Editorial Type:
Article
Article Type:
Research Article

Removal of the Vertical Structure Ambiguity in the Thermodynamic Retrieval from Multiple-Doppler Radar Synthesized Wind Fields

Yu-Chieng Liou aDepartment of Atmospheric Sciences, National Central University, Zhongli, Taoyuan City, Taiwan

Search for other papers by Yu-Chieng Liou in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0002-8448-8554
and
Yung-Lin Teng aDepartment of Atmospheric Sciences, National Central University, Zhongli, Taoyuan City, Taiwan

Search for other papers by Yung-Lin Teng in
Current site
Google Scholar
PubMed Close
Online Publication:
24 May 2023
Print Publication:
01 Jun 2023
DOI:
https://doi.org/10.1175/MWR-D-23-0001.1
Page(s):
1427–1442
Open access
View license

Abstract

It has been long recognized that in the retrieved thermodynamic fields using multiple-Doppler radar synthesized winds, an unknown constant exists on each horizontal level, leading to an ambiguity in the retrieved vertical structure. In this study, the traditional thermodynamic retrieval scheme is significantly improved by the implementation of the Equation of State (EoS) as an additional constraint. With this new formulation, the ambiguity of the vertical structure can be explicitly identified and removed from the retrieved three-dimensional thermodynamic fields. The only in situ independent observations needed to perform the correction are the pressure and temperature measurements taken at a single surface station. If data from multiple surface stations are available, a strategy is proposed to obtain a better estimate of the unknown constant. Experiments in this research were conducted under the observation system simulation experiment (OSSE) framework to demonstrate the validity of the new approach. Problems and possible solutions associated with using real datasets and potential future extended applications of this new method are discussed.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu-Chieng Liou, tyliou@atm.ncu.edu.tw

Abstract

It has been long recognized that in the retrieved thermodynamic fields using multiple-Doppler radar synthesized winds, an unknown constant exists on each horizontal level, leading to an ambiguity in the retrieved vertical structure. In this study, the traditional thermodynamic retrieval scheme is significantly improved by the implementation of the Equation of State (EoS) as an additional constraint. With this new formulation, the ambiguity of the vertical structure can be explicitly identified and removed from the retrieved three-dimensional thermodynamic fields. The only in situ independent observations needed to perform the correction are the pressure and temperature measurements taken at a single surface station. If data from multiple surface stations are available, a strategy is proposed to obtain a better estimate of the unknown constant. Experiments in this research were conducted under the observation system simulation experiment (OSSE) framework to demonstrate the validity of the new approach. Problems and possible solutions associated with using real datasets and potential future extended applications of this new method are discussed.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu-Chieng Liou, tyliou@atm.ncu.edu.tw
Keywords: Radars/Radar observations; Optimization; Variational analysis

1. Introduction

The so-called thermodynamic retrieval represents a technique by which thermodynamic variables (i.e., pressure and temperature perturbations) can be derived from the Doppler radar synthesized three-dimensional winds. These thermodynamic fields are found to be important for weather diagnoses and the initialization of high-resolution numerical models for improving the accuracy of convective-scale forecasts. The method proposed by Gal-Chen (1978, hereafter G78) has been particularly useful, because the boundary condition for solving a Poisson equation to obtain the pressure perturbations could naturally be determined from the wind fields synthesized using Doppler radars. This method has been widely employed to study the structure of various weather phenomena such as a deep moist convection (Hane et al. 1981), a dry boundary layer (Gal-Chen and Kropfli 1984), tornadic thunderstorms (Hane and Ray 1985), convection embedded in a squall line (Lin et al. 1986), and a frontal rainband (Parsons et al. 1987).

Lin et al. (1993) was the first to apply G78 to initialize a numerical model using radar-synthesized winds and retrieved thermodynamic fields. The results for the storm in their simulation exhibited good agreement with the observations. Weygandt et al. (2002) employed the G78 method and a single-Doppler velocity retrieval scheme designed by Shapiro et al. (1995) to initialize a numerical model and showed that the general features of a storm’s evolution could be well captured within 35 min. Liou et al. (2014) studied the impact of using the retrieved thermodynamic variables by G78 on short-term quantitative precipitation forecasts during the 2008 Southwestern Monsoon Experiment (SoWMEX) field experiment. Shimizu et al. (2019) adopted the nudging technique to assimilate the high-temporal-resolution (1-min) radar-synthesized winds and G78-derived thermodynamic parameters into a numerical model and investigated the improvement on very-short-range (<1 h) rainfall forecast for a severe storm case. Roux et al. (1984) followed the concept of G78, but derived two separate equations from it for the retrieval of the pressure and temperature perturbations, respectively. Their approach was employed by Protat et al. (1998) to investigate the interactions among various scales of motion, and by Foerster and Bell (2017) to retrieve the thermodynamic fields in rapidly rotating vortices. Hauser et al. (1988) presented a different approach in which the momentum, thermodynamic, and microphysical equations were all utilized so that the microphysical and thermodynamic variables could be retrieved simultaneously from wind fields specified by Doppler radar observations.

However, it should be mentioned that the solutions obtained using the aforementioned algorithms are not unique and would still remain valid upon adding an arbitrary constant at each horizontal layer. In G78, this unknown constant takes the form of the horizontal average of the pressure perturbations (details can be found in section 3). Under this limitation, the interpretation of the vertical structure of the thermodynamic fields becomes ambiguous.

Efforts have been made to resolve the ambiguous vertical structure problem. Roux (1985) was able to compute the vertical derivatives of the horizontal mean pressure and temperature by employing a simplified thermodynamic equation for the potential temperature and making a series of assumptions for the complicated microphysical processes. If an initial value for pressure and temperature at a given altitude is available, vertical integrations can be conducted to deduce the profiles of the mean pressure and temperature, which are then added to the perturbations to get the complete three-dimensional thermodynamic fields. Similar to Roux (1985), Roux (1988) later proposed another approach in which a unique solution of the pressure and temperature perturbations could also be solved up to a volume-wide constant. This missing constant can be computed from comparison with a surface measurement. Also, by including a thermodynamic equation, the retrieval schemes developed by Roux and Sun (1990), Sun and Houze (1992), and Roux et al. (1993) could provide the gradients of pressure and temperature perturbations in any direction. Geerts and Hobbs (1991), on the other hand, utilized sounding, radar reflectivity, aircraft data, surface observations, and several assumptions such as geostrophic balance to specify the boundary conditions when solving the equations so that unique solutions for the buoyancy and pressure fields can be obtained. Liou (2001) and Liou et al. (2003) also incorporated a simplified thermodynamic equation along with the three momentum equations into a cost function, which was variationally minimized to solve for the potential temperature perturbations and pressure gradients. The source and sink terms associated with the microphysical processes were summed up into one term and treated as a retrievable parameter. By taking advantage of the information provided by a numerical model, the three-dimensional π′ and θ′ were considered in Xu et al. (2010) as increments with respect to model-generated background fields and constrained by the background information and the momentum equations simultaneously during the retrieval. While assimilating G78-derived thermodynamic fields into a numerical model to improve the forecast, Liou et al. (2014) and Shimizu et al. (2019) found it was feasible to use the model outputs to estimate the horizontal average of the pressure perturbations when no other in situ observational data were available. This is equivalent to updating only the pressure perturbations of the model simulation, while keeping its horizontal averages intact.

To resolve the problem associated with the unknown constant on each layer (i.e., the horizontal average of the pressure perturbations in G78), G78 pointed out that mathematically a single-point independent measurement of the pressure for each altitude would be sufficient to determine the unknown horizontal average for that layer. To achieve this, G78 proposed to use a well-instrumented aircraft (maybe available in a field experiment but not in daily routine observations) to provide independent observation of the thermodynamic variable for each layer. On the other hand, Protat et al. (1998) and Liou et al. (2014) suggested that in situ pressure and temperature data could be acquired using radiosonde. However, compared with radar observations, the temporal resolution of radiosonde, even during field experiments, is low. Furthermore, as indicated in G78, great uncertainty may occur in the thermodynamic parameterization of the moist processes, which plays an essential role in the thermodynamic equation. Thus, the goal of this research is to develop a new approach for the estimation of the unknown constant at each height without relying upon aircraft, radiosonde observations or the thermodynamic equation.

