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A Melting Layer Model for Passive/Active Microwave Remote Sensing 
Applications- Part I: Model Formulation and Comparison with Observations 



William S. Olson 1 , Peter Bauer 2 , Nicolas F. Viltard\ Daniel E. Johnson 4 , 

and Wei-Kuo Tao 5 



'JCET, University of Maryland Baltimore County 

2 Deutsche Forschungsanstalt fuer Luft- und Raumfahrt 

3 Centre detudes des Environnements Terrestre et Planetaires 

4 Science Systems and Applications, Inc. 

laboratory for Atmospheres, NASA/Goddard Space Flight Center 



to be submitted to the Journal of Applied Meteorology 
January 4, 2000 



Abstract 

In this study, a 1-D steady-state microphysical model which describes the vertical 
distribution of melting precipitation particles is developed. The model is driven by the 
ice-phase precipitation distributions just above the freezing level at applicable gridpoints 
of "parent" 3-D cloud-resolving model (CRM) simulations. It extends these simulations 
by providing the number density and meltwater fraction of each particle in finely- 
separated size categories through the melting layer. The depth of the modeled melting 
layer is primarily determined by the initial material density of the ice-phase precipitation. 

The radiative properties of melting precipitation at microwave frequencies are 
calculated based upon different methods for describing the dielectric properties of mixed- 
phase particles. Particle absorption and scattering efficiencies at the Tropical Rainfall 
Measuring Mission Microwave Imager frequencies (10.65 to 85.5 GHz) are enhanced 
greatly for relatively small (-0.1) meltwater fractions. The relatively large number of 
partially-melted particles just below the freezing level in stratiform regions leads to 
significant microwave absorption, well-exceeding the absoiption by rain at the base of 
the melting layer. Calculated precipitation backscatter efficiencies at the Precipitation 
Radar frequency (13.8 GHz) increase in proportion to the particle meltwater fraction, 
leading to a "bright-band" of enhanced radar reflectivities in agreement with previous 
studies. 



The radiative properties of the melting layer are determined by the choice of 
dielectric models and the initial water contents and material densities of the "seeding" 
ice-phase precipitation particles. Simulated melting layer profiles based upon snow 
described by the Fabry-Szyrmer core-shell dielectric model and graupel described by the 
Maxwell-Garnett water matrix dielectric model lead to reasonable agreement with radar- 
derived melting layer optical depth distributions. Moreover, control profiles that do not 
contain mixed-phase precipitation particles yield optical depths that are systematically 
lower than those observed. Therefore, the use of the melting layer model to extend 3-D 
CRM simulations appears justified, at least until more realistic spectral methods for 
describing melting precipitation in high-resolution, 3-D CRM's are implemented. 



1. Introduction 

Several methods for estimating instantaneous precipitation rates from spaceborne 
passive microwave radiometer measurements have been developed and applied with 
moderate success; Smith et al. (1994). In recent years, these rainfall retrieval methods 
have been primarily applied to observations of the Special Sensor Microwave/Imager 
(SSM/I), a component of several of the Defense Meteorological Satellite Program's 
polar-orbiting platforms. Intercomparisons of the methods have demonstrated the utility 
of physically-based rainfall algorithms, which rely on physical models relating the three- 
dimensional structure of precipitating clouds to the upwelling radiances measured by the 
SSM/I; ref. Ebert and Manton (1998). The launch of the Tropical Rainfall Measuring 
Mission (TRMM) Microwave Imager (TMI) in November, 1997, improved upon the 
capability of SSM/I by extending the channel frequency range (10.65 - 85.5 GHz) and 
resolution (5 km minimum footprint). In addition, a 13.8 GHz weather radar called the 
Precipitation Radar (PR) was included on the TRMM platform to provide detailed 
vertical structure (250 m resolution) of the observed precipitation. 

By extending the passive radiometer physical models to include a description of the 
radar response to precipitation, retrieval procedures combining both radiometer and radar 
observations are achieved. Such physically based methods have already been developed 
and applied to TMI and PR data; e.g. Haddad, et al. (1997). However, the physical 
models supporting these retrieval methods do not include an explicit representation of 



melting precipitation particles, which have an important radiative effect in stratiform rain 
regions. 

Previous studies by Schols et al. (1997), Meneghini and Liou (1996, 1997), and Bauer 
et al. (1999a), have indicated the possibility of significant absorption of microwaves by 
partially-melted particles, in addition to the commonly-observed enhanced radar 
reflectivity of the melting layer. Although the high radar reflectivities of melting 
hydrometeors typically occupy only a thin layer (-500 m), and thus might be ignored, the 
microwave absorption within the melting layer has more serious consequences for both 
radiometric and radar remote sensing of stratiform precipitation. Absorption and 
emission of microwaves by the melting layer might account for a significant portion of 
the emission observed by spacebome passive radiometers- emission that might otherwise 
be attributed to liquid precipitation in physically-based retrieval methods. Likewise, the 
2-way path attenuation of radar pulses by the melting layer must be included in 
calculations of the total attenuation of reflectivities measured by spaceborne radar. 

In the present study, a one-dimensional, steady-state microphysical model of melting 
ice hydrometeors is developed, following the work of Bauer et al. (1999b). Although 
simplified, the model yields the volume fractions of ice, air, and water of melting 
particles of all species and sizes at a fine grid spacing in the vertical. This model is used 
in the manner of a parameterization to describe the vertical distribution of melting 



precipitation at applicable horizontal gridpoints of fully three-dimensional cloud 
resolving model simulations. 

The extinction optical depth and reflectivity simulations are compared to the 
corresponding PR-derived quantities in an attempt to determine which modeling 
assumptions lead to simulations most consistent with the observations. 



2. Model Description 

a. Microphysics of the Melting Model 

Calculation of the electromagnetic properties of melting layer requires, at the very 
least, a specification of the volume fractions of ice, air, and liquid of each melting particle 
within the ensemble of particles which comprise the layer. From these volume fractions, 
the dielectric properties of the particles may be computed according to different 
approximate formulae; see section lc. In addition, accurate simulations of upwelling 
radiances from cloud and precipitation distributions require that these distributions be 
specified in three dimensions at a resolution greater than 12 km in the horizontal, as 
indicated by Kummerow (1998). Such fields can be obtained from 3-D cloud resolving 
model (CRM) simulations; however, the microphysical schemes utilized in CRM's 
typically do not contain an explicit description of partially-melted precipitation particles; 
e.g. Lin et al. (1983), Rutledge and Hobbs (1983; 1984), due to the large computational 
requirements. Also, it will be shown in section lc that the height- variation of melting 



particle dielectric properties is significant on scales -50 m, which is usually a much finer 
resolution than that provided by many CRM's. Therefore, in the present study a one- 
dimensional melting layer model is developed to describe the melting process at 
applicable horizontal gridpoints of a "parent" 3-D CRM simulation. 

The melting model described in this study is only applicable to regions where 
updrafts and downdrafts are relatively weak in the vicinity of the freezing level. Under 
these conditions precipitation will fall steadily through the air column and only melting, 
sublimation/deposition, evaporation/condensation, and aggregation of particles need be 
considered. Each gridpoint of the parent model domain is first examined to find the 
maximum magnitude of the vertical velocity in the layer between the freezing level and 
two kilometers below it. If the magnitude of the vertical velocity does not exceed 0.5 m 
s" 1 in this layer, and if ice-phase precipitation exists just above the freezing level, then the 
1-D melting model is applied to the gridpoint in question. 

The formulation of the melting layer model follows the simplified approach of Bauer 
et al. (1999b). The temperature, relative humidity, vertical air motion, and hydrometeor 
fields at each qualified gridpoint from the parent 3-D CRM simulation are first 
interpolated to the 50 m resolution melting model vertical grid; see Fig. 1 . The top of the 
melting model domain is assumed to coincide with the parent model grid-level at, or just 
above, the C level, and the bottom of the model domain is set 3 km below this level. 



The size distributions of precipitation species (rain, snow, graupel) in the melting 
model are represented by number densities, n(i,z), at intervals of 0.1 mm diameter. The 
index / indicates the particle diameter interval, and z is the depth below the model top. A 
particle of a given size is characterized by its ice mass, m ice (i,z), liquid water mass, 
m liq (i,z), and volume of air, V air (i,z). 

At the top of the melting model domain the particle size distributions and 
composition are inherited from the parent model, therefore 

«('".0) = n Mt (D mit ), (1) 

where n M ,(D ini , ,) is the parent model particle number density at the top of the melting 
model domain, and n(/,,0) is the melting model number density for an initial particle 
diameter D mili and depth z = below the model top. Initially, ice precipitation species are 
assumed to be spherical and dry mixtures of ice and air (no melt water or collected 
water); therefore, the mass of ice and liquid in a given particle are 

P m ( A„/,, ) — g- 1 - . ice species 
m^frO) = { (2) 



ice \ 



0, rain 



and 



0, 



™«*0'.0) =•{ 



/r£L, 3 



'»« 



ice species 



rain, 



(3) 



where p m (D inin ) is the initial ice particle material density, which may be a function of 
diameter, and p liq is the density of water, 1 .0 g cm' 3 . The small mass of air in the particle 
is neglected, but the volume of air within the particle can be calculated from 



kD 



rice rm \ init l J 



V.„(i\0) = { 



0, 



, ice species 



ram 



(4) 



where p, C( , is the density of pure ice, 0.917 g cm" 1 . A description of the evolution of the 
precipitation particles as they fall from level z to a new depth z-Az follows. The diameter 
index / and level argument z have been omitted unless they are required for clarity. 
A particle of a given diameter D is assumed to fall at a terminal velocity given by 



w r = 



4p„,g 
3p u C D 



,1/2 



D 



Ml 



(5) 



where p„, is the material density of the particle, 



m ™ +m * , (6) 



Hba.+HS.+ V 

w air 
rice rliq 



g is the acceleration of gravity, p a is the density of air, and C D is the particle drag 
coefficient. Since the parent model simulation may have different fallspeed formulations, 
the drag coefficient C D is determined such that the total mass flux of a given ice (rain) 
species is equal to the mass flux of that species in the parent model at the top (bottom) of 
the melting model domain. The parameterization (5) for terminal velocity is used 
because the net particle density of an ice particle may change as the particle melts, and 
(5) accounts for the change in terminal velocity as a function of particle density. A 
completely melted ice particle assumes the drag coefficient of rain. 

