Difference between revisions of "RadxRate equations"

Latest revision as of 15:54, 2 February 2021

RadxRate estimates the precipitation rate at each gate within a three-dimensional radar volume. RadxRate includes several equations for estimating precipitation rate, which can be tuned by the user for a specific environment and precipitation type. All coefficients for the equations are edited in the rate parameter file. This page will walk through these equations.

Polarimetric-based estimates

R(Z)

R(Z) equations are a simple way of estimating precipitation rates from the horizontal reflectivity factor (Z) and take the following form:

[math]\displaystyle{ R(Z)=aZ^b }[/math]

Many variations of this equation exist, depending on the precipitation type (e.g., convective or stratiform, tropical or mid-latitude). An example of the estimated precipitation from northern Taiwan using a tropical version of the equation ([math]\displaystyle{ R(Z)=0.027366Z^{0.69444} }[/math]) is shown in the image below. Since this example is from the lowest elevation angle, the echoes within ~150 km should all correspond to rain. R(Z) coefficients have been estimated for dry snow ([math]\displaystyle{ R(Z)=0.0365Z^{0.625} }[/math]), but aren't shown here given the warm environment near Taiwan.

R(Z, ZDR)

R(Z) relationships are known to be overly simplistic, as reflectivity depends on both hydrometeor size and concentration and single polarization only captures hydrometeor size in one dimension. Including ZDR helps mitigate the impact of big drops on Z. R(Z, ZDR) relationships take the form:

[math]\displaystyle{ R(Z,Z_{DR}) = aZ^bZ_{DR}^c }[/math]

An example of the estimated precipitation using coefficients from Berkowitz et al. (2013) is shown below ([math]\displaystyle{ R(Z,Z_{DR}) = 0.0067Z^{0.927}Z_{DR}^{-3.43} }[/math]).

R(KDP)

KDP has been used to estimate precipitation due to its relationship to the rain water content that slows down the propagating wave. Benefits of R(KDP) algorithms include insensitivity to radar calibration, attenuation, partial beam blockage, and ground clutter (Zrnić and Ryzhkov 1996), though care must be taken in the choice of filters used to calculate KDP and in Mie-scattering regimes. In RadxRate, R(KDP) takes the following form:

[math]\displaystyle{ R(K_{DP}) = sign(K_{DP})a\left|K_{DP}\right|^b }[/math]

An example of the estimated precipitation using coefficients from Berkowitz et al. (2013) is shown below ([math]\displaystyle{ R(K_{DP}) = sign(K_{DP})44.0\left|K_{DP}\right|^{0.822} }[/math]). Note that R(KDP) output has more missing data than other algorithms.

R(KDP, ZDR)

Like R(Z, ZDR) algorithms, R(KDP, ZDR) algorithms combine the rain water content and particle size information that can be inferred from KDP and ZDR. In LROSE, these algorithms take the following form:

[math]\displaystyle{ R(K_{DP},Z_{DR}) = sign(K_{DP})a\left|K_{DP}\right|^bZ_{DR}^c }[/math]

An example of the estimated precipitation using coefficients from Bringi and Chandrasekar (2001) and Brandes et al. (2002) is shown below ([math]\displaystyle{ R(K_{DP},Z_{DR}) = sign(K_{DP})90.8\left|K_{DP}\right|^{0.93}Z_{DR}^{-2.86} }[/math]). Note that R(KDP, ZDR) output has more missing data than other algorithms.

PID-based estimates

In addition to the aforementioned relationships that estimate precipitation rates directly from polarimetric variables, RadxRate includes relationships that prescribe different relationships based on the dominant hydrometeor type inferred from the polarimetric radar data. For example, these PID-based algorithms will use different equations for gates dominated by large raindrops and dry snow. All require the PID algorithm to be applied to the radar volume, which is done in an earlier step of RadxRate.

NCAR Hybrid

The NCAR Hybrid method applies some of the aforementioned precipitation rate equations to different PID categories. The method is described in the logical decision tree below (Dixon et al. 2015) and an example of output using the NCAR Hybrid method is also shown (right image).

NCAR Weighted-PID

The NCAR Weighted-PID method applies some of the aforementioned precipitation rate equations to different PID categories, which are weighted by the interest values assigned by the PID algorithm. The method is described in the logical decision tree below (Dixon et al. 2015) and an example of the NCAR Weighted-PID method is also shown (right image).