The rest of the manuscript is organized as follows. Section 2 explains the preparation of the datasets used in this study. Section 3 gives a brief review of the G78 algorithm. The Equation of State (EoS), which links π′ and θc′, is introduced in section 4. Section 5 provides a detailed description regarding how to implement the EoS as an additional condition to constrain the thermodynamic variables and shows the improvement in retrieved vertical structure of the thermodynamic fields. Section 6 suggests a proper strategy to deal with the scenario when in situ pressure and temperature measurements can be taken at multiple surface stations and observations are affected by errors. Problems and possible solutions associated with using real-world data and potential future extended applications of the proposed new retrieval scheme are discussed in section 7, followed by a summary in section 8.

2. Preparation of the datasets

The retrieval experiments carried out for this study are conducted under the observation system simulation experiment (OSSE) framework. The Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008) is utilized to generate the pseudo observational datasets. The model domain contains 81 × 81 grid points along the horizontal, and 100 σ layers in the vertical direction. The horizontal grid spacing is 1.0 km. The background thermodynamic field characterizes a moderately unstable environment, in which the hodograph exhibits a quarter-circle shear profile (Weisman and Rotunno 2000). The Kessler warm rain microphysics scheme (Kessler 1969) is selected for the simulation. The 1.5-order turbulent kinetic energy closure scheme is applied for the subgrid parameterization.

In G78, the perturbations of the thermodynamic parameters are defined as deviations from the “basic” states, which are functions of height only. Thus, in this research the model is initialized by superimposing a perturbation (θ′) to a horizontally homogeneous potential temperature field using the following formula:
θ′=θmaxcos⁡2(π2Rad),
Rad=(x−x010.0)2+(y−y010.0)2+(z−z01.5)2,
where (x0 = 44.0 km, y0 = 26.7 km, z0 = 1.5 km) and θmax represent the center point and peak value (=5.0 K) of the perturbation, respectively. The (1) and (2) are applied only when Rad ≤ 1. This type of approach for OSSE tests has been adopted by many previous studies (e.g., Gao et al. 1999; Mewes and Shapiro 2002; Tong and Xue 2005).

The structure of the storm is shown in Figs. 1–3. It can be seen in Fig. 1b that at T = 70 min, the time when the data are collected for conducting the retrieval experiments, the storm has experienced a splitting process (Klemp and Wilhelmson 1978). The maximum updraft in Figs. 2 and 3 exceeds 20 and 10 m s−1, respectively.

Fig. 1.
Fig. 1.

The storm structure on a horizontal plane at Z = 1.0 km after (a) 30 and (b) 70 min of the model simulation. The arrows represent horizontal winds, the contour lines are vertical velocities (1.0 m s−1 interval) with updrafts (downdrafts) represented by solid (dashed) lines, and the color shading stands for radar reflectivity (in dBZ).

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

Fig. 2.
Fig. 2.

The storm structure on an X–Z vertical cross section at Y = 30 km after 70 min of the model simulation. The arrows represent the u–w winds, the contour lines are vertical velocities (3.0 m s−1 interval) with updrafts (downdrafts) represented by solid (dashed) lines, and the color shading stands for radar reflectivity (in dBZ).

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

Fig. 3.
Fig. 3.

As in Fig. 2, but for the storm structure over a Y–Z vertical cross section at X = 60 km, with the arrows representing the υ–w winds.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

3. Revisiting the G78 method

The G78 thermodynamic retrieval starts with the equations of motion:
−F≡1θυ0[∂u∂t+V⋅∇u−fυ+turb(u)]=−∂π′∂x,
−G≡1θυ0[∂υ∂t+V⋅∇υ+fu+turb(υ)]=−∂π′∂y,
−H≡1θυ0[∂w∂t+V⋅∇w+turb(w)+g(qr+qs)]=−∂π′∂z+gθc′θυ0θ0,
where the subscript 0 stands for a horizontally homogeneous basic state and varies only with height, which can be determined by a prestorm environmental sounding. The perturbations from the basic state are expressed with variables with a prime. In (3)–(5), (u, υ, w) represent the Cartesian wind components; f is the Coriolis parameter; g stands for the gravitational acceleration; and turb(⋅) denotes a subgrid-scale turbulence parameterization operator. The mixing ratio of rainwater (qr) and snow (qs) can be estimated using the radar reflectivity data through empirical equations such as those found in Tong and Xue (2005). However, more sophisticated ways to retrieve better estimates of the hydrometeor mixing ratio using dual-polarimetric radar data can be found in Carlin et al. (2016). The parameter π is a normalized pressure defined as
π=Cp(PP00)R/Cp,
where P is the pressure; P00 equals 1000 hPa; R (=287 J kg−1 K−1) is the gas constant; and Cp (=1005 J kg−1 K−1) is the specific heat capacity under a constant pressure. The virtual potential temperature (θυ) and virtual cloud potential temperature perturbation ( θc′) as defined in Roux (1985) are expressed by
θυ=θ(1.0+0.61qυ),
θc′=θ′+(0.61qυ′−qc)θ0,
where θ is potential temperature; qυ′ denotes the perturbation of the water vapor mixing ratio; and qc is the cloud water mixing ratio. The virtual cloud potential temperature perturbation ( θc′) is treated as a retrievable parameter. The values of F, G, and H can be obtained once the three-dimensional air motion is obtained through multiple-Doppler radar synthesis (e.g., Liou and Chang 2009). In this study they are computed using the data generated by the model simulation described in section 2. It should be mentioned that in G78, a perfect dataset was prepared to detect the programming error and to better evaluate the performance of the retrieval method itself (see Gal-Chen 1978, Experiment 1). In this research a similar approach is adopted. Thus, the temperature and pressure fields produced by WRF are used to compute F, G, and H in (3)–(5) directly. Nevertheless, in section 6 an experiment is designed in which the F, G, and H are superimposed by perturbations to imitate the overall errors introduced by various factors, including the uncertainty introduced by using wind fields to compute F, G, and H.
According to G78, a cost function J is first formulated over a given horizontal plane using (3) and (4) to represent the differences between ∂π′/∂x and F, as well as ∂π′/∂y and G:
J=∫∫[(∂π′∂x−F)2+(∂π′∂y−G)2]dxdy.
A set of π′ which minimizes J is found by using variational analysis. This is equivalent to solving a Poisson equation for π′:
∂2π′∂x2+∂2π′∂y2=∂F∂x+∂G∂y
subject to the Neumann boundary conditions:
∂π′∂x=Fat x=xw,xe,
∂π′∂y=Gat y=ys,yn,
where xw and xe stand for the east and west boundaries and ys and yn for the south and north boundaries, respectively. Due to the use of the Neumann boundary conditions, the solutions of (10) are not unique, and can be satisfied only up to an arbitrary constant. It is proposed in G78 that this unknown constant can be cancelled by subtracting the horizontal average from the solutions. Thus, the unique solution from (10) is π′ − 〈π′〉, where 〈⋅〉 represents a horizontal average. After rewriting (5), making a horizontal average, and computing the difference, we have
∂π′∂z=gθc′θυ0θ0+H,
∂⟨π′⟩∂z=g⟨θc′⟩θυ0θ0+⟨H⟩,
∂(π′−⟨π′⟩)∂z=gθc′−⟨θc′⟩θυ0θ0+(H−⟨H⟩).
By substituting the retrieved π′ − 〈π′〉 into (12c), one can solve for θc′−⟨θc′⟩.

By definition, 〈π′〉 is a constant at each horizontal level and varies only with height. Thus, in theory, a single-point of independent in situ observation of π′ at each horizontal level combined with the G78-retrieved π′ − 〈π′〉 at the same point is sufficient to determine the 〈π′〉 of that level. By applying this 〈π′〉 to the entire horizontal plane the π′ itself can be computed at all grid points on that level. This procedure is repeated on each level, until eventually a complete set of π′ in the three-dimensional space can be obtained. Once the π′ field is known, the three-dimensional θc′ can also be computed using (12a). However, as discussed in section 1, using aircraft or sounding as suggested in previous studies is difficult to provide frequent in situ observation of π′ on each level. This leads to the following section in which a new approach is proposed. Finally, it is worth mentioning that according to Foerster and Bell (2017), even if π′ and θc′ are solutions that can minimize all three momentum equations shown by (3) to (5) simultaneously, π′ + C(z) and θc′+(θ0θυ0/g)(dC/dz) are still valid solutions where C is an unknown function varying only with height. The search for the unique solution requires additional information and constraints.

