parameter,

*g*stands for the gravity, and turb( ) represents a subgrid-scale turbulence parameterization operator, which are parameterized using a simple first-order closure scheme in this study. The mixing ratio of rainwater (

*q*

_{r}) and snow (

*q*

_{s}) can be estimated using the radar reflectivity data through empirical equations (e.g., Tong and Xue 2005). The variable

*π*stands for a normalized pressure called the Exner function, and is defined as follows:

*P*is the pressure,

*P*

_{0}equals 100 kPa,

*R*is the gas constant, and

*C*

_{p}refers to the specific heat capacity at a constant pressure. The virtual potential temperature (

*θ*

_{υ}) and virtual cloud potential temperature perturbation (

*θ*is potential temperature,

*q*

_{c}is the cloud water mixing ratio. The virtual cloud potential temperature perturbation

The values of *F*, *G*, and *H* can be obtained once the three-dimensional air motion is obtained by WISSDOM.

*S*stands for the total effect from the temporal variation, diffusion, and the source/sink of

*S*term is treated as a retrievable parameter in this study, and no additional parameterizations are applied.

The weighting coefficients *α*_{1}–*α*_{4} are used to balance the contribution from each term. They are determined following the principles discussed in Liou (2001).

As described in Liou (2001), in order to minimize the cost function *J* expressed in (15a) in a three-dimensional space, the quasi-Newtonian conjugate-gradient algorithm (Liu and Nocedal 1988) is employed. This method requires information about the cost function gradients with respect to the control variables (i.e., *π*′, *S*) at each grid point. To perform the minimization immediately above the terrain, Fig. 3 shows that for a given straight line connecting either the lateral or top/bottom boundaries of the analysis domain, it is divided by terrain into segments. The grid points along each segment are categorized into interior, starting, and ending points. A starting (ending) point is the grid point nearest to a mountain of its right (left) in the *x* and *y* directions. As for the vertical direction, the starting point is the first grid above the mountain, while the ending point is usually at the top of the analysis domain. The gradients in each category are derived separately.

*x*direction:

*x*direction.

*y*direction:

*y*direction.

*z*direction:

*z*direction, which are equivalent to the bottom (top) boundary points in the vertical direction.

It should be mentioned that according to the definition described in this section, when conducting thermodynamic retrieval, the grid points marked by “T” in Fig. 1 would be the ending points along the *x* direction, and starting points along the *z* direction. The retrieval is performed only at these grid points located in the flow region (marked by black and blue triangles in Fig. 1). The IBM is only utilized in WISSDOM to provide boundary conditions for wind fields. It is not applied in TPTRS.

Starting from an initial guess for the thermodynamic fields, along with the gradients computed using (16)–(19), an iterative process is performed to minimize the cost function shown in (15). Since (15) is formulated using data only at these grid points located in the flow region, the solution yielded by TPTRS is the thermodynamic field in a three-dimensional space with its bottom boundary following the surface of the terrain.

## 3. Thermodynamic retrieval results from OSSE tests

An observing system simulation experiment (OSSE) is first conducted to investigate the performance of this newly developed thermodynamic retrieval system. The “observations” are provided by a simulation conducted using the Weather Research and Forecasting (WRF) Model. The simulated three-dimensional wind components and rainwater mixing ratio are utilized to generate pseudo radar observational datasets.

*h*is the crest height, (

*x*

_{c},

*y*

_{c}) stands for the center of the mountain, and

*a*and

*b*are the mountain half-width along the east–west, north–south directions. Two mountains are placed inside the model domain centered at (23, 20 km), (47, 20 km), with the peak height set to be 1.5 and 2.5 km, respectively. The mountain half widths along all directions for both mountains are specified to be 5.0 km. To initiate the simulation, thermal bubbles are placed in the model domain by superimposing the perturbation (

*θ*′) to the potential temperature field using the following formula:

*x*

_{0},

*y*

_{0},

*z*

_{0}) represent the center of the perturbation. Note that (21) is applied only when Rad ≤ 1. The storm starts from two thermal bubbles centered at (12, 20 km), (38, 20 km), with the maximum magnitudes

