# [***Paper Review***] A Polarimetric Radar Forward Operator for Model Evaluation

# ***1.*** Introduction
## ***1.1*** 模式對流預報能力
- **短延時降水預報主要由對流降水預報能力主導**
Improved short-range forecasts of precipitation require improvements in the representation of convective events in numerical weather prediction.
(Fritsh et al. 1998)

- **解析度必須達到水平尺度 $100\ m$ 才能精確模擬對流動力**
Resolutions of at least 100m will be required to accurately simulate convective dynamics.
(Bryan et al. 2003; Craig and Dornbrack 2008)

- **高解析中尺度模式能直接模擬對流動力，表現較積雲參數法好**
High Resolution mesoscale models performed better than single column models with a convection scheme.
(Bechtold et al. 2000; Guichard et al. 2004)

- **當解析度越高，微物理參數法的結果會越顯著影響對流動力**
Microphysical processes such as the formation of heavily rimed ice hydrometeors (graupel and hail) feed back onto the storm dynamics.
(Brandes et al. 2006)

- **微物理參數法對水物形成、消散、交互作用的掌握程度，皆會顯著影響對流的強度及生命週期**
The lifecycle and intensity of convections strongly depend on the ability of the microphysical parameterization scheme to represent the processes of formation, decomposition, and interaction of the different hydrometeor types.

- **冰相粒子的掌握程度會顯著影響對流的形成及發展**
The formation and distribution of precipitation has been found to be extremely dependent on the treatment of ice phase hydrometeors.
(Ferrier et al. 1995; Gilmore et al. 2004; Colle et al. 2005; Garvert et al. 2005)

## ***1.2*** 模式 QPF 校驗
- **在模式QPF (主要為微物理參數法) 校驗方面，地面測站僅能提供二維降水資料，而雷達資料可以提供更高時空解析度的三維水物資訊**
When it comes to **QPF (micropysics scheme)** verification, radar can provide 3D information at a high temporal and spatial resolution while rain gauge can only offer 2D information.
(Keil 2003)

- **各雙偏極化參數間的組合，可以有效描述掃描體積內的主要水物種類**
Different polarimetric radar quantities offers the unique possibility of classifying the predominant hydrometeor type within the resolution volume.
(Holler et al. 1994; Vivekanandan et al. 1999; Zrnic et al. 2001)

- **雷達觀測參數與模式輸出參數不同** → **兩種校驗方法**
Polaremetric radar systems do not provide explicit measurements of the quantities appearing in the microphysical equations or models themselves.
→ observation-to-model approach & **model-to-observation approach (using *forward operator*)**

- **透過雙偏極化參數的校驗，可以比較模式與觀測間的微物理過程差異，檢驗降水強度及水物分布**
Comparison of synthetic and observed polarimetric quantities evaluate not only precipitation intensity but also micropysical processes, which includes hydrometeor spatial distribution and classification.

## ***1.3*** Radar Forward Operator
- **Radar Forward Operator 明確描述電磁波傳遞過程及水物散射過程**
Radar forward operator compute the scattering processes **explicitly** and also consider propagation effects.

- **為了得到較好的較驗結果，Forward Operator 的假設必須盡可能與模式達到一致**
For a successful evaluation of model physics, the link between the model and the forward operator has to conform as closely as possible to the model assumption.

- **Forward Operator 的自由參數與水物的形狀、密度、掉落特徵等有關**
The "free" parameters are mainly associated with the microphysical properties of precipitating ice hydrometeors, such as shape, density, and falling behavior.

- **利用敏感度試驗檢驗各個自由參數對雙偏極化參數的影響程度**
Sensitivity tests will be used to identify the subset of parameters that make the dominant contributions to the radar signal.

- **利用前人的研究 (Holler et al. 1994) 剃除不合理的自由參數組合**
The parameter ranges from a simple hydrometeor classification scheme (Holler et al. 1994) will be used to exclude unreasonable combinations of parameters.

# ***2.*** The polarimetric radar forward operator **SynPolRad**
## ***3 Stages in SynPolRad***


1. Calculate the **interactions** (scattering and attenuation) between radar beam and hydrometeors using **T-matrix**.
(Bringi et al. 1986)

2. Calculate the **propagation** (attenuation and refraction) of radar beam.
(following Hasse and Crewell 2000)

3. **Interpolate** observed and synthetic polarimetric radar quantities to the same coordinate system.