4. Implementation of the EoS

In this study, the EoS is implemented as an additional constraint, since it links the absolute pressure with the temperature fields. According to the derivation performed in the appendix, the perturbation form of the EoS can be written as follows:
θc′=T1˜π′+T˜2(0.61qυ′−qc)−T3˜ρ′−T4˜qυ′,
T1˜=[(P00P0)κ1ρ0R(1+0.61qυ0)−RCpθ0P0]P0R(P00P0)κ,
T˜2=θ0,
T3˜=(P00P0)κT0ρ0,
T4˜=(P00P0)κ0.61T01+0.61qυ0,
where (14a), (14c), and (14d) are from the appendix, but the approximation sign is replaced by an equal sign. Note that terms T1˜, T˜2, T3˜, and T4˜ are functions of height only.

It is known that in reality, there may not be proper instruments available to routinely measure vapor and cloud water mixing ratio, thus qυ′ and qc in the aforementioned equations are assumed to be zero. Moreover, in Eq. (13) an additional unknown variable ρ′ is introduced. The role played by ρ′ in the retrieval procedure and the impact of neglecting qυ′ and qc on the estimation of the unknown constants are examined in the next section.

5. Methodology of the new thermodynamic retrieval scheme

a. Method I and results

A procedure, designated Method I, for determining the vertical profile of the unknown 〈π′〉 and ⟨θc′⟩ is first designed to examine the effect of using the EoS.

  • Step 1: Assuming that F, G, and H in (3)–(5) are available, the G78 method is used to retrieve π′ − 〈π′〉 and θc′−⟨θc′⟩ at each grid point in the three-dimensional space.

  • Step 2: If there are in situ measurements of π′ and θc′ on the lowest layer taken by a surface station located at grid point A with the coordinates (xA, yA, 1), where 1 stands for the index of the lowest vertical layer, one can use (13) to compute the ρ′ at this grid point A, and combine the observed π′ and θc′with the G78-retrieved π′ − 〈π′〉 and θc′−⟨θc′⟩ from step 1 to get 〈π′〉 and ⟨θc′⟩ of the lowest layer. At this stage, two tests are performed. In the first test (TEST1), the correct ρ′ is assumed to be known in the air column above grid point A. In the second test (TEST2), the ρ′ at (xA, yA, 1) is applied to the entire air column above grid point A. The purpose of these two tests is to evaluate the role played by ρ′, as well as qυ′ and qc in the EoS. Note that the latter two variables are assumed to be zero in both tests. It should also be emphasized that the assumption of using the same ρ′ for the entire air column in TEST2 is made only temporarily. The results from TEST2 are used as guidance for further improvements, which are introduced in the next section.

  • Step 3: Using θc′and H on the lowest layer at (xA, yA, 1) to estimate the right-hand-side of (12a), which is then integrated upward starting from the π′ on the lowest layer at (xA, yA, 1) to obtain the π′ on the second layer at (xA, yA, 2). By combining this with the G78 retrieved π′ − 〈π′〉 at this grid point (xA, yA, 2), one obtains the 〈π′〉 of the second layer.

  • Step 4: Using π′ on the second layer at point (xA, yA, 2) and the EoS shown in (13), one can estimate θc′ at the same grid point. By combining this with the G78 retrieved θc′−⟨θc′⟩ at this point (xA, yA, 2), one obtains the ⟨θc′⟩ of the second layer. As mentioned earlier, when applying (13), ρ′ is assumed to be either known (TEST1) or from the lowest layer (TEST2), and qυ′ and qc are neglected.

  • Step 5: Steps 3 and 4 are repeated from the second layer to the top layer to obtain the vertical profiles of 〈π′〉 and ⟨θc′⟩ for the entire analysis domain.

It should be emphasized that once 〈π′〉 and ⟨θc′⟩ are obtained for a given layer, they can always be combined with the G78 retrieved π′ − 〈π′〉 and θc′−⟨θc′⟩ to compute the π′ and θc′, followed by getting ρ′ at all grid points on the same layer.

Using the datasets generated by the WRF model at T = 70 min for numerical experiments, two grid points located well outside and very close to the main convective system are selected to start the upward integration for TEST1 and TEST2 using Method I. A further examination reveals that at the grid point far from the main convective region, the |θc′| on most levels aloft is significantly smaller than that above the grid point near the storm, and is less than 10% of the maximum |θc′| on the same level (not shown).

By comparing the solid lines displayed in Fig. 4b with Fig. 4c, and Fig. 4e with Fig. 4f, it is shown that in TEST1 (assuming correct ρ′ is known in the air column aloft), the true vertical profiles of 〈π′〉 and ⟨θc′⟩ (see Figs. 4a,d) can be very accurately retrieved regardless of which grid point is selected to start the upward integration. However, by comparing the dashed lines in Figs. 4b with Fig. 4c, and Fig. 4e with Fig. 4f, it is found that in TEST2 (applying the ρ′ estimated at the surface grid point to the entire air column aloft), the retrieved 〈π′〉 and ⟨θc′⟩ profiles become dependent on the location of the grid point. The result obtained at the point far from the convection area is much better than that near the convection. Since the difference between TEST1 and TEST2 is the assumption about ρ′, these results indicate that ρ′ plays a critical role in determining the quality of the retrieval. In addition, TEST1 indicates that the impact of neglecting qυ′ and qc on the retrievals is limited. This is encouraging, since in real world scenarios, regular observations of qυ′ and qc in a three-dimensional space are usually unavailable.

Fig. 4.
Fig. 4.

The true and retrieved profiles using Method I: (a) true 〈π′〉; (b) retrieved 〈π′〉 profiles with the upward integration starting from a surface grid point located far from the main convection; and (c) as in (b), but with the upward integration starting from a surface grid point near the convection. The solid and dashed lines in (b) and (c) denote the results from TEST1 and TEST2, respectively. (d)–(f) As in (a)–(c), but illustrating the true and retrieved ⟨θc′⟩ profiles. Inside each figure panel, “Retv” stands for retrieved results, and F and N represent that the results are obtained at a grid point far from or near the storm, respectively.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

The grid point A mentioned in step 2 can be any grid point on the lowest layer, from which one can perform the upward vertical integration and obtain a set of 〈π′〉 and ⟨θc′⟩ profiles. Therefore, for the procedure described for TEST2 of Method I is applied to every grid point on the lowest layer. Figure 5 shows the distribution of root-mean-square error (RMSE) in the retrieved 〈π′〉 and ⟨θc′⟩ profiles summed over all heights plotted with respect to the magnitudes of |π′|, |θc′| and |ρ′| at each grid point on the lowest layer. It can be clearly seen that better retrievals occur if the vertical integration starts from those lowest layer grid points with smaller |ρ′|. In addition, compared to |π′|, Figs. 5c and 5d reveal that small |θc′| can also be utilized as a suitable index for obtaining better retrievals.

Fig. 5.
Fig. 5.