*u*,

*υ*,

*w*) along with rainwater (

*q*

_{r}) are plugged into (8)–(10) to generate

*F*,

*G*, and

*H*, which are then used as inputs for (15). Equation (15) is variationally minimized to produce a set of optimally determined pressure and temperature fields (

*π*′,

Figure 4 shows the horizontal and vertical wind fields, the “true” and retrieved thermodynamic fields by TPTRS on a horizontal plane at *Z* = 1.0 km. At this height, as illustrated in Fig. 4a, an updraft and downdraft occur on the windward and leeside of both mountains, which are associated with an accumulation of cold and warm air, respectively, as shown in Fig. 4b. Negative pressure perturbations are found surrounding the mountains. Figure 4c shows that with correct wind information the retrieved thermodynamic fields are in good agreement with the true solutions not only qualitatively, but also quantitatively.

*Y*= 0 km, as well as their retrieved counterparts. It can be seen clearly from Fig. 5a, that the airflow is lifted by the mountains. Figure 5b shows that positive temperature perturbations, originating from the thermal bubbles, occupy the windward side of the mountains above the height

*Z*~1.0 km, and can also be found above the mountain peaks. They are associated with negative pressure perturbations. Along the top periphery of the warm regions, one can identify positive pressure perturbations. It is known that the buoyancy field (

*B*), which can be approximated by the temperature distribution, also contributes to the generation of the pressure perturbations. According to Markowski and Richardson (2011), we have

*B*is negative along the top boundary of the warm regions, thus one can use (22) to reasonably explain the existence of the positive pressure perturbations.

Figure 5c depicts the retrieved temperature and pressure perturbation fields over the mountainous area obtained using the wind information. The results indicate that the aforementioned thermodynamic features adjacent to the mountains are accurately retrieved by the proposed algorithm. Figure 6 displays the true and retrieved residual terms (*S*), as defined in (14). Similar to the experimental results reported in Liou (2001), the retrieved *S* term agrees well with its true counterpart. Nevertheless, the successful recovery of the residual term over terrain is particularly encouraging. It is pointed out in Liou (2001) that this variable represents the total effects of the local temporal change, turbulence, and source/sink to the temperature. Therefore, if further parameterization to separate each term’s contribution is performed, the retrieved *S* can be used as a guide to constrain the total amount.

A quantitative comparison is obtained by computing the spatial correlation coefficient (SCC) and root-mean-square errors (RMSEs) between the true and retrieved fields over the three-dimensional domain. The SCC values for the pressure and temperature are 0.96 and 0.94, while the RMSEs are 0.03 hPa and 0.06 K, respectively. These statistics indicate a successful recovery of the thermodynamic fields over terrain using wind information.

## 4. Thermodynamic retrieval results from a real case study

### a. 2008 SoWMEX field experiment and verification

The Southwestern Monsoon Experiment (SoWMEX) field experiment was conducted in Taiwan from May to June 2008. Its scientific goal was to explore the mechanisms leading to the heavy rainfall in Taiwan and the vicinity that occurs during the Asian summer monsoon season (Jou et al. 2011). It is expected that the knowledge obtained from this experiment can be applied to improve model forecasts of precipitation.

The case selected for this study took place during intensive observing period (IOP) 8 at 1500 UTC 14 June 2008. Figure 7 shows that on that day, a stationary front was located to the northwest of Taiwan, and the composite radar reflectivity displayed in Fig. 8 suggests a prefrontal squall line system. Figure 9 shows the accumulated rainfall generated by Taiwan Central Weather Bureau (CWB) Quantitative Precipitation Estimation and Segregation Using Multiple Sensor (QPESUMS) system (Zhang et al. 2008), which basically represents the radar-measured precipitation calibrated by rain gauge observations. Figure 9 reveals a northeast–southwest-oriented rainband that extended from the ocean to the land in southern Taiwan. A wide range of heavy precipitation can be found over the land, with the 3-h rainfall accumulation from 1200 to 1500 UTC exceeding 50 mm. In some areas of the southern plain of Taiwan more than 200 mm of rainfall fell within 24 h (not shown).