CSU HIDRO

Like the NCAR Hybrid method, the CSU HIDRO method applies the aforementioned precipitation rate equations to different PID categories. The method is described in the logical decision tree below (Cifelli et al. 2011) and an example of the CSU HIDRO method is also shown (right image).

Bringi

Like the NCAR Hybrid method, the Bringi method applies the aforementioned precipitation rate equations to different PID categories. The method is described in the logical decision tree below (Bringi et al. 2009) and an example of the Bringi method is also shown (right image). We note that the equations mentioned in the linked citation were optimized for a C-band radar.

References

Berkowitz, D. S., J. A Schultz, S. Vasiloff, K.L. Elmore, C.D. Payne and J.B. Boettcher, 2013: Status of Dual Pol QPE in the WSR-88D Network. AMS 27th conference on hydrology, Austin, Texas, 2.2. Link

Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41, 674–685. Link

Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar. Principles and Applications. Cambridge University Press, 636 pp. Book

Bringi, V. N., Williams, C. R., Thurai, M., & May, P. T. (2009). Using Dual-Polarized Radar and Dual-Frequency Profiler for DSD Characterization: A Case Study from Darwin, Australia, Journal of Atmospheric and Oceanic Technology, 26(10), 2107-2122. Link

Cifelli, R., Chandrasekar, V., Lim, S., Kennedy, P. C., Wang, Y., & Rutledge, S. A. (2011). A New Dual-Polarization Radar Rainfall Algorithm: Application in Colorado Precipitation Events, Journal of Atmospheric and Oceanic Technology, 28(3), 352-364. Link

Dixon, M. J., J. W. Wilson, T. M. Weckwerth, D. Albo, and E. J. Thompson, 2015: A dual-polarization QPE method based on the NCAR particle ID algorithm: Description and preliminary results. 37th Conf. on Radar Meteorology, Norman, OK, Amer. Meteor. Soc., 9A.1. Link

Ryzhkov, A. V., Giangrande, S. E., & Schuur, T. J. (2005). Rainfall Estimation with a Polarimetric Prototype of WSR-88D, Journal of Applied Meteorology, 44(4), 502-515. Link

Zrnić, D. S., and A. Ryzhkov. 1996. Advantages of rain measurements using specific differential phase. J. Atmos. Oceanic Technol. 13:454–464. Link