The scatterplots of |π′| and |θc′| at each grid point on the first layer with respect to the root-mean-square error (RMSE) in the retrieved 〈π′〉 and ⟨θc′⟩ profiles using the procedure described for TEST2, Method I: (a) |π′| vs RMSE of retrieved 〈π′〉 profile, (b) |π′| vs RMSE of retrieved ⟨θc′⟩ profile, (c) |θc′| vs RMSE of retrieved 〈π′〉 profile, and (d) |θc′| vs RMSE of retrieved ⟨θc′⟩ profile. Each data point is also colored based on its |ρ′|, whose value is illustrated by the color bar.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

Note that when using the EoS, (13) introduces an additional variable ρ′ which shows its importance from the experiments of TEST1 and TEST2, but is usually unknown except at the lowest surface grid points where the in situ measurements of pressure and temperature are taken. Therefore, it is desirable to find places where the influence from ρ′ on (13) is minimal. On the other hand, using the data generated by the model simulation introduced in section 2, Fig. 6 reveals that on the surface layer, |ρ′| is highly correlated with |θc′|, with smaller |ρ′| associated with smaller |θc′|. This is consistent with the analyses in Panofsky and Dutton (1984, p. 22). The strong correlation between |θc′| and |ρ′| at the surface layer explains why the results from TEST2 (see Fig. 5) reveal that the possibility of getting better retrievals of 〈π′〉 and ⟨θc′⟩ profiles is higher if the upward vertical integrations are performed starting from those surface grid points where the |θc′| s are smaller (thus smaller |ρ′|).

Fig. 6.
Fig. 6.

The scatterplot of |θc′| with respect to |ρ′| on the lowest surface layer based on the data from the model simulation described in section 2.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

b. Method II and results

However, a close examination of the results produced by the aforementioned TEST2 and Method I procedures finds that the surface grid point from which the vertical integration yields the best retrievals does not necessarily have the smallest |θc′| on the entire surface layer. As a result, Method II, which is a modified version of Method I, is developed as explained below.

It is known that the 〈π′〉 and ⟨θc′⟩ of the lowest first layer can be inferred by combining the in situ observations of π′ and θc′ at a single point on that layer with the G78-retrieved π′ − 〈π′〉 and θc′−⟨θc′⟩ at the same grid point. This set of 〈π′〉 and ⟨θc′⟩ can be further applied to all other grid points on the first layer to get the π′ and θc′ over the entire layer. With this information, the upward vertical integrations start from only those grid points on the first layer whose |θc′| are smaller than a certain threshold, and end at the top of the domain. Then, an average is made over the multiple profiles produced by each vertical integration starting from each of those selected points. Note that in real case applications, observational uncertainty of in situ measurements of π′ and θc′ can also cause random errors in the computed |θc′|. Thus, by averaging the multiple profiles produced by Method II can be another reason to support the use of Method II over Method I.

A series of tests are conducted to determine the proper threshold. Figure 7 illustrates the retrieved profiles from Method II at T = 70 min of the model simulation. The threshold used to select the surface grid points for the upward integration is that their |θc′| s are the smallest 10% of all the grids on the first layer. It is demonstrated that the profiles of 〈π′〉 and ⟨θc′⟩ are retrieved with good accuracy by using Method II but start to deviate from the true ones on the middle and high levels.

Fig. 7.
Fig. 7.

The true and retrieved: (a) 〈π′〉 and (b) ⟨θc′⟩ profiles obtained using Method II at T = 70 min of the model simulation. The threshold used to select the surface grid points for the upward integration is that their |θc′| values are the smallest 10% of all grid points on the surface layer. The solid and dashed lines indicate true and retrieved (retv) profiles, respectively.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

c. Method III and results

As shown in Fig. 7, notable differences between the retrieved 〈π′〉 and ⟨θc′⟩ profiles and the true solutions can be identified above approximately 5 km height. This is because the upward vertical integrations start from a group of selected surface grid points where the |θc′| s are smaller than a prescribed threshold, but this does not guarantee that the |θc′| s at the same locations (i.e., the same horizontal coordinates) on the middle and high levels aloft are still smaller than the same prescribed threshold. Therefore, in Method III a stepwise approach is proposed, in which the upward integrations are performed only within a limited depth (D) each time. At the end of the integrations, since the θc′ at all grid points on an intermediate level have already been retrieved, a new set of grid points whose |θc′| satisfy the same prescribed criterion at this height is found, from which the upward integrations continue for another distance D. This procedure is repeated until the stepwise upward vertical integrations reach the top of the domain.

Experiments in which a few integration depths within the range from 1.0 to 5.0 km were tested. It is found that the results are not very sensitive to the selection of the integration depth within this range. Thus, with the stepwise integration depth specified to be D = 1.0 km each time, Fig. 8 shows that the retrieved 〈π′〉 and ⟨θc′⟩ profiles on the middle and higher layers can be significantly improved compared with those shown in Fig. 7. It should also be mentioned that by substituting the retrieved 〈π′〉 and ⟨θc′⟩ into (12b), both sides of the equation can be balanced.

Fig. 8.
Fig. 8.

The true and retrieved: (a) 〈π′〉 and (b) ⟨θc′⟩ profiles obtained using Method III at T = 70 min of the model simulation with the stepwise upward integration depth specified to be 1.0 km. The solid and dashed lines stand for true and retrieved (retv) profiles, respectively.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

Using this set of retrieved 〈π′〉 and ⟨θc′⟩ profiles, Fig. 9 depicts the true and retrieved thermodynamic fields over a vertical cross section at Y = 30 km and T = 70 min using G78 technique and the Method III proposed in this study. It can be clearly seen that the vertical structure of the absolute thermodynamic fields (π′ and θc′), instead of their deviations from their horizontal averages, are correctly resolved. A quantitative comparison indicates that the RMSE calculated over the three-dimensional space can be reduced from 0.084 to 0.025 J kg−1 K−1 for π′, and from 0.54 to 0.17 K for θc′, respectively. In addition, the three-dimensional spatial correlation coefficient (SCC) can also be improved from 0.83 to 0.98 for both thermodynamic variables. Finally, since π′ and θc′ are already retrieved, the three-dimensional air density perturbation ρ′ can also be computed by using (13) and is illustrated in Fig. 10. The agreement between the true and retrieved ρ′ fields also justifies the neglect of qυ′ and qc when applying (13).

Fig. 9.
Fig. 9.

The thermodynamic fields over an east–west-oriented vertical cross section at Y = 30 km and T = 70 min: (a),(b) the true π′ and θc′, (c),(d) the deviation of the π′ and θc′ from their horizontal average retrieved by G78 method, and (e),(f) the retrieved π′ and θc′ after using Method III. The contour intervals for π′ and θc′ are 0.1 J kg−1 K−1 and 1.0 K, respectively. Solid and dashed lines denote positive and negative values, respectively.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for the (a) true and (b) retrieved air density perturbation (ρ′) field. The value of ρ′ has been multiplied by 103 kg m−3. The contour lines are plotted every 5 × 10−3 kg m−3. Solid and dashed lines denote positive and negative values, respectively.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

Compared with sounding data, surface stations are able to take pressure and temperature measurements at high temporal resolutions reaching a few minutes. Therefore, the surface observations are always available for resolving the vertical structure ambiguity problem once the G78 thermodynamic retrieval is completed using multiple-Doppler synthesized wind fields. Figure 11 depicts the retrieved 〈π′〉 and ⟨θc′⟩ profiles at T = 40 min and T = 100 min, which are 30 min earlier and later than those shown in Fig. 8, and represent the developing and decaying stages of the storm, respectively. It can be seen that true profiles at different times can be recovered reasonably well at a high temporal resolution. However, the errors in the retrieved 〈π′〉 profile at higher altitudes illustrated in Fig. 11a are relatively pronounced. Yet, the correct vertical trend of the 〈π′〉 profile is still well captured.

Fig. 11.
Fig. 11.

The true (solid lines) and retrieved (dashed lines; retv) profiles using Method III: (a),(b) 〈π′〉 and ⟨θc′⟩ profiles at T = 40 min; and (c),(d) 〈π′〉 and ⟨θc′⟩ profiles at T = 100 min.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

6. Observational errors and multiple surface stations

Although the new method proposed in this study requires only the in situ pressure and temperature measurements from a single surface station, in real scenarios it would not be uncommon for multiple surface stations to exist in the analysis domain. Thus, in this experiment, 20 virtual surface meteorological stations are randomly deployed in the domain, and at each station the observational errors within the range ±0.1 hPa and ±0.1 K are added to the pressure and temperature measurements, respectively. Furthermore, to imitate the uncertainty embedded in F, G, and H [see (3)–(5)] caused by various factors such as the observational errors of radar data and the uncertainty of using wind fields to compute the middle terms of (3)–(5), before they are inputted to the G78 retrieval algorithm the F and G are randomly perturbed by up to 5% of their magnitudes, and H is perturbed by up to 10% of its magnitude.