Data observed by three S-band radars are utilized. They are the S-band dual-polarization Doppler radar (S-POL), operated by the National Center for Atmospheric Research (NCAR), and RCCG and RCKT operated by the CWB of Taiwan. The latter two are Doppler radars. Figure 10 depicts the locations of these three radars, surface stations, along with wind profiler, PingTung and LiouGuei radiosonde stations from which the observational data are adopted for verification.

Figure 11 shows a comparison of the retrieved horizontal wind against the wind profiler measurements from the surface to *Z* = 8 km. It can be seen from the WISSDOM-retrieved results (Fig. 11a) that the wind direction is southerly at the lower layers, but shifts to southwesterly and westerly at the middle and upper layers, respectively. The wind speed varies within the range of 5.0 and 15.0 m s^{−1}. The vertical distribution of the retrieved wind field at this location is verified satisfactorily by the independent wind profiler observations as illustrated in Fig. 11b.

Figure 12 displays the profiles of pressure and temperature perturbations up to *Z* = 5.0 km observed by the PingTung and LiouGuei radiosondes and their retrieved counterparts. The observations indicate that the pressure perturbation is rather weak, while the temperature perturbation is generally negative at lower layers, but becomes positive at higher altitudes, varying from −1.5 to +1.5 K. These features are reasonably recovered by TPTRS. It should be pointed out since the locations of the radiosondes do not match the grid points exactly. Thus, the grid point closest to the radiosonde is selected and the comparison is conducted using the retrieved temperature and pressure right above this grid. The horizontal drift of the radiosonde during its ascending process is not considered. We believe these factors may contribute to the differences between the retrieved and observed profiles. Nevertheless, the retrieved profiles have similar orders of magnitude and trends as the observations.

### b. Interpretation of the retrieved thermodynamic field

This section discusses the structure of the TPTRS-retrieved thermodynamic fields using WISSDOM-synthesized wind fields. Note that the kinematic and thermodynamic fields are obtained with a horizontal and vertical resolution of 1.0 and 0.25 km, respectively. Figure 13 depicts the horizontal velocity and divergence field at *Z* = 0.5 km, and the column vector (CV) radar reflectivity (defined as the composite maximum reflectivity of each column). The prevailing wind blows from the ocean to the southwestern plain of the island, but is deflected as it approaches the CMR. A northeast–southwest-oriented band of intensive reflectivity can be clearly identified, within which one finds strong convergence. Based on the assumption of radar reflectivity conservation, Liou and Luo (2001) and Liou (2007) developed a method to objectively determine the moving speed of a weather system. By adopting this method, it is found that the squall line system is moving roughly toward the east, with a direction 255.8° and at a speed of 13.3 m s^{−1}.

Figure 14 shows the pressure and temperature perturbation fields obtained by TPTRS. A northwest–southeast-oriented band of negative temperature perturbation (cold pool), roughly perpendicular to the moving direction of the squall line, is obtained along the coast. The northern part of the cold pool is produced by the passage of the precipitating convection when moving from the ocean to the land, while the southern part of the cold pool, with a stronger intensity than the northern part, is triggered by the precipitation that has already existed for several hours near the coast (see Fig. 9). The intensity of the cold pool reaches −3.5 K. There is a positive temperature perturbation over the southwest plain area and near the foothills of the CMR. It should also be noticed that over the ocean, the southern section of the rainband (*X* ~ 70 km, *Y* = 0 ~ 40 km) is found to be at the same location as the positive temperature perturbation, implying that convection driven by warmer air is still in the early stage of development. Finally, positive (negative) pressure perturbation is generally associated with cold (warm) region.

Figure 15 depicts the storm-relative *u*–*w* wind field and radar reflectivity over a vertical cross section. The orientation of this vertical cross section is parallel to the motion of the system, and the variables displayed are the mean results from an average over a 20 km horizontal distance (see Fig. 14). The wind field exhibits a typical squall line structure, with evident front-to-rear and rear-to-front flows. The leading edge is located at approximately *X* = 65 km, where the updraft can reach 2.0 ms^{−1}. On the western side of the CMR (*X* ~ 110 km), upward motion is triggered by the orographic lifting, while downward motion can be identified on the eastern side of the CMR (*X* ~ 140 km).

