Parametrized Fuzzy Parameters

The hydrometeor classification membership functions are parametrized either as 1D or 2D beta functions.

The evolution of the 2D membership beta function parameters y=y(ZH) where y=m, a, or b were optimized using the CSU approach.

The parameter dependencies are expressed as fifth order polynomials, with coefficients listed in Parametrized Evolution of the Parameter m of the 2D Membership Functions to Parametrized Evolution of the Parameter b of the 2D Membership Functions.

Parametrizing 2D membership function with polynomials simplifies the implemented software, as the approach is common to CSU and the JPOLE algorithms.

The following table shows the beta function parameters of all the 1D membership functions. It also gives insight into the 2D membership functions, expressing their projections at the reference levels of the auxiliary input variables. For example, the parameter values of 2D membership functions of MBF(Zdr, ZH) visualize the shape of the MBF at ZH = 0 dBZ.

Parameters for 1D and 2D Membership Functions at Specified Reference Planes of the Second Function Variable
MBF Hydrometeor Class ID m a b
MBF( ZH [dB]) 1D
- 1) rain 30.0 31.0 40.0
- 2) wet snow 25.0 21.0 40.0
- 3) snow 0.0 36.0 40.0
- 4) graupel/small hail 45.0 11.0 20.0
- 5) hail 57.5 14.0 20.0
- 6) hail, rain mixture 60.0 11.0 20.0
MBF(height [km]) 2D @ melting level height (MSL) = 2.5 km (warm season)
- 1) rain 0.0 2.3 5.0
- 2) wet snow 2.2 0.5 5.0
- 3) snow 10.0 7.7 60.0
- 4) graupel/small hail 10.0 9.5 60.0
- 5) hail 0.0 15.0 20.0
- 6) hail, rain mixture 0.0 3.0 20.0
MBF(height [km]) 1D @ melting level height < 0 km (cold season, storm height =5 km)
- 1) rain 0.0 0.5 5.0
- 2) wet snow 0.0 2.5 5.0
- 3) snow 10.0 15.0 20.0
- 4) graupel/small hail 10.0 15.0 20.0
- 5) hail 10.0 15.0 20.0
- 6) hail, rain mixture 10.0 15.0 20.0
MBF(Zdr [dB]) 2D
@ ZH = 30 dBZ 1) rain 0.5 0.7 10.0
@ ZH = 27.5 dBZ 2) wet snow 1.1 1.4 15.0
@ ZH = 0 dBZ 3) snow 0.5 0.75 8.0
@ ZH = 42.5 dBZ 4) graupel/small hail 0.43 0.9. 12.5
@ ZH = 7.5 dBZ 5) hail -0.76 1.28 15.9
@ ZH = 60.0 dBZ 6) hail, rain mixture 0.94 2.0 15.0
MBF(Kdp [dgr/km]) 2D
@ ZH = 30 dBZ 1) rain 0.05 0.2 11.1
@ ZH = 25 dBZ 2) wet snow 0.25 1.2 10.0
@ ZH = 7.5 dBZ 3) snow 0.0 0.25 10.0
@ ZH = 42.5 dBZ 4) graupel/small hail 0.27 0.77 12.5
@ ZH = 57.5 dBZ 5) hail 0.5 1.0 10.0
@ ZH = 60.0 dBZ 6) hail, rain mixture 2.37 2.40 15.0
MBF (RhoHV []) 1D
1) rain 1.00 0.04 10.0
- 2) wet snow 0.88 0.11 20.0
- 3) snow 1.0 0.06 10.0
- 4) graupel/small hail 0.96 0.04 10.0
- 5) hail 0.90 0.045 10.0
- 6) hail, rain mixture 0.80 0.13 30.0
In the following tables, parametrization is expressed with 5th order polynomials of the difference with respect to the reference planeo of the quoted auxiliary observable. The polynomial coefficients P0 ... P4 are in increasing order, P0 values shown in Parameters for 1D and 2D Membership Functions at Specified Reference Planes of the Second Function Variable. The Max/Min column expresses the boundaries that the parametrized polynomial values must fall within.