As the ice precipitation particle of a given diameter falls, the rate of increase of the 
mass of melt water contained in the particle is determined by 

(F^-F evap )/L f , r>273.16K;F_>F„ ap 

dm ^={ (7) 



dt 

0, otherwise 



Here, F sens is the rate of sensible heat transfer to the particle, F evap is the rate of latent heat 
loss due to the evaporation of melt water, and Z^is the latent heat of fusion, 3.34 x 10 5 j 



kg" 1 . Note that melting can only occur if the temperature of the particle environment 
exceeds 273. 16 K and the rate of sensible heat transfer exceeds the rate of heat loss due to 
evaporation. Equation (2) follows the development of Ferrier (1994) but neglects 
collection terms that may also affect the particle heat balance. The sensible heat transfer 
rate is given by 

F sens = 2nDK(T-T )P, (8) 

where K is the thermal conductivity of the air, Tis the air temperature, T is the assumed 
melting particle temperature, 273. 16 K, and (3 is the ventilation factor, 

P = 0.78+ 0.31 Sc ]n Re U2 . (9) 

Here, Sc is the Schmidt number, which is given by 

S c = lil(p.v), (10) 

and the Reynolds number is given by 
Re = -w T Dp a /n, (11) 



10 



where /d is the dynamic viscosity of air and y/is the diffusivity of water vapor in air. The 
physical parameters K, fi, and i//are evaluated using the formulae in Pruppacher and Klett 
(1978). The rate of heat loss due to melt water evaporation is 

F n „ = 2nDy,p a [q vs {T )- q v ](3L v , (12) 

where q v and q vs are the mixing ratio of the air and the saturation mixing ratio at the 
surface of the particle, respectively, and L v is the latent heat of vaporization, 2.5 x 10 6 j 
kg"'. In (12), the heat loss due to evaporation is limited by the rate of transfer of water 
molecules from the particle surface to the atmosphere. 



W*v. r>273.16K;F_>F erap 

dm„,„.„„ _ f 



mexap 



dt 

F sens /L v , T>213A6K;F sens <F e 



(13) 



evap 



In the case of rain, or if the ice particle has completely melted, evaporation of liquid 
is described by (Ferrier, 1994), 



dm^ = -2xD(S-l)P 
dt AB 



where the supersaturation ratio, S, is 



11 



S = qjq„, (15) 



and 



Is 1 

AB = v , + — - — . (16) 

KKT 2 vpq v , 



Also, at temperatures below freezing, sublimation/deposition of water vapor from/onto an 
ice particle's surface is given by (Ferrier, 1994), 



dm sllh _ -2kD{S,-\)P 
dt AB ; 



where 



(17) 



S, = qjq ys „ (18) 



is the supersaturation ratio with respect to ice, and 



AS, = -£-r+ . (19) 



Here, q vsi is the water vapor saturation mixing ratio with respect to ice, and L s is the latent 
heat of sublimation, 2.83 x 10 6 j kg" 1 . 



12 



In the processes of melting or sublimation/deposition, it is assumed that the volume 
proportion of air to ice in a given ice or melting particle is maintained. Therefore, at 



temperatures above freezing 



dt 



f. 

J at 
rict 



din 



melt 



dt 



(20) 



while at sub-freezing temperatures 



dV 



asd 



f 

J at 



dt 



dm 



sub 



dt 



(21) 



where 



J at 



m I p. 

ice I r tee 



(22) 



is the volume ratio of air to ice in the particle. 



In addition to phase change processes, ice and melting particles can collide and 
combine to form aggregate particles, while pure liquid drops (rain) are immediately shed 
in collisions. Self-collection of raindrops is not considered here, since this process is 
adequately represented in the parent models, and the focus of this study is the evolution 
of melting precipitation particles. Which types of ice species may aggregate is 
determined according to microphysics of the parent model simulation (see Section 2b). 



13 



Also, in order to maintain consistency with the original definitions of the particle species, 
a given particle can collect another particle only if its size is greater than the collected 
particle. This criterion ensures that the resultant aggregate particle has material 
properties closer to those of the collecting particle. The rate of change in the number 
density, n a , of accreted particles due to collection is given by 

^ = -I,L(U)n^nSi)K[DXj) + D a (i)] 2 \ WTc (j)- WT M^ (23) 

j 

where £.„ is the collection efficiency of the particle interaction, n c is the number density 
of collecting particles, D c and D a are the diameters of the collecting and accreted 
particles, respectively, w Tc and w Ta are the terminal velocities of the collecting and 
accreting particles, respectively, and A is the distribution diameter interval of the 
collecting particles. The summation is over all valid collecting particle sizes. 

The total number of particles of a given size accreted by a single collecting particle 
per unit time is given by 

^- = L(iJ)n.(j) n [D c (i) + D a (j)f \ Wn .(i) - Wra (j)\ A/4. (24) 

It follows that the rate of change of ice and liquid water mass of a single collecting 
particle are 



14 



dt 



= ^L m a,ce{j) 



dN a {j) 
dt 



(25) 



and 



^® = IX,M dNM 



dt 



dt 



(26) 



respectively, where m a ice and m a Uq are the masses of ice and liquid water in each accreted 
particle, and the summations are over all accreted particles. Also, the volume of air in the 
collecting and accreted particles is assumed to be conserved during collection; therefore 



where V aair is the volume of air in the accreted particle. 



dV„ 



dt 4" dt ' 



(27) 



The combined effects of melting, sublimation/deposition, evaporation, and 
aggregation are now considered. For ice or melting precipitation species, 



m ke {i,z-Az) = m ke (i,z) + \-8 



dm 



melt 



dt 



-(1-5) 



dm 



sub 



dt 



dm. 



dt 



>At, 



(28) 



m ltq {i,z-&z) 



^'M^-^t^h 



(29) 



15 



and 



O'-z-Az) = V a , r (i,z) + \5 



dt 



+0-«) 



dV, 



asii 



dt 



dV„ 



dt 



>At, 



(30) 



where 



At = - Az/[w T + w], 



(31) 



and 



* = { 



0, 7<273.16K 



1, r>273.16K. 



(32) 



Here, At is the time required for the particle to fall to the next level, and w is the vertical 



air velocity of the particle environment. For raindrops, 



m 



M 



{i,z-Az) = m,Ji,z) + {- 



dm 



evap 



dt 



At, 



(33) 



which governs both completely melted ice particles and raindrops from the parent model 
simulation that are present at the top of the melting model domain. 

The change in the number density of ice and melting precipitation particles is 
calculated in three steps. First, the reduction of particles due to accretion is computed; 



16 



n(i,z-Az) = n(i,z) + \^^]At. (34) 



Then n(i,z-AzY is adjusted for the loss of particles due to evaporation or sublimation 
according to the governing equations (28)-(33). Since the melting model is steady-state, 
the number flux of particles at level z, less the flux eliminated by accretion, evaporation, 
or sublimation, should equal the number flux of particles at the next level. This condition 
is satisfied if the number flux of "surviving" particles at z, n(i,z-AzY • (Wj(i,z) + w(z)) is 
set equal to the number flux at the next level, n(i,z-Az) • (wj(i,z-Az) + w(z-Az)). It follows 
that 



„(«-Ac) = nt,,^ , M ^:t )] A^ (35) 

[w T (i,z - Az) + w(z - Az)\ 



which yields the number density of particles at the next level. Note that Wj(i,z-Az) is 
obtained from (5) using the particle composition calculated at the next level. 

The foregoing equations provide a description of the melting process that is sufficient 
for determining the bulk electromagnetic properties of the melting layer. However, due to 
differences in model physics and vertical resolution, as well as the lack of horizontal 
advection of precipitation into or out of the one-dimensional melting model domain, the 

17 



profiles of precipitation produced by the melting model may differ from those computed 
in the parent model. To help correct for these differences, the size distribution of each 
precipitation particle species in the melting model is scaled, such that the total mass flux 
of the resulting precipitation distributions is equal to the parent model total mass flux 
interpolated to the melting model grid; 

»«*■('. z) = 7(z)n(i,z), (36) 



with the scaling factor 



yfe) = X^«/ I 



species / ^species |_ i 



X"('.z) (™i« ('.*)+"»«, ('.*)) {w T (i,z)+w(z))A 



(37) 



Here, M par {z) is the mass flux of a given precipitation species in the parent model 
interpolated to level z. From a theoretical perspective, the correction (36) may seem 
heavy-handed. However, the objective of the present work is to develop a description of 
the melting band with sufficient detail for radiative transfer calculations, and one that is 
compatible with the parent 3-D model simulations, as opposed to a stand-alone model. In 
this sense, the melting model can be viewed as a local parameterization of processes not 
resolved by the parent model. 