Figure 16 displays the retrieved thermodynamic fields over the same vertical cross section. A cold pool is identified near the ground extending from *X* ~ 60 to *X* ~ 120 km. It is triggered by evaporative cooling of the rainband (see Fig. 9) and is associated with a positive pressure perturbation. An area of positive temperature perturbation, residing from *X* ~ 0 km to *X* ~ 40 km, is caused by the transportation of warm air due to the persistent southwesterly flow, leading to a pressure deficit greater than −0.60 hPa. Regions with local minimum pressure can be found at (*X* ~ 85 km, *Z* ~ 7 km), while local maximum pressure can be identified at (*X* ~ 75 km, *Z* ~ 6 km) and (*X* ~ 145 km, *Z* ~ 3 km). They are collocated with the top boundary of regions with temperature minimum and maximum, respectively. These features can be explained by the equation shown in (22).

## 5. Summary

A new Terrain-Permitting Thermodynamic Retrieval Scheme (TPTRS) is developed, which can be used to retrieve the three-dimensional pressure and temperature fields immediately over complex terrain using wind information synthesized from multiple Doppler radars. The correctness of the retrieval code and accuracy of the proposed algorithm are validated using datasets generated by a numerical model under the OSSE framework. The retrieval scheme is applied using data collected during IOP 8 of the 2008 SoWMEX to retrieve the thermodynamic fields of a prefrontal squall line after it has made landfall in southern Taiwan, and reasonable results are obtained.

The terrain-resolving capability of WISSDOM and the proposed TPTRS introduced in this study are able to provide three-dimensional high-resolution pressure and temperature along with the wind fields over complex terrain, allowing us to obtain better diagnoses of the thermodynamic and kinematic structure of a heavy rainfall system, especially in mountainous areas. These observed and retrieved meteorological fields are also candidates ready to be assimilated into a numerical weather prediction model to improve its forecast skill.

This research is supported by the Ministry of Science and Technology of Taiwan under MOST107-2111-M-008-040, MOST107-2625-M-008-008, and by the Central Weather Bureau (CWB) under MOTC-CWB-107-M-02. The authors thank CWB for providing the radar and surface station data.

## REFERENCES

Armijo, L., 1969: A theory for the determination of wind and precipitation velocities with Doppler radars.

,*J. Atmos. Sci.***26**, 570–573, https://doi.org/10.1175/1520-0469(1969)026<0570:ATFTDO>2.0.CO;2.Brandes, E. A., 1977: Flow in severe thunderstorms observed by dual-Doppler radar.

,*Mon. Wea. Rev.***105**, 113–120, https://doi.org/10.1175/1520-0493(1977)105<0113:FISTOB>2.0.CO;2.Brandes, E. A., 1984: Relationships between radar-derived thermodynamic variables and tornadogenesis.

,*Mon. Wea. Rev.***112**, 1033–1052, https://doi.org/10.1175/1520-0493(1984)112<1033:RBRDTV>2.0.CO;2.Chang, W.-Y., W.-C. Lee, and Y.-C. Liou, 2015: The kinematic and microphysical characteristics and associated precipitation efficiency of subtropical convection during SoWMEX/TiMREX.

,*Mon. Wea. Rev.***143**, 317–340, https://doi.org/10.1175/MWR-D-14-00081.1.Chen, X. C., K. Zhao, J. Sun, B. Zhou, and W.-C. Lee, 2016: Assimilating surface observations in a four-dimensional variational Doppler radar data assimilation system to improve the analysis and forecast of a squall line case.

,*Adv. Atmos. Sci.***33**, 1106–1119, https://doi.org/10.1007/s00376-016-5290-0.Doviak, R. J., P. Ray, R. G. Strauch, and L. J. Miller, 1976: Error estimation in wind fields derived from dual-Doppler radar measurement.

,*J. Appl. Meteor.***15**, 868–878, https://doi.org/10.1175/1520-0450(1976)015<0868:EEIWFD>2.0.CO;2.Gal-Chen, T., 1978: A method for the initialization of the anelastic equations: Implications for matching models with observations.