Using observational data from each surface station and Method III, we retrieve a set of 〈π′〉 and 〈θ′〉 profiles. Therefore, 20 sets of profiles are obtained, which are shown in Fig. 12 along with their average. Figure 12 illustrates that by averaging the 20 retrieved vertical profiles can effectively reduce the errors and produce a final set of 〈π′〉 and ⟨θc′⟩ profiles with reasonable accuracy.

Fig. 12.
Fig. 12.

Twenty retrieved (a) 〈π′〉 and (b) ⟨θc′⟩ profiles (light gray) at T = 70 min obtained by Method III with the upward integrations starting from 20 surface points where the in situ pressure and temperature are measured with observational errors. The black dashed line represents their average, and the black solid line is the true profile.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

7. Problems and possible solutions associated with using real datasets and potential future extended applications

When applying the proposed method in this research to real case studies, the most critical issue is the radar data coverage. It is assumed in our idealized OSSE tests that complete multiple-Doppler synthesized winds are available, so that one can compute the F, G, and H fields throughout the three-dimensional space. However, in reality the radar data are available only within the multiple-Doppler synthesis regions, which typically would not extend all the way to the ground. This session discusses possible solutions.

The first author (YCL) has developed a system called WISSDOM (WInd Synthesis System using DOppler Measurements; Liou et al. 2012, 2014, 2019), which uses the radial velocities observed by multiple-Doppler radars to synthesize three-dimensional wind fields. By adopting the Immersed Boundary Method similar to Tseng and Ferziger (2003) to deal with the bottom boundary conditions, WISSDOM is capable of performing the wind synthesis directly over complex terrain. For data-void regions, WISSDOM takes wind fields from selected numerical model outputs to fill in the gaps. The selection procedures are explained in the following:

  1. Suppose one needs to use WISSDOM to conduct wind synthesis at T = T0. Thus, the Doppler radial velocity data from each radar closest to T = T0 are collected and available. If needed, the frozen turbulence assumption or more complicated advection-correction techniques such as Shapiro et al. (2010a,b) are applied to deal with the nonsimultaneity of radar data.

  2. From numerical model simulations, output the wind fields within a time window (e.g., ±4 h) centered at T = T0. The model simulations can also be from a large set of ensemble forecasts.

  3. Projecting the model wind fields from (ii) to the location of each radar, and produce the model-generated radial velocity data for each radar station.

  4. For each radar, comparing the model-generated radial wind fields within the time window against the real radar-observed radial wind fields measured at T = T0, then computing the RMSE, and finally making a summation for all radars.

  5. Choosing the model wind field with the smallest total RMSE as the background wind field to fill in the radar data-void regions.

Since the radar observations can provide rather accurate information about real wind field for the regions reachable by the radars, the above procedures are designed to search for the model wind field which is the most compatible one with the multiple-radar observations. This selected model wind field is adopted to fill in the radar data-void regions. In the real case situation, the WISSDOM will be applied to produce the three-dimensional wind fields, from which one can compute the F, G, and H fields over the entire domain.

In addition, the first author (YCL) also developed a system called Terrain-Permitting Thermodynamic Retrieval Scheme (TPTRS) by which one can directly use the wind fields from WISSDOM to retrieve the three-dimensional π′ − 〈π′〉 and θc′−⟨θc′⟩ fields immediately above complex terrain (Liou et al. 2019). This is achieved by employing the quasi-Newtonian conjugate-gradient algorithm to minimize the cost function. Thus, one does not need to use the G78 method to solve a Poisson equation for the pressure perturbation on a horizontal plane within which “holes” with irregular boundaries exist due to the penetration of the mountains through the analysis domain. It is proposed to combine Method III designed in this study with WISSDOM and TPTRS to perform the thermodynamic retrieval.

In theory, the vertical integration designed in Method III can start from any level, and be conducted along not only the upward but also the downward direction. Thus, the in situ pressure and temperature observations can be obtained from a mountain station located at a height where the radar data coverage may be more complete than that in lower layers. The vertical integrations can be performed upward and downward from this height. Figure 13 demonstrates the concept of our design. Similarly, if an airplane with radars on board (e.g., NCAR ELDORA) is equipped with temperature and pressure sensors, then Method III with downward vertical integration can also be applied to airborne dual-Doppler radar data to compute the unknown 〈π′〉 and ⟨θc′⟩ profiles in the retrieved thermodynamic fields. Tsai et al. (2023) derived high-spatial-resolution (50-m) three-dimensional low level (<3 km) wind fields under clear-air condition by using a newly modified WISSDOM and data from scanning Doppler lidars. An extension of their work can be the use of TPTRS and Method III developed in this study to infer the three-dimensional thermodynamic fields in the boundary layer.

Fig. 13.
Fig. 13.

The conceptual diagram of combining WISSDOM, TPTRS, and Method III. The relative positions of radars, dual-Doppler lobes, terrain, and a mountain station (represented by a red dot), where the in situ pressure and temperature observations can be taken, are displayed.

Citation: Monthly Weather Review 151, 6; 10.1175/MWR-D-23-0001.1

8. Summary

By applying the EoS to bridge the temperature and pressure perturbations, the vertical ambiguity that exists in the G78-retrieved thermodynamic variables with the form of an unknown constant on each layer can be explicitly estimated and removed. The only in situ observations needed to start the computation are the pressure and temperature measured at a single surface station. In addition, a strategy is also suggested for when multiple surface stations are available. With the unknown constant estimated and adjusted at each horizontal layer, the complete three-dimensional thermodynamic fields with correct vertical structures can be obtained. Problems and possible solutions arising from using real datasets as well as potential future extended applications of the new method are discussed. Testing the new method proposed in this study with real observational datasets is an ongoing work.

Complete high-resolution three-dimensional pressure and temperature fields with correct horizontal and vertical structures would be very useful in diagnosing the thermodynamic characteristics of heavy rainfall systems. They can also be considered as another type of “observations” and assimilated into a numerical model to improve its forecasting skill (Ke et al. 2022).

Acknowledgments.

This research was supported by the National Science and Technology Council of Taiwan under 110-2625-M-008-001, 110-2111-M-008-032, and 111-2111-M-008-020.

Data availability statement.

The datasets used in this research are generated through numerical simulations performed by the Weather Research and Forecasting (WRF) Model, which is developed by the National Center for Atmospheric Research (NCAR). The model setups and simulation results are available from the corresponding author Dr. Yu-Chieng Liou (tyliou@atm.ncu.edu.tw).

APPENDIX

Derivation of the Equation of State in Perturbation Form

Starting from the Equation of State (EoS) for moist air,
P=ρRT(1+0.61qυ),
where P is the pressure, ρ represents the air density, T stands for the temperature, qυ denotes the mixing ratio of water vapor, and R (=287 J kg−1 K−1) refers to the gas constant for dry air.
The parameters in (A1) can be decomposed into the summation of their basic states and perturbations, represented by a subscript “0” and a prime, respectively:
P′+P0=(ρ0+ρ′)R(T0+T′)(1+0.61qυ0+0.61qυ′).
Note that the basic states are horizontally homogeneous, and vary only with height. Assume that the basis state also satisfies the following formula:
P0=ρ0RT0(1+0.61qυ0).
By subtracting (A3) from (A2), the temperature perturbation can be written as
T′=P′−RT0(ρ′+0.61ρ′qυ+0.61qυ′ρ0)ρR(1+0.61qυ).
The perturbations of potential temperature and Exner function can also be expressed by
θ′=θ−θ0=T(P00P)κ−T0(P00P0)κ=(T0+T′)(P00P0+P′)κ−T0(P00P0)κ=(P00P0)κT′−θ0P0P′κ
and
π′=π−π0=Cp(PP00)κ−Cp(P0P00)κ=Cp(P0+P′P00)κ−Cp(P0P00)κ=Cp(P0P00)κ(P′P0),
where P00 = 1000 hPa and κ = R/Cp. Note that the last equal sign replaces an approximation sign.
From (A6), one obtains the pressure perturbation:
P′=π′Cp(P00P0)κP0κ.
The virtual cloud potential temperature perturbation ( θc′) shown by (6) in this context is
θc′=θ′+(0.61qυ′−qc)θ0.
Substituting (A5) into (A8), one arrives at
θc′=(P00P0)κT′−θ0P0P′κ+(0.61qυ′−qc)θ0.
Finally, substituting (A4) and (A7) into (A9) to replace T′ and P′, we have
θc′=T1˜π′+T˜2(0.61qυ′−qc)−T3˜ρ′−T4˜qυ′,
where
T1˜=[(P00P0)κ1ρR(1+0.61qυ)−RCpθ0P0]P0R(P00P0)κ∼[(P00P0)κ1ρ0R(1+0.61qυ0)−RCpθ0P0]P0R(P00P0)κ,
T˜2=θ0,
T3˜=(P00P0)κT0ρ∼(P00P0)κT0ρ0,
T4˜=(P00P0)κ0.61T0ρ0ρ(1+0.61qυ)∼(P00P0)κ0.61T01+0.61qυ0.
Note that ρ and qυ in the denominator of (A11a), (A11c), and (A11d) are approximated by ρ0 and qυ0, respectively, so that terms T1˜, T˜2, T3˜, and T4˜ all become functions of height only.

