# Dual-frequency Spaceborne Doppler Radar: analysis of performances in estimating latent heat fluxes  {#dual-frequency-spaceborne-doppler-radar-analysis-of-performances-in-estimating-latent-heat-fluxes .unnumbered}

Simone Tanelli^a^ , Jonathan Meagher^a^, Eastwood Im^a^ , Luca
Facheris^b^

^a^ Jet Propulsion Laboratory, California Institute of Technology,
Pasadena CA, USA\
^b^ Dipartimento di Elettronica e Telecomunicazioni, Università di
Firenze, Firenze, Italy

**ABSTRACT**

Knowledge of the global distribution of the vertical velocity of
precipitation is important for estimating latent heat fluxes, and
therefore in the general study of energy transportation in the
atmosphere. Such knowledge can only be acquired with the use of
spaceborne Doppler precipitation radars. Recent studies have shown that
the average vertical velocity can be measured to acceptable accuracy
levels by appropriate selection of radar parameters. Furthermore,
methods to correct for specific errors arising from Non-Uniform Beam
Filling effects and pointing uncertainties have recently been developed.

As detailed in the Global Precipitation Mission (GPM) preparatory
studies, the use of a dual-frequency precipitation radar allows improved
estimation of the main parameters of the hydrometeor size distribution
(bulk quantity and one shape parameter). Such parameters, in turn, lead
to improved estimates of latent heat fluxes. In this paper we address
the performances of a dual- frequency Doppler Precipitation Radar (DDPR)
in estimating the latent heat fluxes from the measured rainfall vertical
velocity and DSD parameters.

**Keywords**: Spaceborne precipitation radar, Doppler

# INTRODUCTION

The heating inside precipitating cloud systems plays a major role in
defining the dynamics of the atmosphere. In particular, several studies
showed that determination of the temporal and spatial variations of
vertical heating could be a vital aid for numerical weather and climate
prediction^1^. In general, the only direct means to obtain the latent
heating profile is through knowledge of the vertical velocity field and
the ability to classify the precipitation into convective, stratiform
and transitioning modes^2^. However, aside from isolated wind profilers,
there are currently no systems capable of sampling the vertical
component of the wind over any significant spatial extent. It is
therefore impossible to validate the vertical-wind assumptions (obtained
from cloud models, or vorticity methods applied to ground based Doppler
radar data) which are required to characterize convection in synoptic
and global atmospheric models.

By design, a Dual-Frequency Doppler Precipitation Radar (DDPR) would
detect hydrometeors and estimate their vertical speed, i.e. it would
measure the reflectivity and vertical velocity of hydrometeors with a
high spatial resolution. In addition, the dual-frequency measurements
(at 14 and 35 GHz) were adopted to estimate the first-order statistics
of the raindrop size distribution, this allows one to diagnose the wind
divergence and latent heating fields directly from the measurements.
However, the technological challenges presented by a DDPR have been, in
the past, the main reason of concern for the design of a spaceborne
mission. Because of such concerns, no spaceborne Doppler radar for
atmospheric observation is yet operating. However, the technology to
develop such instrument has developed significantly in the last ten
years and spaceborne Doppler technology for atmospheric radars is now
available. In fact, a Doppler Cloud Profiling Radar is currently under
study in Japan for an ESA program ^3^. In this paper we will address the
overall feasibility of a DDPR and will provide the preliminary
guidelines to assess its performances in estimating the vertical
profiles of latent heat.

# HEATING IN CLOUD SYSTEMS

The heating budget equations ^4^ define \'apparent heat source\' *Q~1~*
and the \'apparent moisture sink\' *Q~2~* averaged over an area *A*
including a generic cloud system:

$Q_{1} = \frac{\partial\overline{s}}{\partial t} + \nabla \cdot \overline{s}\overline{V} + \frac{\partial\overline{s}\overline{\omega}}{\partial p} = Q_{R} + L_{v}(\overline{c} - \overline{e}) + (L_{v} + L_{f})(\overline{d} - {\overline{s}}_{}) + L_{f}(\overline{f} - \overline{m}) - \nabla \cdot \overline{s}'\overline{V}' - \frac{\partial\overline{s'\omega'}}{\partial p}$
()