,*Mon. Wea. Rev.***106**, 587–606, https://doi.org/10.1175/1520-0493(1978)106<0587:AMFTIO>2.0.CO;2.Gal-Chen, T., and R. A. Kropfli, 1984: Buoyancy and pressure perturbations derived from dual-Doppler radar observations of the planetary boundary layer: Applications for matching models with observations.

,*J. Atmos. Sci.***41**, 3007–3020, https://doi.org/10.1175/1520-0469(1984)041<3007:BAPPDF>2.0.CO;2.Hane, C. E., and P. S. Ray, 1985: Pressure and buoyancy fields derived from Doppler radar data in a tornadic thunderstorm.

,*J. Atmos. Sci.***42**, 18–35, https://doi.org/10.1175/1520-0469(1985)042<0018:PABFDF>2.0.CO;2.Hane, C. E., R. B. Wilhelmson, and T. Gal-Chen, 1981: Retrieval of thermodynamic variables within deep convective clouds: Experiments in three dimensions.

,*Mon. Wea. Rev.***109**, 564–576, https://doi.org/10.1175/1520-0493(1981)109<0564:ROTVWD>2.0.CO;2.Hsu, S. A., E. A. Meindl, and D. B. Gilhousen, 1994: Determining the power-law wind-profile exponent under near-neutral stability conditions at sea.

,*J. Appl. Meteor.***33**, 757–765, https://doi.org/10.1175/1520-0450(1994)033<0757:DTPLWP>2.0.CO;2.Jou, B. J.-D., W.-C. Lee, and R. H. Johnson, 2011: An overview of SoWMEX/TiMREX and its operation.

, 2nd ed. C.-P. Chang, Ed., World Scientific, 303–318.*The Global Monsoon System: Research and Forecast*Lee, J.-T., D.-I. Lee, C.-H. You, H. Uyeda, Y.-C. Liou, and I.-S. Han, 2014: Dual-Doppler radar analysis of a near-shore line-shaped convective system on 27 July 2011, Korea: a case study.

,*Tellus***66A**, 23453, https://doi.org/10.3402/tellusa.v66.23453.Lee, J.-T., K.-Y. Ko, D.-I. Lee, C.-H. You, and Y.-C. Liou, 2018: Enhancement of orographic precipitation in Jeju Island during the passage of Typhoon Khanun (2012).

,*Atmos. Res.***201**, 58–71, https://doi.org/10.1016/j.atmosres.2017.10.013.Lin, Y.-J., T.-C. Chen Wang, and J. H. Lin, 1986: Pressure and temperature perturbations within a squall-line thunderstorm derived from SESAME dual-Doppler data.

,*J. Atmos. Sci.***43**, 2302–2327, https://doi.org/10.1175/1520-0469(1986)043<2302:PATPWA>2.0.CO;2.Liou, Y.-C., 2001: The derivation of absolute potential temperature perturbations and pressure gradients from wind measurements in three dimensional space.

,*J. Atmos. Oceanic Technol.***18**, 577–590, https://doi.org/10.1175/1520-0426(2001)018<0577:TDOAPT>2.0.CO;2.Liou, Y.-C., 2007: Single-Doppler retrieval of the three-dimensional wind in a deep convective system based on an optimal moving frame of reference.

,*J. Meteor. Soc. Japan***85**, 559–582, https://doi.org/10.2151/jmsj.85.559.Liou, Y.-C., and I.-S. Luo, 2001: An investigation of the moving frame single-Doppler wind retrieval technique using Taiwan Area Mesoscale Experiment low-level data.

,*J. Appl. Meteor.***40**, 1900–1917, https://doi.org/10.1175/1520-0450(2001)040<1900:AIOTMF>2.0.CO;2.Liou, Y.-C., and Y.-J. Chang, 2009: A variational multiple-Doppler radar three-dimensional wind synthesis method and its impact on thermodynamic retrieval.

,*Mon. Wea. Rev.***137**, 3992–4010, https://doi.org/10.1175/2009MWR2980.1.Liou, Y.-C., T.-C. Chen Wang, and K.-S. Chung, 2003: A three-dimensional variational approach for deriving the thermodynamic structure using Doppler wind observations—An application to a subtropical squall line.