@@ Line 1: / Line 1: @@
 === '''Overview''' ===
-The goal of RadxRate is to estimate the precipitation rate at each gate within a three-dimensional radar volume. RadxRate includes several equations for estimating precipitation rate, which can be tuned according to the specific environment and precipitation type. This page will walk through these equations.
+RadxRate estimates the precipitation rate at each gate within a three-dimensional radar volume. RadxRate includes several equations for estimating precipitation rate, which can be tuned by the user for a specific environment and precipitation type. All coefficients for the equations are edited in the [http://wiki.lrose.net/index.php/RadxRate#4.29_Rate-specific_parameter_file rate] parameter file. This page will walk through these equations.
 ===== '''Polarimetric-based estimates''' =====
 ====== '''R(Z)''' ======
-Probably the most straightforward method of estimating precipitation rates, precipitation (mostly rainfall) can be estimated using an equation in the following form:
+R(Z) equations are a simple way of estimating precipitation rates from the horizontal reflectivity factor (Z) and take the following form:
-Many variations of this equation exist, depending on the precipitation conditions (e.g., convective vs stratiform, tropical vs midlatitude). An example of the estimated precipitation using a tropical version of this equation from northern Taiwan is shown in the image below. Since this particular example is from the lowest elevation angle, the echoes within ~150 km should all correspond to rain. Coefficients for Z-R relationships have been developed for dry snow, but aren't shown here given the warm environment near Taiwan.
+<div style='text-align: center;'><math>R(Z)=aZ^b</math></div>
+Many variations of this equation exist, depending on the precipitation type (e.g., convective or stratiform, tropical or mid-latitude). An example of the estimated precipitation from northern Taiwan using a tropical version of the equation (<math>R(Z)=0.027366Z^{0.69444}</math>) is shown in the image below. Since this example is from the lowest elevation angle, the echoes within ~150 km should all correspond to rain. R(Z) coefficients have been estimated for dry snow (<math>R(Z)=0.0365Z^{0.625}</math>), but aren't shown here given the warm environment near Taiwan.
 [[File:Rate_rzh.png|600px]]
 ====== '''R(Z, ZDR)''' ======
-Z-R relationships are known to be overly simplistic, as reflectivity depends on both hydrometeor size and concentration and single polarization only captures hydrometeor size in one dimension. Relationships that take both Z and ZDR, and thus the shape of particles, into account have been developed, which take the form:
+R(Z) relationships are known to be overly simplistic, as reflectivity depends on both hydrometeor size and concentration and single polarization only captures hydrometeor size in one dimension. Including ZDR helps mitigate the impact of big drops on Z. R(Z, ZDR) relationships take the form:
+<div style='text-align: center;'><math>R(Z,Z_{DR}) = aZ^bZ_{DR}^c</math></div>
-An example of the estimated precipitation using coefficients from [http://wiki.lrose.net/index.php/RadxRate_equations#References Berkowitz et al. (2013)] is shown below.
+An example of the estimated precipitation using coefficients from [http://wiki.lrose.net/index.php/RadxRate_equations#References Berkowitz et al. (2013)] is shown below (<math>R(Z,Z_{DR}) = 0.0067Z^{0.927}Z_{DR}^{-3.43}</math>).
 [[File:Rate_rzhzdr.png|600px]]
 ====== '''R(KDP)''' ======
-KDP has also been used to estimate precipitation rates due to its relationship to the rain water content that slows down the propagating wave. RadxRate enables the use of KDP in precipitation rate estimation using the following form:
+KDP has been used to estimate precipitation due to its relationship to the rain water content that slows down the propagating wave. Benefits of R(KDP) algorithms include insensitivity to radar calibration, attenuation, partial beam blockage, and ground clutter ([http://wiki.lrose.net/index.php/RadxRate_equations#References Zrnić and Ryzhkov 1996]), though [http://wiki.lrose.net/index.php/KDP_estimation care must be taken] in the choice of filters used to calculate KDP and in Mie-scattering regimes. In RadxRate, R(KDP) takes the following form:
-An example of the estimated precipitation using coefficients from [http://wiki.lrose.net/index.php/RadxRate_equations#References Berkowitz et al. (2013)] is shown below.
+<div style='text-align: center;'><math>R(K_{DP}) = sign(K_{DP})a\left|K_{DP}\right|^b</math></div>
+An example of the estimated precipitation using coefficients from [http://wiki.lrose.net/index.php/RadxRate_equations#References Berkowitz et al. (2013)] is shown below (<math>R(K_{DP}) = sign(K_{DP})44.0\left|K_{DP}\right|^{0.822}</math>). Note that R(KDP) output has more missing data than other algorithms.
 [[File:Rate_rkdp.png|600px]]
 ====== '''R(KDP, ZDR)''' ======
+Like R(Z, ZDR) algorithms, R(KDP, ZDR) algorithms combine the rain water content and particle size information that can be inferred from KDP and ZDR. In LROSE, these algorithms take the following form:
+<div style='text-align: center;'><math>R(K_{DP},Z_{DR}) = sign(K_{DP})a\left|K_{DP}\right|^bZ_{DR}^c</math></div>