$Q_{2} = - L_{v}\left( \frac{\partial\overline{q}}{\partial t} + \nabla \cdot \overline{q}\overline{V} + \frac{\partial\overline{q}\overline{\omega}}{\partial p} \right) = L_{v}(\overline{c} - \overline{e}) + L_{v}(\overline{d} - {\overline{s}}_{}) + L_{v}\nabla \cdot \overline{q}'\overline{V}' + L_{v}\frac{\partial\overline{q'\omega}}{\partial p}'$
()

where *s* is the dry static energy, *t* is time, *∇⋅**V*** is the wind
divergence, *p* is pressure, ω is the vertical p-velocity (*i.e.*,
*dp/dt*) , *Q~R~* is the heating rate due to radiation, *L~v~* is the
latent heat of vaporization, *L~f~* is the latent heat of fusion, *q* is
the specific humidity and *c*,*e*,*d*,*s~\*~* ,*f*,*m* are the rates of
condensation, evaporation, deposition, sublimation, freezing and
melting, respectively. The overbar represents the horizontal average
over the area *A*, and the prime represents deviation from the average.
The heating quantities are often expressed in terms of K/day by dividing
them by the specific heat of air *c~p~* (constant pressure). These
equations have been applied to several diagnostic budget studies in the
last thirty years with the purpose of estimating the mechanisms of heat
transfer from a cloud system to the large scale environment.

The first level discrimination of the profiles of latent heating depends
on the classification of the precipitating system. In fact, early
results showed that the heating profiles in stratiform and convective
systems are markedly different. Furthermore, it was found that heating
profiles of different stratiform systems are generally consistent with
one another and from place to place ^5^. In general, the mean vertical
velocity is upward above the freezing level and downward below it. The
magnitude of such velocities is in the tens of cm/s and the vertical
profiles are quite consistent throughout the globe and across the
seasons. These velocities are consistent with the studies that predict
heating in the upper troposphere (mainly due to latent heat release
during condensation and deposition) and cooling in the lower troposphere
(absorption of latent heat by melting within the melting layer and
evaporation below it).

On the other hand, it was observed that this is not the case for
convective systems: not only the heating profiles of convective
precipitation were found to vary from place to place ^5^ but field
campaigns have shown^6^ that the convective regimes (and the associate
heating profiles) change significantly at the same location, in the same
season, and even within the same mesoscale system, hence adding further
uncertainty to the characterization of convection. Maxima of vertical
velocity of several m/s (absolute value) are found sometimes in the
lower troposphere and sometimes in the upper troposphere, and the
corresponding vertical profiles of latent heating derived from
diagnistic studies are markedly different. Part of these inconsistencies
is due to different observational methods (not all of them being direct
measurements of vertical velocity), but significant differences derive
from the actual variabily of the characteristics of convection even
within the same system. For example, updrafts and downdrafts are often
coupled, but the relative intensity and position (i.e., altitude at
which they occur and horizontal distance) are very variable. Their
contributions to the overall latent heating profile are significantly
different in fact condensation and freezing result from updrafts while
evaporation and melting are associated to downdrafts.

Equation (1) can be reshaped ^7^ to isolate the contributions from three
portions of a large area *A* that includes a generic Mesoscale
Convective System (MCS): the environment (cloud-free, indicated by the
subscript *e*), the stratiform portion of the cloud system (small
vertical air motions, indicated by the subscript *s*) and the convective
portion of the cloud system (the convective cells embedded in the
stratiform area, indicated by

the subscript-index *i* in order to discriminate each convective cell):

$\begin{array}{r}
\frac{\partial\overline{s}}{\partial t} + \nabla \cdot \overline{s}\overline{V} + \frac{\partial s_{e}\overline{\omega}}{\partial p} - A_{e}Q_{\text{Re}} = \\
A_{s}\left\lbrack Q_{\text{Rs}} + L_{v}({\overline{c}}_{s} - {\overline{e}}_{s}) + (L_{v} + L_{f})({\overline{d}}_{s} - {\overline{s}}_{{}_{s}}) + L_{f}({\overline{f}}_{s} - {\overline{m}}_{s}) - \frac{\partial\omega_{s}}{\partial p}(s_{s} - s_{e}) + \frac{\partial A_{s}}{\partial t}\frac{(s_{s} - s_{e})}{A_{s}} \right\rbrack + \\
 + \sum_{i}^{}{A_{i}\left\lbrack \underset{(i)}{Q_{\text{Ri}}} + \underset{(\text{ii})}{L_{v}({\overline{c}}_{i} - {\overline{e}}_{i})} + \underset{(\text{iii})}{(L_{v} + L_{f})({\overline{d}}_{i} - {\overline{s}}_{{}_{i}})} + \underset{(\text{iv})}{L_{f}({\overline{f}}_{i} - {\overline{m}}_{i})} - \underset{(v)}{\frac{\partial\omega_{\text{is}}}{\partial p}(s_{i} - s_{e})} + \underset{(\text{vi})}{\frac{\partial A_{i}}{\partial t}\frac{(s_{i} - s_{e})}{A_{i}}} \right\rbrack}
\end{array}$ ()

The six terms in the contribution of each portion (*i.e.*, inside the
square brackets) are: (i) net radiative heating, (ii) latent heat
exchange due to condensation and evaporation of hydrometeors, (iii)
latent heat exchange due to deposition and sublimation of hydrometeors,
(iv) latent heat exchange due to melting and freezing of hydrometeors,
(v) convergence of sensible heat flux, (vi) expansion and contraction of
the cloud cluster.