,*J. Appl. Meteor.***42**, 1443–1454, https://doi.org/10.1175/1520-0450(2003)042<1443:ATVAFD>2.0.CO;2.Liou, Y.-C., S.-F. Chang, and J. Sun, 2012: An application of the immersed boundary method for recovering the three-dimensional wind fields over complex terrain using multiple-Doppler radar data.

,*Mon. Wea. Rev.***140**, 1603–1619, https://doi.org/10.1175/MWR-D-11-00151.1.Liou, Y.-C., T.-C. Chen Wang, Y.-C. Tsai, Y.-S. Tang, P.-L. Lin, and Y.-A. Lee, 2013: Structure of precipitating systems over Taiwan’s complex terrain during Typhoon Morakot (2009) as revealed by weather radar and rain gauge observations.

,*J. Hydrol.***506**, 14–25, https://doi.org/10.1016/j.jhydrol.2012.09.004.Liou, Y.-C., J.-L. Chiou, W.-H. Chen, and H.-Y. Yu, 2014: Improving the model convective storm quantitative precipitation nowcasting by assimilating state variables retrieved from multiple-Doppler radar observations.

,*Mon. Wea. Rev.***142**, 4017–4035, https://doi.org/10.1175/MWR-D-13-00315.1.Liou, Y.-C., T.-C. Chen Wang, and P.-Y. Huang, 2016: The inland eyewall reintensification of Typhoon Fanapi (2010) documented from an observational perspective using multiple-Doppler radar and surface measurements.

,*Mon. Wea. Rev.***144**, 241–261, https://doi.org/10.1175/MWR-D-15-0136.1.Liu, D. C., and J. Nocedal, 1988: On the limited memory BFGS method for large scale optimization. Tech. Rep. NAM 03, Department of Electrical Engineering and Computer Science, Northwestern University, 26 pp. [Available from the Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL 60208.]

Markowski, P., and Y. Richardson, 2011:

. Wiley-Blackwell, 407 pp.*Mesoscale Meteorology in Midlatitudes*Maury, B., 2001: A fat boundary method for the Poisson problem in a domain with holes.

,*J. Sci. Comput.***16**, 319–339, https://doi.org/10.1023/A:1012821728631.O’Brien, J. J., 1970: Alternative solutions to the classical vertical velocity problem.

,*J. Appl. Meteor.***9**, 197–203, https://doi.org/10.1175/1520-0450(1970)009<0197:ASTTCV>2.0.CO;2.Parsons, D. B., C. G. Mohr, and T. Gal-Chen, 1987: A severe frontal rainband. Part III: Derived thermodynamic structure.

,*J. Atmos. Sci.***44**, 1615–1631, https://doi.org/10.1175/1520-0469(1987)044<1615:ASFRPI>2.0.CO;2.Peterson, E. W., and J. P. Hennessey, 1978: On the use of power laws for estimates of wind power potential.

,*J. Appl. Meteor.***17**, 390–394, https://doi.org/10.1175/1520-0450(1978)017<0390:OTUOPL>2.0.CO;2.Protat, A., and I. Zawadzki, 1999: A variational method for real time retrieval of three-dimensional wind field from multiple-Doppler bistatic radar network data.

,*J. Atmos. Oceanic Technol.***16**, 432–449, https://doi.org/10.1175/1520-0426(1999)016<0432:AVMFRT>2.0.CO;2.Protat, A., and I. Zawadzki, 2000: Optimization of dynamic retrievals from a multiple-Doppler radar network.

,*J. Atmos. Oceanic Technol.***17**, 753–760, https://doi.org/10.1175/1520-0426(2000)017<0753:OODRFA>2.0.CO;2.Protat, A., Y. Lemaitre, and G. Scialom, 1998: Thermodynamic analytical fields from Doppler radar data by means of the MANDOP analysis.

,*Quart. J. Roy. Meteor. Soc.***124**, 1633–1668, https://doi.org/10.1002/qj.49712454914.Ray, P. S., R. J. Doviak, G. B. Walker, D. Sirmans, J. Carter, and B. Bumgarner, 1975: Dual-Doppler observation of a tornadic storm.