18 



Finally, from the computed ice and liquid masses and volume of air in a given 
particle, the volume fractions of ice, liquid, and air are calculated; 



fiJU) = "ffiVft, , (38) 

V(i,z) 



m, 



M^IPu* 



V(i,z) 



f a ,r(U) = %^, (40) 



where 

n ,-, z) = ^M + f! !S M + Kir( ,. z) . (41) 

rice riiq 



The quantities n adj ,f jce ,f !iq , and /„,,., are utilized in the following sections to calculate the 
electromagnetic properties of melting precipitation. 



b. Application to Parent 3-D Cloud-Resolving Model Simulations 

In the previous section, a simplified model describing the evolution of melting 
precipitation was developed. Being one-dimensional, this melting model is designed to 



19 



represent the melting layer at a single gridpoint in a parent, 3-D CRM simulation and is 
driven by the distributions of ice precipitation just above the freezing level in the parent 
simulation. In this section the parent model simulations are briefly described, and 
applications of the melting model to a single parent model profile are presented. The 3-D 
CRM model runs serve as the basis of the melting simulations described in the following 
section as well as the simulated radiance and radar reflectivity fields in Part II. 

The parent model simulations of this study are derived from the Goddard Cumulus 
Ensemble (GCE) model and the University of Wisconsin Non-hydrostatic Modeling 
System (UW-NMS); see Table 1. A detailed description of the GCE model may be found 
in Tao and Simpson (1993). The GCE model is non-hydrostatic, with cloud 
microphysics described by an adaptation of the Lin et al. (1983) scheme. In this scheme, 
precipitating particle distributions are represented by inverse exponential distributions 
with fixed intercepts; the slopes of the distributions are adjusted to account for changes in 
water content, which are computed prognostically. The particle material densities'in this 
scheme are also fixed: rain has a density of 1 .0 g cm" 3 , while snow and graupel have 
densities of 0.1 and 0.4 g cm' 3 , respectively. Three GCE model simulations are utilized 
in the present work. Two of these simulations are initialized using temperature, 
humidity, and wind conditions observed 22 February 1993, during the Tropical Ocean- 
Global Atmosphere Coupled Ocean Atmosphere Response Experiment (TOGA 
COARE). A tropical squall line observed to the southwest of the TOGA COARE 

20 



Intensive Flux Array (IFA) was simulated on a 1 km, 128 x 128 grid (TOGA1) and a 3 
km, 128 x 128 grid (TOGA3). Both squall lines were initiated with a spreading cool pool 
centered in the model domain that generated a gust front-like forcing. A third simulation 
is derived from a long-term, forced simulation of convection in the TOGA COARE IFA 
during the period 19-26 December, 1992. The extensive mesoscale convective system 
that occurred on 24 December is represented in the simulated 12 UTC precipitation field 
on a 2 km, 256 x 256 grid (TOGA2). 

The UW-NMS is described in Tripoli (1992a). The microphysical scheme employed 
is from Flatau et al. (1989), with specified parameters tabulated by Panegrossi et al. 
(1998). In the simulations utilized in the present study, the precipitating hydrometeor 
species have inverse exponential size distributions, but unlike the GCE model, the slopes 
of these distributions are fixed, while the intercepts are allowed to vary with water 
content. Rain and graupel have fixed material densities of 1.0 and 0.6 g cm" 3 
respectively, while snow has a size-dependent density given by 



v-O.6 

= 1IU1V/ I 

m-snow 



= 0.0597 D -0 - 6 , (42) 



where the particle diameter D is in mm, and the resulting density is in g cm" 3 . Two UW- 
NMS simulations are used in the current study. The first is a simulation of a 
thunderstorm complex observed during the Cooperative Huntsville Meteorological 



21 



Experiment. Identified here as COHMEX, this simulation was performed on a 1 km, 51 
x 51 grid domain. A description of COHMEX can be found in Mugnai et al. (1990). A 
second simulation was initialized with a balanced vortex and sounding data from the 
environment preceding the passage of Hurricane Gilbert (1988) at Kingston, Jamaica. 
The resulting hurricane simulation (hereafter, HURRICANE) was performed on a two- 
way, quadruply-nested grid, which included a 3.3 km resolution, 62 x 62 inner nest 
during the last 6 hours of simulation time to resolve the inner core region of the storm. 
Details of HURRICANE may be found in Tripoli (1992b). 

Depicted in Fig. 2 is a plan view of the surface rainfall rate field in the TOGA1 
tropical squall line simulation at 180 min into the simulation. On the right side of the 
figure is the bowed convective leading edge of the squall line with rain rates exceeding 
16 mm h" 1 . The leading-edge convection extends to the south and west (north is up in the 
Figure). Less intense and more horizontally uniform stratiform precipitation trails the 
convection to the north and west. 

The crossing of the dotted lines at x = 42 km and y = 15 km in Fig. 2 indicates the 
position of the test profile examined in this section. This gridpoint is located near the 
center of an area of stratiform rain, with a surface rainfall rate of 0.6 mm h" 1 . The melting 
layer model is applied to the TOGA1 simulation at the gridpoint. The top of the melting 
layer model coincides with the 6 km level in the parent model simulation, where the 
temperature is just below freezing (271.4 K), and the snow and graupel water contents are 

22 



0.17 g m" 3 and 0.25 g m' 3 , respectively. The freezing level occurs 0.37 km below the 
melting model top. 

Presented in Fig. 3a and b are the liquid water content and mass flux profiles, 
respectively, of snow, graupel, rain, and total precipitation calculated using the melting 
model. Note that in order to emphasize the melting model physics, subsequent scaling of 
the precipitation mass flux using (36) is omitted in this example. From the top of the 
model domain to the freezing level, sublimation of ice reduces the mass and mass flux of 
both snow and graupel, while aggregation processes are weak (collection efficiency is 
less than 0.1). The melting of ice-phase particles falling below the freezing level is 
initially inhibited by the evaporation of meltwater [see (12)], a process that consumes 
most of the sensible heat flux to each particle while removing little mass. More effective 
aggregation at these temperatures (collection efficiency of 1 .0) causes a slight increase in 
the water content of snow while the water content of graupel decreases; however, the 
total mass flux of snow and graupel together is nearly constant from the freezing level to 
a depth of 0.75 km (about 0.4 km below the freezing level). 

Below a depth of 0.75 km, the increasing humidity of the air effectively shuts off 
meltwater evaporation, and the increasing sensible heat flux from the air causes rapid 
melting of snow and graupel. The denser graupel particles do not melt as quickly as 
snow of the same size, and the graupel completely melt at a depth 0.25 km lower than 
snow. The melted and partially-melted snow and graupel attain higher terminal 

23 



velocities, and therefore the total water content of precipitation decreases rapidly with 
melting, despite the fact that the total mass flux of precipitation is nearly constant. . At 
1 .8 km depth the snow and graupel are completely melted, and the gradual decrease of 
precipitation water content and mass flux below this level is caused by the evaporation of 
rain. 

One of the limiting assumptions in the GCE model microphysics is that both snow 
and graupel particles have material densities independent of particle size. Studies by 
Locatelli and Hobbs (1974), Mitchell et al. (1990), and others suggest that the density of 
snow particles decreases markedly with size, while graupel particles show only a slight 
decrease in density with size. The snow density relationship of Mitchell et al. (1990), and 
a curve Fit to the graupel observations of Locatelli and Hobbs (1974) yield 

P„„ = 0.149 D- 10 , (43) 



and 



P-„, = 0.144 D™', (44) 



where D is in mm and p m . mm . and p m . graupe , are in g cm" 3 . Snow and graupel with these 
density distributions are substituted for the snow and graupel distributions at 6 km at the 
same gridpoint of the TOGA1 simulation, and the number densities of both types of 
particles are scaled to yield the same water contents as those in the original simulation. 

24 



The modified snow and graupel distributions are then used to initialize the melting 
model, with the resulting profiles of precipitation water contents and mass fluxes shown 
in Figs. 3c and 3d, respectively. 

Comparing the model profiles to the profiles based upon the fixed-density model in 
Figs. 3a and 3b, it may be noted that the primary difference is the more rapid melting of 
the variable density particles. The depth of the melting layer contracts from about 1 .0 km 
to 0.5 km. This result is due to the more rapid melting of the larger snow and graupel 
particles, which are less dense than their constant-density counterparts. These larger 
particles require the most time to completely melt and therefore limit the melting time of 
the distribution. The differences in melting rate will have an impact on the radiometric 
properties of the melting band, to be discussed in Section 2c. 



25 



c. Dielectric Properties of Melting Hydrometeors 
i. Maxwell-Garnett Model 

Generally, the effective dielectric constant of an inhomogeneous particle is calculated 
as a function of the individual contributions by ice, water, and air. The most widely used 
formulation follows the Maxwell-Garnett approach (Maxwell-Garnett 1904) which was 
generalized by Bohren and Battan (1982). An inhomogeneous particle is described as a 
matrix material with randomly distributed and oriented elliptical inclusions which 
contribute to the effective dielectric constant of particle in proportion to their volume 
fraction, f mc . 