## ***2.1*** Calculation of the polarimetric quantities ($Z_{HH}$, $Z_{DR}$, and $L_{DR}$)
- Definitions of $Z_{HH}$, $Z_{DR}$, and $L_{DR}$:
$$
Z_{HH}=10\log(z_{HH})
$$
$$
Z_{DR}=10\log(\dfrac{z_{HH}}{z_{VV}})
$$
$$
L_{DR}=10\log(\dfrac{z_{HV}}{z_{HH}})
$$

- Relationship between **transmitted wave $\vec{E^i}$** and **scattered wave $\vec{E^s}$**:
$$
\vec{E^i}=\vec{S}\times\vec{E^i}
$$
$$
\left [
\begin{array}{cc}
E^s_V \\
E^s_H
\end{array}
\right ]=
\left [
\begin{array}{cc}
S_{VV} & S_{VH} \\
S_{HV} & S_{HH}
\end{array}
\right ]
\left [
\begin{array}{cc}
E^i_V \\
E^i_H
\end{array}
\right ]
$$
where $\vec{S}$ is the **backscatter matrix**.

- Definitions of $z_{HH}$, $z_{VV}$, and $z_{HV}$:
$$
z_{HH}=\dfrac{\pi^5|K|^2}{\lambda ^4}\int_{D}D^6N(D)dD\ \times\ \int_{\theta}|S_{11}\sin^2{\theta}+S_{22}\cos^2{\theta}|^2p(\theta)d\theta
$$
$$
z_{VV}=\dfrac{\pi^5|K|^2}{\lambda ^4}\int_{D}D^6N(D)dD\ \times\ \int_{\theta}|S_{11}\cos^2{\theta}+S_{22}\sin^2{\theta}|^2p(\theta)d\theta
$$
$$
z_{HV}=\dfrac{\pi^5|K|^2}{\lambda ^4}\int_{D}D^6N(D)dD\ \times\ \int_{\theta}|S_{11}-S_{22}|^2\cos^2{\theta}\sin^2{\theta}\ p(\theta)d\theta
$$

## ***2.2*** Calculation of the complex dielectric factor $K$
- **$|K|^2$ 值主要由粒子組成成分決定，其變化會顯著影響回波值**
Radar observations are highly sensitive to the complex dielectric factor, which varies significantly depending on the phase and composition of the scattering particle.

- **$|K|^2$ 亦為溫度和雷達波長的常數**
The complex dielectric factor also depends on temperature and wavelength.

- **$|K|^2$ 越大，雙偏極化訊號越明顯**
High dielectric factors cause larger depolarization, while polarimetric signatures are reduced for low-density ice hydrometers.
(Matrosov et al. 1996)

- **SynPolRad 假設所有混相粒子皆為冰/水/空氣均質混合而成**
Ice particles are taken to be homogeneous in SynPolRad.

- **在給定波長及溫度的情況下，先求出純冰相粒子 (Warren 1984) 及純水相粒子 (Ray 1972) 的$|K|^2$，再依據粒子密度及融化率求出混相粒子的$|K|^2$**
Dielectric factors are derived for pure ice (Warren 1984) and water (Ray 1972) at given temperature and wavelength. Then, $K$ is calculated for the given hydrometeor, taking into account its density and degree of melting.

## ***2.3*** Beam propagation and attenuation
### ***2.3.1*** Beam propagation
- **使用地球有效半徑**
Assume **effective Earth radius** $R_{eff}$:
$$
R_{eff}=\dfrac{4}{3}\times R_{E}
$$
where $R_{E}$ is the actual Earth radius.

- **地球有效半徑假設折射指數隨高度為線性遞減**
The approach lies in the assumption of a linear dependence of the atmosphere's refractive index on height.

### ***2.3.2*** Beam attenuation
- **Specific Attenuation** $k\ [dB/km]$

- **Specific Differential Attenuation** $A_{dp}\ [dB/km]$
which is defined as the difference in attenuation in the horizontal and vertical channel.

- **水物造成的衰減利用T-matrix計算**
The attenuation resulting from the presence of hydrometeors are calculated by the T-matrix.
(Vivekanandan et al. 1990)

- **空氣/水氣造成的衰減利用 Liebe et al. (1993) 的方法計算**
Gaseous attenuation due to the presence of molecular oxygen, water vapor, and nitrogen is computed employing the propagation model of Liebe et al. (1993).