,*J. Appl. Meteor.***14**, 1521–1530, https://doi.org/10.1175/1520-0450(1975)014<1521:DDOOAT>2.0.CO;2.Ray, P. S., C. L. Ziegler, W. Bumgarner, and R. J. Serafin, 1980: Single and multiple-Doppler radar observations of tornadic storms.

,*Mon. Wea. Rev.***108**, 1607–1625, https://doi.org/10.1175/1520-0493(1980)108<1607:SAMDRO>2.0.CO;2.Roux, F., 1985: Retrieval of thermodynamic fields from multiple-Doppler radar data using the equations of motion and the thermodynamic equation.

,*Mon. Wea. Rev.***113**, 2142–2157, https://doi.org/10.1175/1520-0493(1985)113<2142:ROTFFM>2.0.CO;2.Roux, F., 1988: The West African squall line observed on 23 June 1981 during COPT 81: Kinematics and thermodynamics of the convective region.

,*J. Atmos. Sci.***45**, 406–426, https://doi.org/10.1175/1520-0469(1988)045<0406:TWASLO>2.0.CO;2.Roux, F., and J. Sun, 1990: Single-Doppler observations of a West African squall line on 27–28 May 1981 during COPT 81: Kinematics, thermodynamics and water budget.

,*Mon. Wea. Rev.***118**, 1826–1854, https://doi.org/10.1175/1520-0493(1990)118<1826:SDOOAW>2.0.CO;2.Roux, F., V. Marecal, and D. Hauser, 1993: The 12/13 January 1998 narrow cold-frontal rainband observed during MFDP/FRONTS 87. Part I: Kinematics and thermodynamics.

,*J. Atmos. Sci.***50**, 951–974, https://doi.org/10.1175/1520-0469(1993)050<0951:TJNCFR>2.0.CO;2.Scialom, G., and Y. Lemaitre, 1990: A new analysis for the retrieval of three-dimensional mesoscale wind fields from multiple Doppler radar.

,*J. Atmos. Oceanic Technol.***7**, 640–665, https://doi.org/10.1175/1520-0426(1990)007<0640:ANAFTR>2.0.CO;2.Shapiro, A., and J. J. Mewes, 1999: New formulations of dual-Doppler wind analysis.

,*J. Atmos. Oceanic Technol.***16**, 782–792, https://doi.org/10.1175/1520-0426(1999)016<0782:NFODDW>2.0.CO;2.Sun, J., and F. Roux, 1988: Thermodynamic structure of the trailing stratiform regions of two West African squall lines.

,*Ann. Geophys.***6**, 659–670.Sun, J., and N. A. Crook, 1996: Comparison of thermodynamic retrieval by the adjoint method with the traditional retrieval method.

,*Mon. Wea. Rev.***124**, 308–324, https://doi.org/10.1175/1520-0493(1996)124<0308:COTRBT>2.0.CO;2.Tong, M., and M. Xue, 2005: Ensemble Kalman filter assimilation of Doppler radar data with a compressible nonhydrostatic model: OSS experiments.

,*Mon. Wea. Rev.***133**, 1789–1807, https://doi.org/10.1175/MWR2898.1.Tsai, C.-L., K. Kim, Y.-C. Liou, G. Lee, and C.-K. Yu, 2018: Impacts of topography on airflow and precipitation in the Pyeongchang area seen from multiple-Doppler radar observations.

,*Mon. Wea. Rev.***146**, 3401–3424, https://doi.org/10.1175/MWR-D-17-0394.1.Tseng, Y., and J. Ferziger, 2003: A ghost-cell immersed boundary method for flow in complex geometry.

,*J. Comput. Phys.***192**, 593–623, https://doi.org/10.1016/j.jcp.2003.07.024.Zhang, J., and Coauthors, 2008: High-resolution QPE system for Taiwan.

, S. K. Park, L. Xu, Eds., Springer-Verlag, 147–162.*Data Assimilation for Atmospheric, Oceanic, and Hydrologic Applications*Ziegler, C. L., P. S. Ray, and N. C. Knight, 1983: Hail growth in an Oklahoma multicell storm.

,*J. Atmos. Sci.***40**, 1768–1791, https://doi.org/10.1175/1520-0469(1983)040<1768:HGIAOM>2.0.CO;2.