REFERENCES

  • Carlin, J. T., A. V. Ryzhkov, J. C. Snyder, and A. Khain, 2016: Hydrometeor mixing ratio retrievals for storm-scale radar data assimilation: Utility of current relations and potential benefits of polarimetry. Mon. Wea. Rev., 144, 2981–3001, https://doi.org/10.1175/MWR-D-15-0423.1.

    • Search Google Scholar
    • Export Citation

  • Foerster, M. A., and M. M. Bell, 2017: Thermodynamic retrieval in rapidly rotating vortices from multiple-Doppler radar data. J. Atmos. Oceanic Technol., 34, 2353–2374, https://doi.org/10.1175/JTECH-D-17-0073.1.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Gao, J., M. Xue, A. Shapiro, and K. K. Droegemeier, 1999: A variational method for the analysis of three-dimensional wind fields from two Doppler radars. Mon. Wea. Rev., 127, 2128–2142, https://doi.org/10.1175/1520-0493(1999)127<2128:AVMFTA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Geerts, B., and P. V. Hobbs, 1991: Organization and structure of clouds and precipitation on the mid-Atlantic coast of the United States. Part IV: Retrieval of the thermodynamic and cloud microphysical structures of a frontal rainband from Doppler radar data. J. Atmos. Sci., 48, 1287–1305, https://doi.org/10.1175/1520-0469(1991)048<1287:OASOCA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Hauser, D., F. Roux, and P. Amayenc, 1988: Comparison of two methods for the retrieval of thermodynamic and microphysical variables from Doppler radar measurements: Application to the case of a tropical squall line. J. Atmos. Sci., 45, 1285–1303, https://doi.org/10.1175/1520-0469(1988)045<1285:COTMFT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ke, C.-Y., K.-S. Chung, Y.-C. Liou, and C.-C. Tsai, 2022: Impact of assimilating radar data with 3D thermodynamic fields in an ensemble Kalman filter: Proof-of-concept and feasibility. Mon. Wea. Rev., 150, 3251–3273, https://doi.org/10.1175/MWR-D-22-0128.1.

    • Search Google Scholar
    • Export Citation

  • Kessler, E., 1969: On the Distribution and Continuity of Water Substance in Atmospheric Circulation. Meteor. Monogr., No. 32, Amer. Meteor. Soc., 84 pp.

  • Klemp, J. B., and R. B. Wilhelmson, 1978: Simulations of right- and left-moving storms produced through storm splitting. J. Atmos. Sci., 35, 1097–1110, https://doi.org/10.1175/1520-0469(1978)035<1097:SORALM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Lin, Y. J., T. C. 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.

    • Search Google Scholar
    • Export Citation

  • Lin, Y., P. S. Ray, and K. W. Johnson, 1993: Initialization of a modeled convective storm using Doppler radar-derived fields. Mon. Wea. Rev., 121, 2757–2775, https://doi.org/10.1175/1520-0493(1993)121<2757:IOAMCS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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. Climatol., 42, 1443–1454, https://doi.org/10.1175/1520-0450(2003)042<1443:ATVAFD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Liou, Y.-C., P.-C. Yang, and W.-Y. Wang, 2019: Thermodynamic recovery of the pressure and temperature fields over complex terrain using wind fields derived by multiple-Doppler radar synthesis. Mon. Wea. Rev., 147, 3843–3857, https://doi.org/10.1175/MWR-D-19-0059.1.

    • Search Google Scholar
    • Export Citation

  • Mewes, J. J., and A. Shapiro, 2002: Use of the vorticity equation in dual-Doppler analysis of the vertical velocity field. J. Atmos. Oceanic Technol., 19, 543–567, https://doi.org/10.1175/1520-0426(2002)019<0543:UOTVEI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Panofsky, H. A., and J. A. Dutton, 1984: Atmospheric Turbulence: Models and Methods for Engineering Applications. John Wiley and Sons, 424 pp.

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Roux, F., J. Testud, M. Payen, and B. Pinty, 1984: West African squall-line thermodynamic structure retrieved from dual-Doppler radar observations. J. Atmos. Sci., 41, 3104–3121, https://doi.org/10.1175/1520-0469(1984)041<3104:WASLTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., S. Ellis, and J. Shaw, 1995: Single-Doppler velocity retrievals with Phoenix II data: Clear air and microburst wind retrievals in the planetary boundary layer. J. Atmos. Sci., 52, 1265–1287, https://doi.org/10.1175/1520-0469(1995)052<1265:SDVRWP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., K. M. Willingham, and C. K. Potvin, 2010a: Spatially variable advection correction of radar data. Part I: Theoretical considerations. J. Atmos. Sci., 67, 3445–3456, https://doi.org/10.1175/2010JAS3465.1.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., K. M. Willingham, and C. K. Potvin, 2010b: Spatially variable advection correction of radar data. Part II: Test results. J. Atmos. Sci., 67, 3457–3470, https://doi.org/10.1175/2010JAS3466.1.

    • Search Google Scholar
    • Export Citation

  • Shimizu, S., K. Iwanami, R. Kato, N. Sakurai, T. Maesaka, K. Kieda, Y. Shusse, and S. Suzuki, 2019: Assimilation impact of the high-temporal-resolution volume scans on quantitative precipitation forecasts in a severe storm: Evidence from nudging data assimilation experiments with a thermodynamic retrieval method. Quart. J. Roy. Meteor. Soc., 145, 2139–2160, https://doi.org/10.1002/qj.3548.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.

  • Sun, J., and R. A. Houze Jr., 1992: Validation of a thermodynamic retrieval technique by application to a simulated squall line with trailing stratiform precipitation. Mon. Wea. Rev., 120, 1003–1018, https://doi.org/10.1175/1520-0493(1992)120<1003:VOATRT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Tsai, C.-L., K. Kim, Y.-C. Liou, and G. Lee, 2023: High-resolution 3D winds derived from a newly modified WISSDOM synthesis scheme using multiple Doppler lidars and observations. Atmos. Meas. Tech., 16, 845–869, https://doi.org/10.5194/amt-16-845-2023.

    • Search Google Scholar
    • Export Citation

  • Tseng, Y.-H., and J. H. 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.

    • Search Google Scholar
    • Export Citation

  • Weisman, M. L., and R. Rotunno, 2000: The use of vertical wind shear versus helicity in interpreting supercell dynamics. J. Atmos. Sci., 57, 1452–1472, https://doi.org/10.1175/1520-0469(2000)057<1452:TUOVWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Weygandt, S. S., A. Shapiro, and K. K. Droegemeier, 2002: Retrieval of model initial fields from single-Doppler observations of a supercell thunderstorm. Part II: Thermodynamic retrieval and numerical prediction. Mon. Wea. Rev., 130, 454–476, https://doi.org/10.1175/1520-0493(2002)130<0454:ROMIFF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., L. Wei, W. Gu, J. Gong, and Q. Zhao, 2010: A 3.5-dimensional variational method for Doppler radar data assimilation and its application to phased-array radar observations. Adv. Meteor., 2010, 797265, https://doi.org/10.1155/2010/797265.