- The attenuated polarimetric quantities can be calculated as
$$
Z_{HH}=Z^o_{HH}-2\int_0^rk\ dr
$$
$$
Z_{DR}=Z^o_{DR}-2\int_0^rA_{dp}\ dr
$$
$$
L_{DR}=L^o_{DR}+\int_0^rA_{dp}\ dr
$$
where r is the distance from the radar.
(Bringi and Chandrasekar 2001)

## ***2.4*** Interpolation of the observations
### ***2.4.1*** PPI/RHI Scan
1. **在模式網格點上計算各個合成雙偏極化參數**
Compute the polarimetric quantities at the grid points of the model.

2. **將合成雙偏極化參數內插至以合成雷達為中心的極座標，以計算折射、衰減等過程**
Interpolate the variables to the polar coordinate system for the simulation of beam propagation including attenuation along its path.

3. **將觀測及合成的雷達參數自極座標轉換回卡氏座標**
Both the observed and the synthetic polarimetric radar data are transferred back to the model grid for evaluation.

### ***2.4.2*** The reason to do "3." (Why not stop at 2.)
- **模擬的極座標解析度較觀測低**
The resolution of radar is finer than the model simulation.

- **觀測的次網格資訊和極值可以被平滑化**
The subgrid variability and extreme values of the observation can be smoothed.

- **卡式座標能簡化統計計算過程**
The regular horizontal resolution can simplify the computation of statistics.

- **模式中雷達在不同位置的模擬結果，可以更有效率的互相比較**
The comparison of simulated observations for different positions of the radar in the model domain can be more efficient.

# ***3.*** Application to the NWP model
## The link between SynPolRad and model
- **SynPolRad Inputs**:
- Directly from NWP model: **PSD**, **density**
- Free parameters: **shape** (axis ratio), **falling behavior** (canting angle $\theta$ ), **degree of melting** (dielectric factor $|K|^2$ )

- **Schematic Diagram** between SynPolRad and model:


- **For rain**:
- **DSD**: from microphysical schemes
- **Density**: $\rho=1\ g/cm^3$
- **Dielectric factor**: $|K|^2=0.93$ (water)
- **Axis ratio**: $\alpha=\alpha(D)$
(Andsager et al. 1999)
- **Canting Angle** $\theta$ : $\theta=10^\circ$, $\sigma_{\theta}=5^{\circ}$
(Chandrasekar et al. 1988; Straka et al. 2000)

## **3.1** Sensitivity of polarimetric quantities to microphysical properties of ice
- **The goal**:
Investigate the importance of individual input parameters for the polarimetric variables and to find relations or dependencies among them.

- **The input parameters**:
- **Temperature**: $T=-10^{\circ}C$

- **Wave Length**: $\lambda=5.45\ cm$ (C-band)
- **Free Parameters**: chosen to represent the full range of ice phase hydrometeors in the atmosphere

- **The results**
1. $N_0$:
- 回波隨 $N_0$ 變大而增大，增大幅度在 $N_0$ 很小時尤其明顯
- $LDR$ 和 $Z_{DR}$ 不受 $N_0$ 影響
- 平均 $Z_{DR}<0$ 乃是因為 $Axis\ Ratio$ 的範圍大部分 $>1$
- 

2. $D_0$:
- 回波隨 $D_0$ 變大而增大
- $LDR$ 不受 $D_0$ 影響
- $Z_{DR}$ 的極值在 $D_0$ 變大時有增大趨勢
- 

3. $\theta_{Max}$:
- 回波不隨 $\theta_{Max}$ 變化
- $LDR$ 隨 $\theta_{Max}$ 增加而變大，在 $\theta_{Max}=60^\circ$ 時達到最大
- $Z_{DR}$ 在 $\theta_{Max}$ 變大時，極值有變小趨勢
- $LDR$ 與 $Z_{DR}$ 為相對的概念：$LDR$ 描述粒子在極化方向的傾斜程度；$Z_{DR}$ 描述粒子在極化方向的排列整齊程度
- 

4. $Axis\ Ratio$:
- 回波在 $Axis\ Ratio=1$ 時有極大值，因球型粒子能最有效率填滿解析體積
- $LDR$ 隨粒子越偏離球形而增加
- $Z_{DR}$ 在 $Axis\ Ratio<1$ (扁平狀) 時 $<0$，在 $Axis\ Ratio>1$ 時 $>0$
- $Z_{DR}$ 有一條 $0$ 值線乃肇因於粒子傾斜時，在極化平面上有機會呈現圓形
- 