„ L J inc j mat J inc J inc //1<\ 

i-f +/ r ' (4 ^ 

J inc J inc ^ 



where 



c = 



2e 

£■ —£ 



£,„c 



£ — £ 

mc mat 



In 






\ £ mal J 



(46) 



Here, e ma „ e inc , and e mix , denote the complex dielectric constant of the matrix, the 
inclusions, and the mixture, respectively, and \n(£ inc /e mal ) is the principal value of the 
complex number £ m j£ mav For three component mixtures, (45) must be applied twice, and 



26 



the resulting dielectric constant of the mixture depends upon the order of application as 
well as the choice of matrix and inclusion materials in each application. 



ii. Meneghini and Liou Models 

Several intercomparisons of available dielectric model formulations have been carried 
out, including (45) and others that treat the mixture as a homogeneous composite. 
Klaassen (1988) demonstrated that even homogeneous formulations such as Debye 
(1929), Bruggeman (1935), and Maxwell-Garnett (1904) deviate strongly from one 
another. To test these formulations, Meneghini and Liao (1996) solved the 
electromagnetic field equations numerically for particles subdivided into a grid of cells 
containing ice, air, or water. The effective dielectric constants of the particles were then 
determined as those which, when input to Mie theory, produced the same extinction and 
backscattering coefficients as the numerical technique. 

For particles containing homogeneous distributions of ice, air, and water, the effective 
dielectric constants were well-approximated by the Maxwell-Garnett formula for water 
inclusions in an ice matrix, £ MGiw (hereafter referenced as MGiw), for ice volume fractions 
greater than 0.2. The Maxwell-Garnett formula for ice inclusions in a water matrix, e MGwj 
(hereafter referenced as MGwi) compared best to their numerical method when the ice 
volume fraction was less than 0.2. On the other hand, if meltwater accumulated at the 



27 



particle surface, the effective dielecric constant was again well represented by e MCwi . The 
distribution of meltwater within the particle, however, depends upon the initial particle 
density and melting stage; therefore, different dielectric models may be optimal for a 
given particle density and melting stage. 

For a spatially homogeneous ice-water mixture, i.e., meltwater pockets randomly 
distributed throughout the particle volume, Meneghini and Liou (1996) developed an 
analytical function to better represent the particle dielectric constant, e ML96 (hereafter 
referenced as ML96), at a frequency of 7.7 GHz. At intermediate melting stages they 
utilized an error function to interpolate between the dielectric constant of an ice-air 
matrix with water inclusions, £ MGiw (initial melting stage), and that of a water matrix with 
ice-air inclusions, e WG „., (later melting stage), as a function of the fractional volume of ice, 

J ice' 

£-196 = 0.5{[l-erf(g)]e WGH , + [l+erf(£)]e MGw }, (47) 

where the error function is defined as 



2 r« 



erf(S) = -^J o exp(-^ 2 )^, (48) 



and 



28 



£ = [/JO-/J-^,] (49) 



2tf 2 



The parameters $, = 0.2 and i!> 2 = 0. 1 yield a good approximation to the effective 
dielectric constants determined from their numerical technique. 

The application of the ML96 formula to frequencies other than 7.7 GHz may not 
seem justified, but the effective dielectric constant calculation is driven by the weights 
obtained from the error function which are closely tied to e MCiK at early melting stages and 
thus represent a rather conservative estimate of the dielectric constant. Also, the ML96 
formula was applied only to mixtures of pure ice and liquid water. In the present study 
fke + fair ls substituted into (49) for f ice , where the Maxwell-Garnett formula for air 
inclusions in an ice matrix is used to compute the dielectric constant of the ice-air 
mixture. 

More recently, Meneghini and Liao (1997; hereafter referenced as ML97), improved 
the numerical accuracy of their dielectric constant calculations, employing a continuous, 
multiple-component particle approach with parameterizations which allow for 
applications to frequencies between 10 and 95 GHz. Their model was based upon the 
assumption that the properties of the particle constituents are homogeneous, isotropic, 
and linearly superimposed. Therefore, 



29 



\ Hi) , \ E a,r) r 

''UqJliq I -p \ ^ air J air I p \ '^ ice J ice 

e = \^d LfW rsm 



where e jce , e air , and e,,- 9 represent the permittivities of pure ice, air and water with volume 
fractions f ice ,f air , and/^, respectively. The terms in brackets denote the average electric 
fields, assuming that the particle can be decomposed into individual cells over which the 
fields are homogeneous. Therefore, the summation of all cell contributions to the total 
field is equivalent to the average field multiplied by the number of cells. The ratio of the 
average electric fields in (50) is computed from the Debye formula assuming a dry 
"snow" particle with/', ce +f' a , r = 1, 



(E a , r ) SL 

\ ice I J air 



£ , — £ 

dry ice 



£ -£, 

air ^drt 



(51) 



where 



£ </n = 


P,cc + lK,cePa-n 


rice ice » dry 


K ice = 


e,n-l 


£*v+2' 



(52) 



(53) 



and 



30 



Pm = m m,ce (54) 



is the density of the dry "snow" particle. Finally, both the real and imaginary parts of 
(E liq )/(E ice ) were computed by ML97 using a conjugate gradient numerical method and 
parameterized as functions of frequency and fractional meltwater. 



iii. Concentric Shell Models 

Another approach to the problem of approximating the dielectric properties of a 
melting ice particle is to construct it from concentric shells of material with different 
dielectric properties. The most common example is a water-coated sphere, which may be 
a sufficient approximation for melting hailstones or graupel with high material densities. 
Regarding snow particles, a more detailed description of each layer is required, and this 
may be obtained through the application of (45) to each shell. 

Shivola and Lindell (1989) developed the theoretical background for various 
dielectric constant profiles through inhomogeneous particles. Their work allowed a more 
sophisticated treatment of continuous dielectric constant variations as a function of radius 
within the particle. The assumed density and dielectric discontinuities in a shell model 



31 



represent a simplification of their approach which seems justified in view of the lack of 
knowledge regarding the spatial variations of density and composition as a function of 
radius in melting panicles. 

For the purpose of modeling radar reflectivities in the melting layer, Fabry and 
Szyrmer (1999; hereafter FS) implemented a core-shell model which was then 
incorporated into the framework of a microphysical melting model by Szyrmer and 
Zawadski (1999). In their model, the particle core consists of ice inclusions in a water 
matrix, which together serve as a matrix for air inclusions. The latter are treated as 
bubbles in a comparatively solid environment. The outer shell is modeled as ice 
inclusions in a water matrix, which together form inclusions in a matrix of air. Thus the 
outer shell represents a rather tenuous collection of melting ice crystals. The dielectric 
constants of the core and shell are calculated using (45). 

FS compared modeled and observed radar reflectivities at 0.9 and 9.4 GHz and 
obtained better agreement using this approach in relation to alternative models based 
upon (45) for different choices of matrix and inclusion materials. FS confirm that the use 
of (45) with a water matrix showed an exaggeration of the melting layer reflectivity, a 
result in concert with studies by Bauer et al. (1999a,b), who noted excessive microwave 
emission from the modeled melting layer when (45) with the water matrix assumption 
was applied. Moreover, these authors concluded that the choice of a particle dielectric 



32 



constant model outweighs other sources of uncertainty such as the assumption of particle 
density. 

Crucial to the FS model is the calculation of particle density change with melting 
stage and the position of the boundary between the particle core and outer shell. Prior to 
melting, the core and outer shell densities of the dry snow particle are 



P dry, core ~ Pdr)- ^dry ' (55) 



and 



n - n O-O 

Pdn, shell ~ P dry t, i \ ' (56) 

(1 " airy) 



where a Jry is the radius of the core region expressed as a fraction of the total radius of the 
particle. Note that (55) and (56) may be derived assuming that density is a continuous 
function of 1/radius within the particle; then p dr> , core and p dr> . sMI are the average densities 
within the particle core and shell, respectively. The bulk material density of a particle 
with these core and shell densities is equal to p dty , regardless of the a dr> . chosen. In FS, 
a dn = 0-5 in (55) and (56). During melting, the densities of the core and shell regions 
change according to 



33 



r dry, core rliq / cn\ 

r I \ ' ( ' 

Jmliq r dry, core \ Jmliq) rliq 



and 



r dry, shell rtiq /CON 

J m, shell ~ ~ Z : ~~f, ~ \~ > (58) 



Jmliq P dry, shell + \* JmliqjPliq 



respectively, where 



Jmliq > (5") 

m ice+™,iq 



the mass fraction of melt water, is assumed to be the same in both the core and shell 
regions. Under these conditions, the core radius fraction of the particle is 



a = 



-|l/3 
rm rm, shell 

m,core rm, shell 



(60) 



where p m is the average density of the particle, given by (6). As the particle melts, the 
core and shell densities approach 1 (completely melted), and the core radius fraction also 
approaches 1 (the melted particle radius). 



34 



If the core region of the particle is treated as a spherical inclusion within a matrix of 
shell material, the effective dielectric constant of the complete particle can be 
approximated by the original Maxwell-Garnett theory. Under this assumption the 
effective dielectric constant is 



c- — p . 3 ' £ 'shell a [ £ core £ shell\ / fi |\ 

t-FS — t shell ~T~ ~ _ rY i \c _p V 

£ core ~*~*- £ shell a [ £ core £ shell} 



where £ core and e shel , are the permittivities of the core and shell regions, respectively. 
However, (61) is only appropriate if the particle dimensions are small compared to the 
wavelength of radiation, a condition that is not always satisfied in the present application. 
The rigorous alternative is to calculate the particle radiative properties by solving the 
electromagnetic field equations for a two-shell system, as in Bohren and Huffman (1983). 
This alternative will be explored in Section 2d. 



iv. Intercomparison of Dielectric Constant Models 

Particle dielectric constants at 10.65, 19.35, 37.0, and 85.5 GHz are calculated for 
melting snow (initial p„, = 0. 1 g cm" 3 ) and graupel (intial p m = 0.4 g cm' 3 ) and converted to 
refractive indices using 



35 



n = 4e. (62) 

Melting snow and graupel refractive indices are presented in Figs. 4 and 5, respectively. 
The five curves in each panel correspond to calculations using the different dielectric 
models described in the previous subsection. The curves converge to points near the 
lower left (no meltwater) and upper right (completely melted) corners of each panel, with 
a "+" indicating a meltwater volume fraction of 0.5. 