+An example of the estimated precipitation using coefficients from [http://wiki.lrose.net/index.php/RadxRate_equations#References Bringi and Chandrasekar (2001)] and [http://wiki.lrose.net/index.php/RadxRate_equations#References Brandes et al. (2002)] is shown below (<math>R(K_{DP},Z_{DR}) = sign(K_{DP})90.8\left|K_{DP}\right|^{0.93}Z_{DR}^{-2.86}</math>). Note that R(KDP, ZDR) output has more missing data than other algorithms.
 [[File:Rate_rkdpzdr.png|600px]]
 ===== '''PID-based estimates''' =====
-In addition to the aforementioned relationships that estimate precipitation rates directly from polarimetric variables, RadxRate includes relationships that prescribe different relationships based on the dominant hydrometeor type inferred from the polarimetric radar data. That is, these PID-based algorithms will apply different equations to gates dominated by large raindrops and dry snow. All require the PID algorithm to be applied to the radar volume, which is done in an earlier step of RadxRate.
+In addition to the aforementioned relationships that estimate precipitation rates directly from polarimetric variables, RadxRate includes relationships that prescribe different relationships based on the dominant hydrometeor type inferred from the polarimetric radar data. For example, these PID-based algorithms will use different equations for gates dominated by large raindrops and dry snow. All require the PID algorithm to be applied to the radar volume, which is done in an earlier step of RadxRate.
 ====== '''NCAR Hybrid''' ======
-The NCAR Hybrid method applies some of the aforementioned precipitation rate equations to different PID categories. The method is described in the logical decision tree below ([http://wiki.lrose.net/index.php/RadxRate_equations#References Dixon et al. 2015]).
+The NCAR Hybrid method applies some of the aforementioned precipitation rate equations to different PID categories. The method is described in the logical decision tree below ([http://wiki.lrose.net/index.php/RadxRate_equations#References Dixon et al. 2015]) and an example of output using the NCAR Hybrid method is also shown (right image).
+[[File:Ncar_hybrid.png|400px]][[File:Rate_rhybrid_2.png|600px]]
-[[File:Ncar_hybrid.png|400px]]
+====== '''NCAR Weighted-PID''' ======
+The NCAR Weighted-PID method applies some of the aforementioned precipitation rate equations to different PID categories, which are weighted by the interest values assigned by the PID algorithm. The method is described in the logical decision tree below ([http://wiki.lrose.net/index.php/RadxRate_equations#References Dixon et al. 2015]) and an example of the NCAR Weighted-PID method is also shown (right image).
-An example of the NCAR hybrid method is shown below.
+[[File:Pid_weighted.png|400px]][[File:Rate_rpid.png|600px]]
-[[File:Rate_rhybrid_2.png|600px]]
 ====== '''CSU HIDRO''' ======
+Like the NCAR Hybrid method, the CSU HIDRO method applies the aforementioned precipitation rate equations to different PID categories. The method is described in the logical decision tree below ([http://wiki.lrose.net/index.php/RadxRate_equations#References Cifelli et al. 2011]) and an example of the CSU HIDRO method is also shown (right image).
+[[File:Hidro_hybrid.jpg|550px]][[File:Rate_rhidro.png|600px]]
-[[File:Rate_rhidro.png|600px]]
 ====== '''Bringi''' ======
+Like the NCAR Hybrid method, the Bringi method applies the aforementioned precipitation rate equations to different PID categories. The method is described in the logical decision tree below ([http://wiki.lrose.net/index.php/RadxRate_equations#References Bringi et al. 2009]) and an example of the Bringi method is also shown (right image). We note that the equations mentioned in the linked citation were optimized for a C-band radar.
-[[File:Rate_rbringi.png|600px]]
+[[File:Bringi_hybrid.gif|500px]][[File:Rate_rbringi.png|600px]]
 ===== '''References''' =====
 Berkowitz, D. S., J. A Schultz, S. Vasiloff, K.L. Elmore, C.D. Payne and J.B. Boettcher, 2013: Status of Dual Pol QPE in the WSR-88D Network. AMS 27th conference on hydrology, Austin, Texas, 2.2. [https://ams.confex.com/ams/93Annual/webprogram/Paper221525.html Link]
+Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41, 674–685. [https://doi.org/10.1175/1520-0450(2002)041%3C0674:EIREWA%3E2.0.CO;2 Link]
+Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar. Principles and Applications. Cambridge University Press, 636 pp. [https://doi.org/10.1017/CBO9780511541094 Book]
 Bringi, V. N., Williams, C. R., Thurai, M., & May, P. T. (2009). Using Dual-Polarized Radar and Dual-Frequency Profiler for DSD Characterization: A Case Study from Darwin, Australia, Journal of Atmospheric and Oceanic Technology, 26(10), 2107-2122. [https://doi.org/10.1175/2009JTECHA1258.1 Link]
@@ Line 57: / Line 86: @@
 Dixon, M. J., J. W. Wilson, T. M. Weckwerth, D. Albo, and E. J. Thompson, 2015: A dual-polarization QPE method based on the NCAR particle ID algorithm: Description and preliminary results. 37th Conf. on Radar Meteorology, Norman, OK, Amer. Meteor. Soc., 9A.1. [https://ams.confex.com/ams/37RADAR/webprogram/Paper275705.html Link]
+Ryzhkov, A. V., Giangrande, S. E., & Schuur, T. J. (2005). Rainfall Estimation with a Polarimetric Prototype of WSR-88D, Journal of Applied Meteorology, 44(4), 502-515. [https://doi.org/10.1175/JAM2213.1 Link]
+Zrnić, D. S., and A. Ryzhkov. 1996. Advantages of rain measurements using specific differential phase. J. Atmos. Oceanic Technol. 13:454–464. [https://doi.org/10.1175/1520-0426(1996)013%3C0454:AORMUS%3E2.0.CO;2 Link]