5. $Ice\ Density$
- 回波、$LDR$、$Z_{DR}$ 皆隨冰密度增加而更顯著
- 

- **To study the impact of varying dielectric factor (fucntion of ice density and water content)**:
- **固定參數**：$N_0=800\ mm^{-1}m^{-3}$、$D_0=5\ mm$、$\theta=40^\circ$、$Axis\ Ratio=0.3$
- **The Results**:
- 回波隨 $Water\ Portion$ 和 $Density$ 增大而變大，其中 $Water\ Portion$ 的影響較顯著
- 圖中鉛值線段，代表在給定密度下，空氣被水填滿後，$|K|^2$ 在其演算法下不改變的情形
- 

- **The summarize of sensitivity experiment**:
- 回波主要由 $PSD\ (N_0\ 、D_0)$ 和 $|K|^2\ (density\ 、degree\ of\ melting)$ 影響
- $LDR$ 及 $Z_{DR}$ 主要由 $canting\ angle\ 及\ axis\ ratio$ 影響
- $LDR$ 與 $Z_{DR}$ 為相對的概念：$LDR$ 描述粒子在極化方向的傾斜程度；$Z_{DR}$ 描述粒子在極化方向的排列整齊程度
- 回波、$LDR$、$Z_{DR}$ 皆隨著 $|K|^2$ 變大而增加

## ***3.2*** Determination of the free parameters for ice
- **利用 $LDR$ 跟 $Z_{DR}$ 的組合來分類水物**
Hydrometeor classification is based only on $LDR$ and $Z{DR}$.

- **利用 Holler et al. (1994) 中的水物分類方法，限制不同水物的 $LDR$ 及 $Z_{DR}$ 區間**
An additional constraint is provided by specification of typical thresholds for the polarimetric quantities $LDR$ and $Z_{DR}$ for different hydrometer types from a hydrometeor classification scheme.
(Holler et al. 1994)

- **Gains from sensitivity experiment**:
$LDR$ and $Z_{DR}$ mainly depend on $|K|^2$, $Axis\ Ratio$, and $\theta$.

- **因此某些固定的 $\theta$ / $Axis\ Ratio$ 組合可以使 $LDR$ 和 $Z_{DR}$ 在個別水物限制的區間中**
A pair of fixed $Axis\ Ratio$ and $\theta$ can be defined such that the resulting values of synthetic $LDR$ and $Z_{DR}$ will always range within the thresholds of the hydrometeor classification.

- **以 Graupel 作為求取 free parameters 的範例**
- **Graupel Assumption**:
- Dry / Density ($\rho=0.2\ g/cm^3$) provided by NWP model → $|K|^2$ is determined
- $Axis\ Ratio$ is independent to $Size$
- $N(D)$, $N_0$, 和 $\lambda$ 已由NWP model COSMO-DE 提供

- **在 Graupel 所限制的 $Z_{DR}$ 和 $LDR$ 過去個案中，作$\theta_{Max}-Axis\ Ratio-Hydrometeor$ 相位圖**:
- 
- 取 $Axis\ Ratio=0.5$, $\theta_{Max}=40^\circ$ 作為 Graupel 的 free parameter

- **各水物在SynPolRad輸入的參數**: $N(D)$, $N_0$, 和 $\lambda$ 已由NWP model 提供

## ***3.3*** Melting ice and brightband effects
- **利用不同比例的冰/水混和來模擬正在溶解的冰相粒子**
NWP models describe melting more crudely as a transition from snow to rain with coexisting snow and rain phase.

- **Illingworth 2004 指出，由於介電係數增加及粒子滾動，溶解層的 $LDR$ 會變大，$>-15\ dB$，因此 SynPolRad 透過調整 $\theta$ 和 $Water\ Portion$ 使溶解層的 $LDR$ 變大**
SynPolRad changes the free parameters in order to reach the typical $LDR$ vales of $-15\ dB$ (Illingworth 2004) resulting from the enhanced tumbling of melting particles.