It may be noted from the figures that the refractive index curves are generally 
bounded by the MGwi and MGiw model curves. The quasi-linear MGwi curve yields the 
maximum rate of increase of the imaginary component of refractive index with melting. 
Conversely, the MGiw model typically yields the smallest rate of increase of the 
imaginary component. The other three models tend to follow the behavior of MGiw but 
with noted differences. ML96 closely follows MGiw up to a volume meltwater fraction 
of 0.5, but then it makes a transition to MGwi for higher meltwater fractions. The ML97 
model yields imaginary refractive indices sometimes even lower than those produced by 
MGiw for very low meltwater fractions, but then gradually diverges from the MGiw 
curve for higher meltwater fractions. In contrast, the FS model yields a higher imaginary 
refractive index component than either ML96 or ML97 for meltwater fractions less than 



36 



0.5, but then approaches the MGiw model refractive index as the meltwater fraction 
increases from 0.5 to 1.0. 

The refractive indices of snow and graupel are subtley different. A close inspection 
of Figs. 4 and 5 reveals greater real and imaginary refractive index components for 
graupel with a given meltwater fraction for all but the MGwi model. The MGwi model 
yields slightly smaller refractive indices for graupel relative to snow. Overall, the 
refractive index differences between snow and graupel decrease as the meltwater fraction 
increases. 

Since the radiative absorptivity of a material increases with the imaginary component 
of the refractive index, pure snow or graupel are expected to be poor absorbers. Once 
snow or graupel begin melting, the MGwi model would produce the greatest increase in 
absorptivity with meltwater fraction. As mentioned earlier, the MGwi model may 
exaggerate the radiative effect of melting except for relatively dense particles (graupel or 
hail) for which meltwater may initially accumulate near the surface of the particle. The 
more rigorous ML97, which describes the dielectric properties of particles with 
homogeneously-distributed meltwater, may also be applicable to denser ice particles, but 
the absorptivity of the particles would generally be less. Snow, having an ice-air density 
which decreases with radius within the particle, is more appropriately described by the 
FS, core-shell model. The bulk refractive index of snow based upon FS indicates 
significantly greater absorptivity for low meltwater fractions in comparison to ML97 or 

37 



MGiw, but less absorptivity for higher meltwater fractions. The impact of dielectric 
constant models on the radiative properties of melting ice particles is next examined. 



d. Radiative Properties 

Plotted in Fig. 6 are the absorption and scattering efficiencies of the FS melting 
particles at 10.65, 19.35, 37.0, and 85.5 GHz. The efficiencies are calculated based upon 
the analytical solution for electromagnetic waves interacting with a dielectric sphere 
having a core and outer shell (ref. Bohren and Huffman, 1983). The absorption and 
scattering efficiencies, multiplied by a particle's geometric cross section, are proportional 
to the radiative power absorbed and scattered by the particle at these TMI channel 
frequencies. Presented in each plot are efficiency curves for five different meltwater 
water volume fractions spanning the range from "dry" snow (p ice . air = 0.10 g cm" 3 ) to pure 
liquid composition. At all frequencies, and for all particle compositions, the efficiencies 
increase with increasing particle size until the particle radius is approximately equal to 
the wavelength of radiation. At larger particle radii, maximum efficiencies are first 
attained by the pure liquid particles, and then by particles with decreasing meltwater 
fractions. Dry snow particles have very low absorption and scattering efficiencies, with 
the exception of relatively large particles at 37.0 and 85.5 GHz. 

For relatively small volume fractions of meltwater (less than -0.10), absorption and 
scattering efficiencies of the particles generally increase with meltwater fraction. The 

38 



increase of efficiency with meltwater fraction is greatest for the smallest meltwater 
fractions, a nonlinear sensitivity that has been noted by other investigators; e. g. 
Meneghini and Liao (1996). Note that the absorption efficiencies of particles that are 
relatively large with respect to wavelength attain maximum values for intermediate 
meltwater fractions (-0.4), while the efficiencies of pure liquid drops of the same size are 
less. The same trend is not seen in particle scattering efficiencies, which nearly always 
increase with increasing meltwater fraction. The implication of these trends is that the 
radiometric absorption/emission and scattering of only slightly melted particles can be 
significant compared to pure liquid drops of the same size, and that the 
absorption/emission of particles with significant melted fractions can actually exceed that 
of pure liquid drops. 

The individual particle properties are integrated over the size distribution of each 
particle species in the test profiles depicted in Fig. 3a to produce the profiles of bulk 
absorption and scattering coefficients in Figs. 7a and 7b, respectively. These bulk 
absorption and scattering coefficients are based upon application of the FS, core-shell 
dielectric model to both snow and graupel. For comparison, absorption and scattering 
coefficients based upon the MGwi model are plotted in Figs. 7c and 7d, respectively. 

It may be inferred from Fig. 7a that the bulk absorption by melting snow and graupel 
is greater than the absorption by rain alone at the base of the melting layer. The higher 
water contents of precipitation (Fig. 3a) as well as the elevated absorption efficiencies of 

39 



partially melted precipitation (Figs. 6a, c, e, and g) contribute to greater absorption in the 
melting layer. Maximum absorption occurs just below the freezing level based upon the 
FS, core-shell model, and it is about an order of magnitude greater than the absorption by 
rain below the melting layer at 10.65 GHz. Maximum absorption is about 5, 3 and 2 
times the absorption by rain at 19.35, 37.0 and 85.5 GHz, respectively. Microwave 
scattering is also maximized in the melting layer. However, at frequencies less than 85.5 
GHz scattering is generally much less than absorption, and the scattering peaks are 
broader, extending through the depth of the melting layer (Fig. 7b). The broader 
scattering peaks result from the less rapid, monotonic increase of scattering efficiency 
with melted particle fraction, as seen in Figs. 6b, d, f, and h, whereas the absorption 
efficiency increases more rapidly with melted fraction for a wide range of particle sizes 
(Figs. 6a, c, e, and g). At 85.5 GHz, absorption and scattering in the melting layer are 
comparable. The absorption and scattering coefficients based upon the MGwi model 
exhibit similar trends, with a slightly higher and narrower peaks of absorption given by 
MGwi. The scattering profiles produced by the two models are nearly identical. 

In Fig. 8, the extinction efficiency and normalized backscatter cross-section based 
upon the FS, core-shell model are plotted for particles with various volume fractions of 
meltwater at 13.8 GHz, the operating frequency of the PR. The extinction efficiency, 
multiplied by the particle's geometric cross-section, is proportional to the radar 
attenuation by the particle. Similarly, the normalized backscatter cross-section multiplied 

40 



by the particle geometric cross-section is proportional to the radar power reflected by the 
particle. The extinction efficiencies at 13.8 GHz (Fig. 8a) follow the same trends as the 
absorption efficiencies at 10.65 GHz (Fig. 6a). It may be inferred that radar attenuation 
becomes significant with the onset of particle melting. On the other hand, particle 
backscatter cross-sections increase almost in proportion to the particle melted fraction. 

The individual particle extinction efficiencies and backscatter cross-sections are 
integrated over the size distribution of each particle species in the test profiles depicted in 
Fig. 3a to produce the profiles of bulk extinction coefficient and radar reflectivity in Fig. 
9a and 9b, respectively. These profiles are based upon the FS, core-shell dielectric model, 
and the contributions from snow, graupel, and rain are indicated. For comparison, 
extinction coefficients and radar reflectivities based upon the MGwi model are plotted in 
Figs. 9c and 9d, respectively. 

Since microwave scattering is almost negligible in comparison to absorption at 13.8 
GHz, the profiles of total extinction have almost the same form as the profiles of 
absorption shown in Fig. 7a. Note that the peak extinction by snow is almost the same as 
that of graupel, even though the water content of graupel is always greater than that of 
snow (Fig. 3a). This result is explained by the greater proportion of large particles, which 
have greater extinction efficiencies, in the snow distribution according to the GCE model 
microphysics. The greater proportion of large snow particles offsets the slightly greater 
extinction efficiency of graupel. On the other hand, due to their lower density, snow 

41 



particles melt more rapidly than graupel particles of the same size, leading to a relatively 
narrow peak of snow extinction. The peak reflectivity of the melting band reaches nearly 
37 dBZ about 0.6 km below the freezing level, where the slower-melting graupel makes a 
greater contribution to the reflectivity. The peak reflectivity of the melting layer greatly 
exceeds the reflectivity of rain below the melting layer (-27 dBZ). Note also that the 
peak of reflectivity is at a slightly lower altitude than the peak of extinction. This effect 
is explained by the more gradual increase of backscatter efficiency (relative to extinction 
efficiency) with meltwater fraction; see Fig. 8. The greater refractive indices produced 
by the MGwi model for melting particles leads to generally greater extinction and 
reflectivity of the melting layer (Figs. 9c and 9d). Ice species with initial densities which 
decrease with size (Fig. 3c, d) melt more rapidly, resulting in narrower peaks of 
extinction and reflectivity (Fig. 10a, b). 