Parametrized Evolution of the Parameter m of the 2D Membership Functions
m ID P1 P2 P3 P4 Max/Min
MBF(height [km]) (warm season)
- 1) 0.0 0 0 0 -
- 2) 0.0 0 0 0 -
ML-2.5 3) 1.0 0 0 0 -
- 4) 0.0 0 0 0 -
- 5) 0.0 0 0 0 -
- 6) 0.0 0 0 0 -
MBF(Zdr [dB])
(ZH-30) ±30 dBZ 1) 4.75E-02 1.76E-03 6.99E-06- 4.50E-07 5.5/0.1
(ZH-27.5) ±17.5 dBZ 2) 4.62E-03 6.42E-04 2.22E-05 0.0000E+0 02.5/1.1
(ZH-0) ±327 dBZ 3) 0 0 0 0 0.5/-0.5
(ZH-42.5) ±12.5 dBZ 4) 3.37E-02 9.64E-04 0 0 1.5/0.155
(ZH-57.5) ±12.5 dBZ 5) -1.73E-02 1.51E-03 -8.33E-05 4.17E-06 5.5/0.1
(ZH- 60) ±10 dBZ 6) -5.41E-02 -5.71E-03 1.67E-04 0 5.5/0.1
MBF(Kdp [dgr/km])
(ZH-30) ±35 dBZ 1) -4.96E-03 2.06E-03 1.91E-04 3.99E-06 14/0.05
(ZH-27.5) ±17.5 dBZ 2) 0 0 0 0 0.25/0.25
(ZH-0) ±327 dBZ 3) 0 0 0 0 0/0
(ZH-42.5) ±12.5 dBZ 4) 6.87E-02 1.00E-03 -1.48E-040 0 2.0/0.3
(ZH-57.5) ±12.5 dBZ 5) 0 0 0 0 0.5/0.5
(ZH- 60) ±10 dBZ 6) 3.08E-01 8.57E-03 3.33E-04 0 10./0.0
Parametrized Evolution of the Parameter a of the 2D Membership Functions
a ID P1 P2 P3 P4 Max/Min
MBF(height [km]) (warm season)
- 1) 1.0 0 0 0 -
- 2) -1.0 0 0 0 -
ML-2.5 3) 0.0 0 0 0 -
- 4) -1.0 0 0 0 -
- 5) 0.0 0 0 0 -
- 6) 1.0 0 0 0 -
MBF(Zdr [dB])
(ZH-30) ±30 dBZ 1) 2.59E-02 4.36E-04 -4.89E-06 -5.62E-08 8/0.4
(ZH-27.5) ±17.5 dBZ 2) -9.11E-03 2.1429E-04 3.84E-05 0.0000E+00 2.5/1.1
(ZH-0) ±327 dBZ 3) 0 0 0 0 0.5/-0.5
(ZH-42.5) ±12.5 dBZ 4) 3.37E-02 9.64E-04 0 0 2.0/0.5
(ZH-57.5) ±12.5 dBZ 5) 5.78E-03 4.53E-03 2.78E-05 1.25E-05 0/-2
(ZH- 60) ±10 dBZ 6) 4.17E-03 -2.38E-02 -1.67E-04 1.50E-04 3.0/0.5
a ID P1 P2 P3 P4 Max/Min
MBF(Kdp [dgr/km])
(ZH-30) ±35 dBZ 1) 1.11E-02 2.95E-03 1.04E-04 6.11E-07 8/0.2
(ZH-27.5) ±17.5 dBZ 2) 0 0 0 0 0.4/0.4
(ZH-0) ±327 dBZ 3) 0 0 0 0 0.25/0.25
(ZH-42.5) ±12.5 dBZ 4) 6.87E-02 1.00E-03 -1.48E-04 0 2.0/0.3
(ZH-57.5) ±12.5 dBZ 5) 0 0 0 0 1/1
(ZH- 60) ±10 dBZ 6) 2.97E-01 8.86E-03 2.67E-04 0 10./0.1
Parametrized Evolution of the Parameter b of the 2D Membership Functions
b ID P1 P2 P3 P4 Max/Min
MBF(height [km]) (warm season)
- 1) 0 0 0 0 -
- 2) 0 0 0 0 -
ML-2.5 3) 0 0 0 0 -
- 4) 0 0 0 0 -
- 5) 0 0 0 0 -
- 6) 0 0 0 0 -
MBF(Zdr [dB])
(ZH-30) ±30 dBZ 1) 2.29E-01 5.49E-03 -5.83E-05 0 25/8
(ZH-27.5) ±17.5 dBZ 2) 0 0 0 0 20/5
(ZH-0) ±327 dBZ 3) 0 0 0 0 8/8
(ZH-42.5) ±12.5 dBZ 4) 0 0 0 0 12.5/12.5
(ZH-57.5) ±12.5 dBZ 5) 2.103E-18- 1.42E-01 2.47E-19 6.67E-04 17/7
(ZH- 60) ±10 dBZ 6) -5.92E-17 1.67E-02 5.92E-19 -6.67E-04 20/5
MBF(Kdp [dgr/km])
(ZH-30) ±35 dBZ 1) 2.29E-01 5.49E-03 -5.83E-05 0 16./9.
(ZH-27.5) ±17.5 dBZ 2) 0 0 0 0 10/10
(ZH-0) ±327 dBZ 3) 0 0 0 0 10/10
(ZH-42.5) ±12.5 dBZ 4) 2.57E-01 0 0 0 17.0/9
(ZH-57.5) ±12.5 dBZ 5) 0 0 0 0 10/10
(ZH- 60) ±10 dBZ 6) -1.67E-01 2.44E-17 6.67E-03 0 25/8.0