- **$\theta$ 和 $Water\ Portion$ 改變後的值由敏感度實驗決定**:
- 敏感度實驗輸入**參數範圍** ($\theta_{Max}$、$\Delta\theta$、$Water\ Portion$):

- 敏感度實驗結果：
由前兩張圖決定 $\theta_{Max}=60^\circ$、$\Delta\theta=45^\circ$
由後兩張圖決定 $Water\ Portion=36\%$

# ***4.*** Testing SynPolRad
## ***4.1*** Case Description
- **Case**:
- **Time**: 12 August 2004
- **Type**: squall line

- **Model**:
- **Model Name**: COSMO-DE
- **Model Domain**: 100 $\times$ 100 $\times$ 40
- **Horizontal Resolution**: 2.8 km
- **Model Start Time**: 0000 UTC

- **Observation**:
- **Time**: 1900 UTC
- **Scan**: $1^\circ$ PPI scan by POLDIRAD
- **Reflectivity**:

## ***4.2*** Tests for Rain/Snow/Graupel respectively
- **The Goal**:
To test the consistency between SynPolRad and the NWP model.
- **Model output to SynPolRad**: $q_{rain}$, $q_{snow}$, $q_{graupel}$, and $q_{precipitation}$

- **The Results**:

- **Discusstion of the Results**:
- **Rain**:
- 成功模擬出兩個對流中心
( $Z=55\ dBZ$, $Z_{DR}=3\ dB$, $LDR=-25\ dB$ )
- 對流胞的上衝流使 $0^\circ C$ 線以上也有雨滴
- $Z_{DR}$ 較大的地方被分類為 $"Heavy\ Rain"$
- $0^\circ C$ 線以上被分類為 $"Snow"$ 乃是因 classification scheme 只用溫度來作為 $"Snow"$ 和 $"Rain"$ 的區分

- **Snow**:
- 輸入的 mixing ratio 較雨少，$Z$ 也較雨低
- 有模擬出雷達亮帶
- $Z_{DR}$ 和 $LDR$ 的極值出現在亮帶
( $Z_{DR}=2.5\ dB$, $LDR=-15\ dB$ )
- 因 $Z_{DR}$ 和 $LDR$ 在亮帶增大，影響到水物分類的結果

- **Graupel**:
- 雖然 mixing ratio 較雨多，但 $|K|^2$ 較小使的回波較低
- 有模擬出雷達亮帶
- $Z_{DR}$ 和 $LDR$ 的極值出現在亮帶
- 因 $Z_{DR}$ 和 $LDR$ 在亮帶增大，影響到水物分類的結果

## ***4.3*** Testing for the Whole
- **Comparison of PPI Scan**:
- 左圖為觀測，中圖與右圖為兩個不同雷達位置的模擬結果(三角形為雷達位置，有考慮衰減)
- 對流位置大致掌握，層狀降水強度高估、範圍低估

- **Comparison of RHI Scan**:
- **Reflectivity**:
- 水相粒子高估，冰相粒子低估
- 層狀區和亮帶高估
- 亮帶的高估使右圖對流區受衰減而嚴重低估
- 模式高估對流深度及亮帶高度

- $Z_{DR}$:
- 觀測中 $Z_{DR}$ 較大的區域在降水較強的地方及亮帶
- 觀測中對流後方因衰減使 $Z_{DR}$ 減小至$<0$
- 模擬中亮帶 $Z_{DR}$ 高估

- $LDR$:
- 觀測中 $LDR$ 在對流區及亮帶較強，前者因對流區內有較多的 $graupel$ 及 $hail$
- 模式中有掌握亮帶 $LDR$ 的趨勢
- 模式中沒有掌握對流區 $LDR$ 的趨勢，因模式並沒有模擬 $hail$ 或 $high-density\ graupel$

- **Hydrometeor Classification**:
- 觀測中，上層為雪、下層為雨，亮帶及對流區有 graupel 及 hail
- 模式在冰相粒子方面將雪誤判為軟雹 (主要肇因於 $LDR$ ) ，顯示模式微物理參數法尚無法有效掌握冰相粒子
- 模式並沒有預報 $hail$

# ***5.*** Discussion and conclusions
- 在單一水物分類上，SynPolRad 可以模擬出與 NWP model output 大致一致的結果，而這也是必須的，代表 SynPolRad 可以有效的以模擬雙偏極化參數反映模式微物理參數法的結果，進而與雷達觀測作比較
- SynPolRad 不只可以透過回波估計降水強度，也可以透過雙偏極化參數估計冰相粒子的分布情形，更能代表整個模式中的降水過程
- 由第***4.3***節的比較可以看出，預報模式對冰相粒子的掌握程度尚且不足
- SynPolRad 尚無法模擬 $K_{DP}$ ，而在雷達實務運作上 $K_{DP}$ 應較 $LDR$ 更為重要
- **SynPolRad 可以建立一套標準作業程序，仔細檢驗大量時間及空間的模式產物，更有效率檢驗微物理參數法的問題所在，進而改進以增進預報能力**