3. Comparison of Melting Band Simulated Attenuation 
to Radar Observations 



The aspect of the melting band model which is perhaps most relevant to radiometer 
and radar remote sensing of precipitation is whether or not the model can simulate the 
extinction of microwave radiances. In the previous section it was demonstrated that 
radiative absorption and scattering per kilometer within the melting layer could be several 
times the absorption and scattering of the fully melted precipitation below. This 

42 



absorption and scattering could have an impact on the upwelling radiances measured by 
passive microwave radiometers. Similarly, the additional extinction in the melting layer 
could lead to greater attenuation of reflectivities below the layer, as observed by 
spaceborne radar. 

Here, the radar "mirror-image" technique (ref. Liao et al., 1999) is applied to 
observations of the Precipitation Radar (PR) in stratiform rain areas to estimate the 
radiative extinction associated with melting precipitation. A schematic of a PR 
reflectivity profile is provided in Fig. 1 1 for the identification of specific reflectivity 
measurements described in the analysis. Mirror-image reflectivities refer to reflectivities 
measured at ranges beyond the range of the surface reflection. These reflectivities are 
produced by radar pulses that have reflected off the earth's surface, are then 
backscattered by a precipitation target, and are finally reflected off the earth's surface a 
second time toward the radar receiver. The present application of the technique to PR 
observations follows. 

First, nadir-view reflectivity profiles and derived products from the PR are collected 
from the period August 4 - 22, 1998, over the region bounded by 20 °S and 20 °N 
latitude, 180 °W and 120 °W longitude. The data are filtered using the qualitative flags in 
the derived TRMM product 2A-23 (Awaka et al. 1998) to select only profiles where 
stratiform rain was present and a radar bright band was detectable. The reflectivities of 
each three consecutive nadir profiles are averaged to reduce noise for subsequent 

43 



processing. The averaged profiles are next analyzed to identify the bright band 
reflectivity bin (maximum reflectivity bin within 1.5 km of the maximum reflectivity 
gradient above the rain layer). The range difference between the radar surface return and 
the bright band is calculated, and then the bright band "mirror image" bin is identified as 
the maximum reflectivity bin within 0.75 km of an equal range displacement beyond the 
surface return. If either the bright band reflectivity (Z^)or its mirror image reflectivity 
(Z bb . m ) are below 25 dBZ, the profile is rejected. An alternate surface range is defined as 
the average of the range of the bright band range and the bright band mirror image range. 
If this surface range deviates by more than 0.125 (half a range bin) from the range of the 
maximum surface return, then the profile is rejected. A "basal" reflectivity (Z ba ) and a 
"basal-mirror" reflectivity (Z ha . m ) are then identified as those corresponding to bins 1.5 
km below the bright band and its mirror image, respectively. Also, a "reference" 
reflectivity (Z rf ) and a "reference-mirror" reflectivity (Z tf . m ) are identified as those of bins 
0.75 km before and 0.75 km beyond the alternate surface range. If any of the 
reflectivities of the basal, basal-mirror, reference, or reference-mirror are less than 20 
dBZ, the profile is rejected. The purpose of the filtering process is to remove profiles 
which (a) do not have an easily identifiable bright band or bright band mirror image, (b) 
have uncertain bright band or mirror image bright band ranges relative to the surface 
range, and (c) do not have basal, basal-mirror, reference, and reference-mirror 
reflectivities which are significantly greater than minimum-detectable (-17 dBZ). 

44 



Profiles that survive the filtering process are analyzed to estimate the optical depth 
between the bright band height and the level 1.5 below the bright band. Although it may 
be argued that the bulk of the extinction due to melting, based upon Figs. 9 and 10, lies 
between the top of the bright band and a kilometer below the top, application of the 
mirror-image technique requires well-defined reference levels, and reflectivity 
measurements that are relatively noise-insensitive. Radar reflectivities above the bright 
band maximum are often close to minimum-detectable, and the bright band maximum 
and its mirror-image supply well-defined height references. A 1.5 km layer depth 
ensures that the melting process is entirely contained within the layer, and puts the base 
reflectivity and its mirror image below the gradient region associated with melting, 
establishing a more certain reflectivity reference. 

Following Liao et al. (1999), the reflectivity double-differences are defined 

n» = {Z bh -Z hh _ m )-(Z tf -Z^_ m ), (63) 

and 



«*. = K-^ fl _ m )-(z^-Z^ m ), (64) 

where all reflectivities are in dBZ. For ideal, beam-filling radar targets and specular 
reflection of the nadir-view radar beam off a flat ocean surface, the measured difference 

45 



Q bb would be four times the radar path attenuation (4-way attenuation) between the bright 
band and the 0.75 km altitude reference level. Similarly, Q ha would be the 4-way 
attenuation between the base level and the reference level. For wind-roughened ocean 
surfaces, Liao et al. (1999) modeled the 4-way attenuation between any altitude h and a 
reference level of 0.75 km, corresponding to a PR-measured double-difference Q, and 
surface radar backscatter cross-section, <f. These 4-way attenuation simulations were fit 
to an empirical function, T[Q., h, o°]. Using this function, measurements Q. bb and Q. ba , and 
an estimate of the surface backscatter cross-section <f from nadir-view PR measurements 
in rain-free regions (TRMM 1C21 product; Meneghini et al. 1999), the optical depth, r, 
between the bright band and base level can be estimated from 

T = ] ^{^ hh A h ^"]-r[n ba ,h hb -l-5km,cj"}}, (65) 

where h bb is the altitude of the bright band. 

Equation (65) is applied to the filtered PR reflectivity profile data from August, 1998, 
and the resulting estimates of x are plotted versus corrected basal reflectivity 
measurements, Z* ba , in Fig. 12. The basal reflectivities are corrected using 



z t = z u + -T 2 "^. (66) 

ba ba in(10) K } 



46 



where the reflectivities are evaluated in dBZ. The correction, (66), compensates for the 
attenuation of the basal reflectivity due to overlying precipitation between the basal level 
and the bright band. This correction makes Z* ba a parameter representative of the 
"output" precipitation below the melting layer which is more or less independent of 
optical depth variations in the precipitation above. In this way, the impact of optical 
depth variations in the melting layer are seen primarily along the optical depth axis of the 
plots in Fig. 12. Error bars are calculated based upon the estimated uncertainties in the 
PR reflectivity data, the altitudes of the bright band and basal reflectivity bins, and the 
surface backscatter cross-section. Since the error bars do not vary greatly over the 
distribution of plotted values, they are plotted for only one representative point in each 
panel. 

Also plotted in the different panels of Fig. 12 are the optical depths corresponding to 
corrected basal reflectivities calculated using the melting layer model developed in the 
present study. The melting model profiles are initialized using the snow and graupel 
water contents just above the freezing level in stratiform areas of the TOGA1 and 
TOGA3 model simulations. Four variations of the melting layer simulations are 
represented in the figure. First, a melting simulation in which partially melted ice 
hydrometeors are "refrozen", such that the meltwater is converted to an ice-air mixture 
with the same material density as the remainder of the frozen particle. Only when a snow 
or graupel particle is completely melted is it converted to pure liquid (raindrop). This 

47 



first simulation therefore represents a control in which the dielectric properties of mixed- 
phase, partially-melted ice particles are not considered. In a second melting simulation 
the dielectric properties of snow are described by the FS core-shell model, and graupel 
dielectric properties are described by the MGwi model. A third melting simulation again 
incorporates the FS core-shell model for snow, but substitutes the ML97 model for 
graupel. A fourth simulation incorporates the FS core-shell and MGwi models for snow 
and graupel, respectively, but alters the density distributions of snow and graupel 
according to (43) and (44). 

Even considering the uncertainties in the PR-observed optical depths, Fig. 12 
indicates a significant range of melting layer optical depths corresponding to corrected 
basal reflectivities between 25 and 40 dBZ. Optical depths between 0.025 and 0.35 are 
derived from the mirror-image technique, and there is a trend of higher optical depths 
with higher basal reflectivities. In contrast, the melting model containing no mixed-phase 
particles (Fig. 12a) produces systematically lower optical depths for a given basal 
reflectivity. This systematic difference can be partly explained by the uncertainty in the 
observed optical depths and natural variations in precipitation particle size distributions 
that are not represented in the melting model. Since the current melting model does not 
include a description of the evolution of the rain drop-size spectrum, the issue of varying 
particle distributions cannot be fully addressed here. 



48 



However, the presence of mixed-phase, melting ice hydrometeors can at least partly 
explain the observed optical depth distribution. Plotted in Fig. 12b are the optical depth - 
corrected basal reflectivity pairs from the melting model simulation based upon the FS 
core-shell model for snow and the MGwi model for graupel. Note that there is a much 
greater breadth of optical depths produced by this simulation and much greater overlap 
with the observed distribution of optical depths, in comparison to the simulation without 
mixed-phase particles. The greater overlap of the distributions is the result of two 
effects: first, there is a general increase in the optical depths of the simulated melting 
layers due to absorption by mixed-phase particles; second, the corrected basal 
reflectivities are slightly lower due to greater attenuation by precipitation above the bright 
band. Still, in the observed distribution at lower basal reflectivities there are a few 
relatively high melting layer optical depths that are not explained by the FS/MGwi 
simulation. 