    • Search Google Scholar
    • Export Citation
Save
Cover Monthly Weather Review
Monthly Weather Review

  • Carlin, J. T., A. V. Ryzhkov, J. C. Snyder, and A. Khain, 2016: Hydrometeor mixing ratio retrievals for storm-scale radar data assimilation: Utility of current relations and potential benefits of polarimetry. Mon. Wea. Rev., 144, 2981–3001, https://doi.org/10.1175/MWR-D-15-0423.1.

    • Search Google Scholar
    • Export Citation

  • Foerster, M. A., and M. M. Bell, 2017: Thermodynamic retrieval in rapidly rotating vortices from multiple-Doppler radar data. J. Atmos. Oceanic Technol., 34, 2353–2374, https://doi.org/10.1175/JTECH-D-17-0073.1.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Gao, J., M. Xue, A. Shapiro, and K. K. Droegemeier, 1999: A variational method for the analysis of three-dimensional wind fields from two Doppler radars. Mon. Wea. Rev., 127, 2128–2142, https://doi.org/10.1175/1520-0493(1999)127<2128:AVMFTA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Geerts, B., and P. V. Hobbs, 1991: Organization and structure of clouds and precipitation on the mid-Atlantic coast of the United States. Part IV: Retrieval of the thermodynamic and cloud microphysical structures of a frontal rainband from Doppler radar data. J. Atmos. Sci., 48, 1287–1305, https://doi.org/10.1175/1520-0469(1991)048<1287:OASOCA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Hauser, D., F. Roux, and P. Amayenc, 1988: Comparison of two methods for the retrieval of thermodynamic and microphysical variables from Doppler radar measurements: Application to the case of a tropical squall line. J. Atmos. Sci., 45, 1285–1303, https://doi.org/10.1175/1520-0469(1988)045<1285:COTMFT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ke, C.-Y., K.-S. Chung, Y.-C. Liou, and C.-C. Tsai, 2022: Impact of assimilating radar data with 3D thermodynamic fields in an ensemble Kalman filter: Proof-of-concept and feasibility. Mon. Wea. Rev., 150, 3251–3273, https://doi.org/10.1175/MWR-D-22-0128.1.

    • Search Google Scholar
    • Export Citation

  • Kessler, E., 1969: On the Distribution and Continuity of Water Substance in Atmospheric Circulation. Meteor. Monogr., No. 32, Amer. Meteor. Soc., 84 pp.

  • Klemp, J. B., and R. B. Wilhelmson, 1978: Simulations of right- and left-moving storms produced through storm splitting. J. Atmos. Sci., 35, 1097–1110, https://doi.org/10.1175/1520-0469(1978)035<1097:SORALM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Lin, Y. J., T. C. 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.

    • Search Google Scholar
    • Export Citation

  • Lin, Y., P. S. Ray, and K. W. Johnson, 1993: Initialization of a modeled convective storm using Doppler radar-derived fields. Mon. Wea. Rev., 121, 2757–2775, https://doi.org/10.1175/1520-0493(1993)121<2757:IOAMCS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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. Climatol., 42, 1443–1454, https://doi.org/10.1175/1520-0450(2003)042<1443:ATVAFD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Liou, Y.-C., P.-C. Yang, and W.-Y. Wang, 2019: Thermodynamic recovery of the pressure and temperature fields over complex terrain using wind fields derived by multiple-Doppler radar synthesis. Mon. Wea. Rev., 147, 3843–3857, https://doi.org/10.1175/MWR-D-19-0059.1.

    • Search Google Scholar
    • Export Citation

  • Mewes, J. J., and A. Shapiro, 2002: Use of the vorticity equation in dual-Doppler analysis of the vertical velocity field. J. Atmos. Oceanic Technol., 19, 543–567, https://doi.org/10.1175/1520-0426(2002)019<0543:UOTVEI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Panofsky, H. A., and J. A. Dutton, 1984: Atmospheric Turbulence: Models and Methods for Engineering Applications. John Wiley and Sons, 424 pp.

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Roux, F., J. Testud, M. Payen, and B. Pinty, 1984: West African squall-line thermodynamic structure retrieved from dual-Doppler radar observations. J. Atmos. Sci., 41, 3104–3121, https://doi.org/10.1175/1520-0469(1984)041<3104:WASLTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., S. Ellis, and J. Shaw, 1995: Single-Doppler velocity retrievals with Phoenix II data: Clear air and microburst wind retrievals in the planetary boundary layer. J. Atmos. Sci., 52, 1265–1287, https://doi.org/10.1175/1520-0469(1995)052<1265:SDVRWP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., K. M. Willingham, and C. K. Potvin, 2010a: Spatially variable advection correction of radar data. Part I: Theoretical considerations. J. Atmos. Sci., 67, 3445–3456, https://doi.org/10.1175/2010JAS3465.1.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., K. M. Willingham, and C. K. Potvin, 2010b: Spatially variable advection correction of radar data. Part II: Test results. J. Atmos. Sci., 67, 3457–3470, https://doi.org/10.1175/2010JAS3466.1.

    • Search Google Scholar
    • Export Citation

  • Shimizu, S., K. Iwanami, R. Kato, N. Sakurai, T. Maesaka, K. Kieda, Y. Shusse, and S. Suzuki, 2019: Assimilation impact of the high-temporal-resolution volume scans on quantitative precipitation forecasts in a severe storm: Evidence from nudging data assimilation experiments with a thermodynamic retrieval method. Quart. J. Roy. Meteor. Soc., 145, 2139–2160, https://doi.org/10.1002/qj.3548.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.

  • Sun, J., and R. A. Houze Jr., 1992: Validation of a thermodynamic retrieval technique by application to a simulated squall line with trailing stratiform precipitation. Mon. Wea. Rev., 120, 1003–1018, https://doi.org/10.1175/1520-0493(1992)120<1003:VOATRT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • Tsai, C.-L., K. Kim, Y.-C. Liou, and G. Lee, 2023: High-resolution 3D winds derived from a newly modified WISSDOM synthesis scheme using multiple Doppler lidars and observations. Atmos. Meas. Tech., 16, 845–869, https://doi.org/10.5194/amt-16-845-2023.

    • Search Google Scholar
    • Export Citation

  • Tseng, Y.-H., and J. H. 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.

    • Search Google Scholar
    • Export Citation

  • Weisman, M. L., and R. Rotunno, 2000: The use of vertical wind shear versus helicity in interpreting supercell dynamics. J. Atmos. Sci., 57, 1452–1472, https://doi.org/10.1175/1520-0469(2000)057<1452:TUOVWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Weygandt, S. S., A. Shapiro, and K. K. Droegemeier, 2002: Retrieval of model initial fields from single-Doppler observations of a supercell thunderstorm. Part II: Thermodynamic retrieval and numerical prediction. Mon. Wea. Rev., 130, 454–476, https://doi.org/10.1175/1520-0493(2002)130<0454:ROMIFF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., L. Wei, W. Gu, J. Gong, and Q. Zhao, 2010: A 3.5-dimensional variational method for Doppler radar data assimilation and its application to phased-array radar observations. Adv. Meteor., 2010, 797265, https://doi.org/10.1155/2010/797265.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    The storm structure on a horizontal plane at Z = 1.0 km after (a) 30 and (b) 70 min of the model simulation. The arrows represent horizontal winds, the contour lines are vertical velocities (1.0 m s−1 interval) with updrafts (downdrafts) represented by solid (dashed) lines, and the color shading stands for radar reflectivity (in dBZ).

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    The storm structure on an X–Z vertical cross section at Y = 30 km after 70 min of the model simulation. The arrows represent the u–w winds, the contour lines are vertical velocities (3.0 m s−1 interval) with updrafts (downdrafts) represented by solid (dashed) lines, and the color shading stands for radar reflectivity (in dBZ).