If the ML97 model is substituted for MGwi to describe the dielectric properties of 
graupel (Fig. 12c), somewhat lower optical depths result, but these are still generally 
higher than those produced by the model with no mixed-phase particles. Even though 
ML97 provides a fairly rigorous description of homogeneous melting particles, the 
radiative extinction produced by these particles appears to be insufficient to explain the 
observed optical depths. Snow and graupel particles with empirical size-dependent 
densities are substituted for the constant-density particles from the GCE parent 

49 



simulations to obtain the modeled optical depths and basal reflectivities of Fig. 12d. 
Note that the modeled optical depths have a greater spread than those produced by the 
constant-density particles at basal reflectivities less than about 33 dBZ, and they are 
generally lower than those produced by the constant-density particles at higher basal 
reflectivities. A simple explanation of these distributions is difficult, since the variable- 
density particles generally produce shallower melting layers with smaller optical depths, 
but at the same time, corrected basal reflectivities are also reduced (Fig. 9a, b vs. Fig. 
10a, b). Overall, the optical depth distributions from the constant- and variable-density 
particles are quite similar. 

Although the foregoing analysis does not confirm the validity of any particular 
melting model simulation, it does suggest that greater consistency between observed and 
simulated radiative properties of the melting layer can be achieved when the dielectric 
properties of mixed-phase, partially-melted precipitation particles are included in the 
simulations. Moreover, the FS, core-shell dielectric model for snow, combined with the 
MGwi dielectric model for graupel, produces the greatest overlap between the observed 
and simulated optical depth/reflectivity distributions. The observed distribution of 
melting layer optical depths includes some relatively large values that are not simulated 
by any of the melting models. These optical depths may be due to uncertainties in the 
observations and/or a lack of realism/generality in the melting simulations, including 
errors in the dielectric modeling of melting particles, precipitation particle size 

50 



distributions which deviate from those observed, and a lack of robust stratiform regions 
overlying drier air which produce greater amounts of ice precipitation but less rainfall. 



4. Summary and Outlook 

In this study, a 1-D steady-state microphysical model which describes the vertical 
distribution of melting precipitation particles is developed. The model is driven by the 
ice-phase particle (snow, graupel) distributions just above the freezing level at applicable 
horizontal gridpoints of parent 3-D cloud-resolving model simulations, and extends these 
simulations by calculating the number density and meltwater fraction of each particle in 
finely-separated size categories through the melting layer. The depth of the modeled 
melting layer is primarily determined by the initial material density of ice-phase particles: 
distributions of constant-density snow (0. 1 g cm' 3 ) and graupel (0.4 g cm' 3 ) generally melt 
over a deeper layer than distributions of snow and graupel having densities that decrease 
with particle size. 

The radiative properties of melting precipitation at microwave frequencies are 
calculated based upon different methods for describing the dielectric properties of mixed- 
phase particles. Particle absorption and scattering efficiencies from 10.65 to 85.5 GHz 
are enhanced greatly for relatively small (-0.1) meltwater fractions. The relatively large 
numbers of these partially-melted particles just below the freezing level in stratiform 
regions lead to significant microwave absorption, exceeding the absorption by rain below 

51 



the melting layer by factors of 10, 5, 3, and 2 at 10.65, 19.35, 37.0 and 85.5 GHz, 
respectively, in one test profile. Calculated backscatter efficiencies at 13.8 GHz increase 
in proportion to the particle meltwater fraction, leading to a "bright-band" of enhanced 
radar reflectivities in agreement with previous studies. 

The radiative properties of melting layer are determined by the choice of dielectric 
models and the initial water contents and material densities of the ice-phase precipitation 
particles. In an attempt to resolve these sources of ambiguity in the melting layer model, 
four sets of melting profiles are generated based upon two tropical squall line CRM 
simulations. The control set contains no mixed phase precipitation, while the other sets 
include mixed-phase particles with radiative properties determined by different dielectric 
models and initial particle density distributions. The set of profiles based upon snow 
described by the Fabry-Szyrmer core-shell dielectric model and graupel described by the 
Maxwell-Garnett water matrix dielectric model, with constant material densities of 0. 1 g 
cm-3 and 0.4 g cm-3, respectively, leads to reasonable consistency with PR-derived 
melting layer optical depth distributions. Snow and graupel with initial size-dependent 
densities produce a similar optical depth distribution if the same dielectric models are 
utilized. A more general conclusion from the intercomparison is that the control profiles 
which do not contain mixed-phase particles yield optical depths that are systematically 
lower than those observed. Therefore, the use of the melting layer model to extend 3-D 



52 



CRM simulations appears justified, at least until more realistic spectral methods for 
describing melting precipitation in 3-D CRM's are implemented. 

Independent in situ measurements of particle size distributions in stratiform regions 
from airborne probe data have recently been performed as part of the TRMM field 
campaigns. Coupled with coincident airborne radiometer and radar observations, these 
data will hopfully lead to a more complete description of the microphysical and radiative 
properties of melting precipitation. A comparison of modeled and observed melting layer 
properties will be the focus of a future investigation by the authors of this study. 

In Part II of this series, the 3-D CRM simulations listed in Table 1, augmented by the 
1-D model in stratiform regions, serve as the basis for calculations of upwelling radiances 
at the TMI frequencies and computations of extinction/reflectivities at the PR frequency. 



Acknowledgments 

The authors would like to thank Brad Ferrier, Christian Kummerow, and Ye Hong 
for their helpful suggestions throughout the course of this study. Robert Meneghini and 
Liang Liao provided their invaluable expertise on the implementation of the radar "mirror 
image" technique. The research effort was supported by the TRMM Science program. 



53 



References 

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the Tropical Rainfall Measuring Mission (TRMM) precipitation radar. Proc. 8 th 

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Bauer, P., J. P. V. Poiares Baptista, and M. Iulis, 1999a: On the effect of the melting 

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54 



Ferrier, B., 1994: A double-moment multiple-phase four-class bulk ice scheme. Part I: 

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Fabry, F., and W. Szyrmer, 1999: Modeling of the melting layer. Part II: 

Electromagnetics. Submitted to J. Atmos. Sci. 
Haddad, Z. S., E. A. Smith, C. D. Kummerow, T. Iguchi, M. R. Farrar, S. L. Durden, M. 
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Liao, L., R. Meneghini, and T. Iguchi, 1999: Simulations of mirror image returns of 
air/space-borne radars in rain and their applications in estimating path attenuation. 
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Lin, Y.-L., R. D. Farley, and H. D. Orville, 1983: Bulk parameterization of the snow 
field in a cloud model. J. Climate Appl. Meteor., 22, 1065-1092. 



55 



Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation 

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Meneghini, R., and L. Liao, 1996: Comparisons for cross sections for melting 

hydrometeors as derived from dielectric mixing formulas and a numerical method. J. 

Appl. Meteor., 35, 1658-1670. 
Meneghini, R., and L. Liao, 1997: Effective dielectric constants of mixed-phase 

hydrometeors. Submitted to 7. Ocean. Atmos. Tech. 
Meneghini, R., T. Iguchi, T. Kozu, L. Liao, K. Okamoto, J. A. Jones, J. Kwiatkowski, 

1999: Use of the surface reference technique of path attenuation estimates from 

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Mitchell, D. L., R. Zhang, and R. L. Petter, 1990: Mass-dimensional relationships for ice 

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Mugnai, A., H. J. Cooper, E. A. Smith, and G. J. Tripoli, 1990: Simulation of microwave 

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Meteor. Soc, 71, 2-13. 
Panegrossi, G., S. Dietrich, F. S. Marzano, A. Mugnai, E. A. Smith, X. Xiang, G. J. 

Tripoli, P. K. Wang, and J. P. V. Poiares Baptista, 1998: Use of cloud model 

microphysics for passive microwave-based precipitation retrieval: Significance of 

56 



consistency between model and measurement manifolds. J. Atmos. Sci., 55, 1644- 

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60. 



57 



Smith, E. A., C. D. Kummerow, and A. Mugnai, 1994: The emergence of inversion-type 

precipitation profile algorithms for estimation of precipitation from satellite 

microwave measurements. Remote Sens. Rev., 11, 21 1-242. 
Szyrmer, W. and I. Zawadski, 1999: Modeling of the melting layer. Part I: Dynamics 

and microphysics. J. Atmos. Sci., in press. 
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description. Terrest. Atmos. Oceanic Sci., 4, 35-72. 
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of a tropical cyclone. Meteor. Atmos. Phys., 49, 229-254. 



58 



Tables 



Table 1. Characteristics of the parent 3-D cloud-resolving model simulations. 

Approx. Horizontal Grid 

Model/Simulation Date Location Resolution Ikml Dimensions Duration 
GCEATOGA1 2/22/93 9S, 159E lxl 128x128x30 6 hours 



GCE/TOGA3 



2/22/93 9S, 159E 3x3 



128 x 128 x 30 6 hours 



GCEfTOGA2 12/19/92 2S, 155E 2x2 256x256x40 7 days 



UW-NMS/ 
COHMEX 



7/1 1/96 35N, 87W 1 x 1 



51x51x42 4 hours 



UW-NMS/ 
HURRICANE 



9/11/88 19N,70W 3.3x3.3' 62x62x42' 6 hours' 



'refers to highest-resolution nested grid 



59 



Figure Captions 



Fig. 1. Schematic of the grid domain of the 1-D melting layer model. 



Fig. 2. Plan view of the surface rainfall rate distribution at 180 min from the TOGA1 
CRM simulation. The location of the test profile is indicated by the cross-hairs. 



Fig. 3. Melting layer model vertical profiles of (a) precipitation water contents, and (b) 
precipitation mass fluxes based upon melting of snow and graupel with size-independent 
densities. Panels (c) and (d) are the same as (a) and (b), but for snow and graupel 
densities that decrease with size. 



Fig. 4. The real and imaginary parts of the refractive indices of melting snow based upon 
different dielectric constant models at (a) 10.65 GHz, (b) 19.35 GHz, (c) 37 GHz, and (d) 
85.5 GHz. The lower-left vertex of the model curves in each panel represents dry snow, 
while the upper-right vertex represents completely melted snow. The "+" on each model 
curve represents a melted particle fraction of 0.5 by volume. 



Fig. 5. Same as Fig. 4 but for melting graupel. 



60 



Fig. 6. Absorption and scattering efficiencies at 10.65 GHz (a and b), 19.35 GHz (c and 
d), 37.0 GHz (e and f), and 85.5 GHz (g and h), plotted as functions of particle radius 
based upon the Fabry-Szyrmer core-shell dielectric constant model for snow particles 
with meltwater volume fractions of 0.0, 0.01, 0.1, 0.4, and 1.0. 



Fig. 7. Melting layer vertical profiles of (a) absorption coefficient, and (b) scattering 
coefficient at 10.65, 19.35, 37.0, and 85.5 GHz, based upon the Fabry-Szyrmer core-shell 
dielectric constant model applied to the snow and graupel distributions plotted in Fig. 3a. 
Absorption coefficient and scattering coefficient profiles in panels (c) and (d) are based 
upon the Maxwell-Garnett water matrix dielectric constant model. 



Fig. 8. Extinction and backscatter efficiencies at 13.8 GHz, plotted as functions of 
particle radius based upon the Fabry-Szyrmer core-shell dielectric constant model for 
snow particles with meltwater volume fractions of 0.0, 0.01, 0.1, 0.4, and 1.0. 



Fig. 9. Melting layer vertical profiles of (a) extinction, and (b) reflectivity at 13.8 GHz, 
based upon the Fabry-Szyrmer core-shell dielectric constant model applied to the snow 
and graupel distributions plotted in Fig. 3a. Extinction and reflectivity profiles in panels 
(c) and (d) are based upon the Maxwell-Garnett water matrix dielectric constant model. 

61 



Fig. 10. Melting layer vertical profiles of (a) extinction, and (b) reflectivity at 13.8 GHz, 
based upon the Fabry-Szyrmer core-shell dielectric constant model applied to the snow 
and graupel distributions plotted in Fig. 3c. 



Fig. 11. Schematic of reference levels utilized to determine melting layer extinction 
optical depths from measurements of the Precipitation Radar. 



Fig. 12. Extinction optical depths of simulated melting layers (black dots), plotted as 
functions of attenuation-corrected basal reflectivities, using (a) no mixed-phase particles, 
(b) Fabry-Szyrmer core-shell model snow and Maxwell-Garnett water matrix model 
graupel, (c) Fabry-Szyrmer core-shell model snow and Meneghini-Liou 1997 model 
graupel, and (d) Fabry-Szyrmer core-shell model snow and Maxwell-Garnett water- 
matrix model graupel (but for snow and graupel densities that decrease with particle 
size). Melting layers are derived from the TOGA1 and TOG A3 cloud-resolving model 
simulations; ref. Table 1. Also, observed melting layer extinction optical depths 
(diamonds) are plotted as functions of attenuation-corrected basal reflectivities. 
Observed optical depths are estimated using the mirror-image technique, applied to 
observations of the Precipitation Radar at low latitiudes. Representative error bars are 
indicated for one pair of observations. 

62 



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0.0 



(a) FS snow 



i — •- 



fliq = 0.00 

f!iq = 0.01 

fliq = 0.10 

fliq = 0.40 

fliq= 1.00 



/ 



/ 




12 3 4 

PARTICLE RADIUS [mm] 



(c) FS snow 




2 3 4 

PARTICLE RADIUS [mm] 



(e) FS snow 




2 3 

PARTICLE RADIUS [mm] 



(g) FS snow 




12 3 4 

PARTICLE RADIUS [mm] 





2.0 




1.8 


cz 


1.6 


a 
o 

O 


1.4 
1.2 


X 

o 

o" 


1.0 
0.8 
0.6 
0.4 




0.2 




0.0 



(b) FS snow 



- 


I ' 




I ' I ' 




- 






/ / 
/ / 
/ / 


- 


- 






y / 

s 


- 


~ 


I _^-=^ 


<-' 


-+- — , "" I 


- 



(d)FS 



2 3 

PARTICLE RADIUS [mm] 



snow 




2 3 4 

PARTICLE RADIUS [mm] 



(f) FS snow 




12 3 4 

PARTICLE RADIUS [mm] 





2.0 




1.8 




1.6 
1.4 
1.2 


s 
Q 


1.0 
0.8 


in 

oo 


0.6 
0.4 




0.2 




0.0 



(h) FS snow 



- 


1 1 1 1 , 1 . 1 1 


- 


^cT .^.ir-^-\ 1,1.1, 



2 3 

PARTICLE RADIUS [mm] 




O 

J 
W 
Q 
O 
2 
o 

to 
> 

< 



o 



o.offiIF5 



-0.5 - 



-1.5 T 



0.0 0.2 0.4 0.6 0.8 

ABSORPTION COEFFICIENT [km-1] 




0.1 0.2 0.3 0.4 0.5 

SCATTERING COEFFICIENT [km-1] 



(c) MGwi 



O 
H 

W 
Q 
O 

s 

o 

H 
to 
> 

H 
< 

►J 

H 
33 

a 
to 




-3.0 *■ 



0.0 



-0.5 



-1.0 



-1.5 



-2.0 



-2.5 - 



(d) MGwi 



-3.0 



II' ' ' ' 


' / I ' I ■ ' 
/ oc 


- 


r! 




- 


\ \ 




- 


\ "■- - 






= / . ; 






/ >' f 






f ' I 




- 


f ,' 




- 


- I I 




- 


1U.CD UriZ 




19.35 GHz 






37.0 GHz 




'.ill 




- 


OJ. J OllZ 

1,1, 



ABSORPTION COEFFICIENT [km-1] 



0.0 0.1 0.2 0.3 0.4 0.5 

SCATTERING COEFFICIENT [km-1] 



X 

u 

a 

DC 

u 

oq 



4.0 
3.5 
3.0 
2.5 
2.0 
1.5 
1.0 
0.5 
0.0 



(a) FS snow 



~ ■ — i — ■ — i — 

fliq = 0.00 

fliq = 0.01 

fliq = 0.10 

fliq = 0.40 

fliq = l.OO 




12 3 4 

PARTICLE RADIUS [mm] 



(b)FSsnow 




12 3 4 

PARTICLE RADIUS [mm] 



E 

a. 
O 
H 

J 

w 

Q 
O 



> 

p 

< 

H 

a 
w 

DC 



^•? 



0.0 



-0.5 



-1.0 



-1.5 



-2.0 



-2.5 



(a)FS 



-3.0 



0C 




Snow 

Graupel 

Rain 

Total 



_L 



0.00 



0.05 



0.10 



0.15 



I 

Oh 

o 

H 
_! 

a 

Q 
O 

S 
o 

H 

> 

P 
< 



ac 
a 



13.8 GHz EXTINCTION COEFFICIENT [km- 



0.20 

-1] 



(b)FS 



-0.5 




-2.0 



-2.5 



-3.0 



25 30 35 

13.8 GHz RADAR REFLECTIVITY [dBZ] 



40 



MGwi 



0.0 ,W MGwi 



E 
^t 

a, 
O 
H 
-J 
w 
Q 
O 

o 

H 
W 
> 

H 

< 



DC 
O 



-1.0 




-2.0 - 



-2.5 - 



0.05 0.10 0.15 

GHz EXTINCTION COEFFICIENT [km 




0.00 0.05 0.10 0.15 0.20 

13.8 GHz EXTINCTION COEFFICIENT [km-1] 



0.0 



(b)FS 



-u.t> 


A \ 




\ \___ 


-1.0 


_ ~ ~~— - _^^ 




__ — — 


-1.5 




-2.0 




-2.5 




t n 


1 i 



20 25 30 35 

13.8 GHz RADAR REFLECTIVITY [dBZ] 



^ 



n 



radar antenna 




'<mmmzr//A 



reflectivity 



altitude of 
reflectivity bin 

h 



l bb 



h 



bb 



0.75 km 
0km 




range 



(a) No Mixed-Phase 




o.o 



25 30 35 40 45 

CORRECTED BASAL REFLECTIVITY [dBZ] 



0.5 



— 0.4 



° 0.3 

< 
u 

P 
a. 
O 0.2 



(b)FS/MGwi 



i 

in 



0.1 



0.0 



' I 


1 


1 ' 


i 










.' 




• Mode 










O Observed 




• • 


- 








»• * 




- 




o 




- 


T 


o 


o<^ 






- 


o 


o 


oc 




- 


HiH 


o <*> 




ts. • • 


_ 


■ 




°<A 


'nrlvs 




- 










~ 


^J^rfl 


^BPfc*^« 


<5 




- 




o 






p *"' ■ i 


1 


i 


i 





25 30 35 40 45 

CORRECTED BASAL REFLECTIVITY [dBZ] 



0.5 



(c) FS/ML97 



0.4 - 



• Model 
O Observed 



* o JgfcS ] 




25 30 35 40 45 

CORRECTED BASAL REFLECTIVITY [dBZ] 



0.5 



— 0.4 

DC 
f- 
cu 
w 

Q 0.3 

< 
u 



(d)FS/MGwi -R„(D) 



• Model 
O Observed 



O 



O 



• • • 

• . »•• • 

• • •* 



o o*> jb r. • • 




25 30 35 40 45 

CORRECTED BASAL REFLECTIVITY [dBZ]