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    As in Fig. 2, but for the storm structure over a Y–Z vertical cross section at X = 60 km, with the arrows representing the υ–w winds.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    The true and retrieved profiles using Method I: (a) true 〈π′〉; (b) retrieved 〈π′〉 profiles with the upward integration starting from a surface grid point located far from the main convection; and (c) as in (b), but with the upward integration starting from a surface grid point near the convection. The solid and dashed lines in (b) and (c) denote the results from TEST1 and TEST2, respectively. (d)–(f) As in (a)–(c), but illustrating the true and retrieved ⟨θc′⟩ profiles. Inside each figure panel, “Retv” stands for retrieved results, and F and N represent that the results are obtained at a grid point far from or near the storm, respectively.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    The scatterplots of |π′| and |θc′| at each grid point on the first layer with respect to the root-mean-square error (RMSE) in the retrieved 〈π′〉 and ⟨θc′⟩ profiles using the procedure described for TEST2, Method I: (a) |π′| vs RMSE of retrieved 〈π′〉 profile, (b) |π′| vs RMSE of retrieved ⟨θc′⟩ profile, (c) |θc′| vs RMSE of retrieved 〈π′〉 profile, and (d) |θc′| vs RMSE of retrieved ⟨θc′⟩ profile. Each data point is also colored based on its |ρ′|, whose value is illustrated by the color bar.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    The scatterplot of |θc′| with respect to |ρ′| on the lowest surface layer based on the data from the model simulation described in section 2.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The true and retrieved: (a) 〈π′〉 and (b) ⟨θc′⟩ profiles obtained using Method II at T = 70 min of the model simulation. The threshold used to select the surface grid points for the upward integration is that their |θc′| values are the smallest 10% of all grid points on the surface layer. The solid and dashed lines indicate true and retrieved (retv) profiles, respectively.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    The true and retrieved: (a) 〈π′〉 and (b) ⟨θc′⟩ profiles obtained using Method III at T = 70 min of the model simulation with the stepwise upward integration depth specified to be 1.0 km. The solid and dashed lines stand for true and retrieved (retv) profiles, respectively.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    The thermodynamic fields over an east–west-oriented vertical cross section at Y = 30 km and T = 70 min: (a),(b) the true π′ and θc′, (c),(d) the deviation of the π′ and θc′ from their horizontal average retrieved by G78 method, and (e),(f) the retrieved π′ and θc′ after using Method III. The contour intervals for π′ and θc′ are 0.1 J kg−1 K−1 and 1.0 K, respectively. Solid and dashed lines denote positive and negative values, respectively.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    As in Fig. 9, but for the (a) true and (b) retrieved air density perturbation (ρ′) field. The value of ρ′ has been multiplied by 103 kg m−3. The contour lines are plotted every 5 × 10−3 kg m−3. Solid and dashed lines denote positive and negative values, respectively.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    The true (solid lines) and retrieved (dashed lines; retv) profiles using Method III: (a),(b) 〈π′〉 and ⟨θc′⟩ profiles at T = 40 min; and (c),(d) 〈π′〉 and ⟨θc′⟩ profiles at T = 100 min.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Twenty retrieved (a) 〈π′〉 and (b) ⟨θc′⟩ profiles (light gray) at T = 70 min obtained by Method III with the upward integrations starting from 20 surface points where the in situ pressure and temperature are measured with observational errors. The black dashed line represents their average, and the black solid line is the true profile.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    The conceptual diagram of combining WISSDOM, TPTRS, and Method III. The relative positions of radars, dual-Doppler lobes, terrain, and a mountain station (represented by a red dot), where the in situ pressure and temperature observations can be taken, are displayed.

Fig. 1.

The storm structure on a horizontal plane at Z = 1.0 km after (a) 30 and (b) 70 min of the model simulation. The arrows represent horizontal winds, the contour lines are vertical velocities (1.0 m s−1 interval) with updrafts (downdrafts) represented by solid (dashed) lines, and the color shading stands for radar reflectivity (in dBZ).
Fig. 2.

The storm structure on an X–Z vertical cross section at Y = 30 km after 70 min of the model simulation. The arrows represent the u–w winds, the contour lines are vertical velocities (3.0 m s−1 interval) with updrafts (downdrafts) represented by solid (dashed) lines, and the color shading stands for radar reflectivity (in dBZ).
Fig. 3.

As in Fig. 2, but for the storm structure over a Y–Z vertical cross section at X = 60 km, with the arrows representing the υ–w winds.
Fig. 4.

The true and retrieved profiles using Method I: (a) true 〈π′〉; (b) retrieved 〈π′〉 profiles with the upward integration starting from a surface grid point located far from the main convection; and (c) as in (b), but with the upward integration starting from a surface grid point near the convection. The solid and dashed lines in (b) and (c) denote the results from TEST1 and TEST2, respectively. (d)–(f) As in (a)–(c), but illustrating the true and retrieved ⟨θc′⟩ profiles. Inside each figure panel, “Retv” stands for retrieved results, and F and N represent that the results are obtained at a grid point far from or near the storm, respectively.
Fig. 5.

The scatterplots of |π′| and |θc′| at each grid point on the first layer with respect to the root-mean-square error (RMSE) in the retrieved 〈π′〉 and ⟨θc′⟩ profiles using the procedure described for TEST2, Method I: (a) |π′| vs RMSE of retrieved 〈π′〉 profile, (b) |π′| vs RMSE of retrieved ⟨θc′⟩ profile, (c) |θc′| vs RMSE of retrieved 〈π′〉 profile, and (d) |θc′| vs RMSE of retrieved ⟨θc′⟩ profile. Each data point is also colored based on its |ρ′|, whose value is illustrated by the color bar.
Fig. 6.

The scatterplot of |θc′| with respect to |ρ′| on the lowest surface layer based on the data from the model simulation described in section 2.
Fig. 7.

The true and retrieved: (a) 〈π′〉 and (b) ⟨θc′⟩ profiles obtained using Method II at T = 70 min of the model simulation. The threshold used to select the surface grid points for the upward integration is that their |θc′| values are the smallest 10% of all grid points on the surface layer. The solid and dashed lines indicate true and retrieved (retv) profiles, respectively.
Fig. 8.

The true and retrieved: (a) 〈π′〉 and (b) ⟨θc′⟩ profiles obtained using Method III at T = 70 min of the model simulation with the stepwise upward integration depth specified to be 1.0 km. The solid and dashed lines stand for true and retrieved (retv) profiles, respectively.
Fig. 9.

The thermodynamic fields over an east–west-oriented vertical cross section at Y = 30 km and T = 70 min: (a),(b) the true π′ and θc′, (c),(d) the deviation of the π′ and θc′ from their horizontal average retrieved by G78 method, and (e),(f) the retrieved π′ and θc′ after using Method III. The contour intervals for π′ and θc′ are 0.1 J kg−1 K−1 and 1.0 K, respectively. Solid and dashed lines denote positive and negative values, respectively.
Fig. 10.

As in Fig. 9, but for the (a) true and (b) retrieved air density perturbation (ρ′) field. The value of ρ′ has been multiplied by 103 kg m−3. The contour lines are plotted every 5 × 10−3 kg m−3. Solid and dashed lines denote positive and negative values, respectively.
Fig. 11.

The true (solid lines) and retrieved (dashed lines; retv) profiles using Method III: (a),(b) 〈π′〉 and ⟨θc′⟩ profiles at T = 40 min; and (c),(d) 〈π′〉 and ⟨θc′⟩ profiles at T = 100 min.
Fig. 12.

Twenty retrieved (a) 〈π′〉 and (b) ⟨θc′⟩ profiles (light gray) at T = 70 min obtained by Method III with the upward integrations starting from 20 surface points where the in situ pressure and temperature are measured with observational errors. The black dashed line represents their average, and the black solid line is the true profile.
Fig. 13.

The conceptual diagram of combining WISSDOM, TPTRS, and Method III. The relative positions of radars, dual-Doppler lobes, terrain, and a mountain station (represented by a red dot), where the in situ pressure and temperature observations can be taken, are displayed.
All Time Past Year Past 30 Days
Abstract Views 99 0 0
Full Text Views 1723 1259 553
PDF Downloads 564 173 14