The relative importance of each term in the right-hand side of Eq. (3)
depends on the size of *A* with respect to *A~cloud~ = A~s~ + ΣA~i~* and
on the assumptions that can be made about the characteristics of the
large-scale environment. For example, assuming that *A~cloud~ / A* is
large enough so that the environmental radiation term is not important,
the typical assumption for tropical systems is that vertical advection
dominates the left hand term and that static energy is almost
altitude-invariant. Under these conditions the net heating represented
by the right hand side is a direct indication of the vertical motion in
the area A ^7^. In general, the most relevant terms on the are the
contributions by latent heat absorption/release (ii) and (iii).

A practical approach to estimate the rates of phase change appearing in
(ii) and (ii) from remote sensors is to derive them from the vertical
variation of the mass flux of hydrometeors. For example the Hydrometeor
Heating (HH) algorithm ^9,10^ assumes that the rate of change from phase
1 to phase 2 can be written as:

$c_{1,2} = \left| \frac{\partial F_{1}(z)}{\partial z} - \frac{\partial F_{2}(z)}{\partial z} \right|$
()

where the subscript indicates the hydrometeors in the corresponding
phase and *F(z)* is the mass flux:

$F(z) = M(z)\left\lbrack v_{T}(z) - w(z) \right\rbrack$ ()

where *M(z)* is the mass content, *w(z)* is the vertical wind velocity
(positive upwards) and *v~T~(z)* is the mass weighted average terminal
velocity of the hydrometeors. The mass content *M(z)* is routinely
estimated through spaceborne radar and/or radiometric measurements of
received power. When measurements of vertical velocity are not available
*w(z)* is estimated from modeled cloud-scale winds and *v~T~(z)* is
derived from estimates of the hydrometeor density and size distribution,
in fact:

$v_{T} = \frac{\text{πρ}}{6M}\int_{}^{}{v_{t}(D)D^{3}N(D)\text{dD}}$ ()

where *v~t~(D)* is the terminal velocity in still air of an hydrometeor
of size *D*, *N(D)* is the particle size distribution, and *ρ* is the
water density. Note that the terminal velocity *v~t~(D)* depends
significantly on the hydrometeor kind and density, therefore it is
crucial to be able to discriminate the amounts of each hydrometeor type
(*e.g.*, rain, snow, graupel, etc.). Furthermore, this approach does not
account for the horizontally advected masses of hydrometeors, therefore
analysis of the three-dimensional spatial distribution of the
hydrometeors is necessary to refine estimates of latent heating
profiles.

Perhaps the most relevant effort in providing large-scale estimates of
the heating in precipitating systems derived from the Tropical Rainfall
Measuring Mission (TRMM). Notably, the TRMM Precipitation Radar (PR) is
a Ku-band, non-Doppler radar paired to a multifrequency radiometer (TMI)
and other passive sensors. Three algorithms for estimating latent heat
have been developed within TRMM: the first is the HH algorithms based of
the vertical mass flux analysis described above. Correcting terms are
added to account for the mass flux of cloud liquid content and the
calibration uncertainties. Since the cloud-scale vertical velocity *w*
is not measured by TRMM, an estimated velocity is regressed from the
vertical profiles relying on information provided by cloud-resolving
model (CRM) simulations. The second is the Convective Stratiform Heating
(CSH) algorithm which needs information only on the surface rainfall
rate, amount of stratiform rain and the type and location of the
observed cloud system. The retrieval of latent heating profiles is
obtained through a look up table that characterizes convective profiles
for different regions of the globe and different seasons. The third is
the Goddard PROFiling Heating algorithm (GPROFH) which retrieves the
vertical profiles of latent heating by finding the CRM-simulated profile
that generates the forward simulated radiometric quantities best fitting
the measurements.

As recently demonstrated ^9^, the three algorithms provide estimates of
magnitude of maximum heating rates consistent among each other (4-6 K
day^-1^) and comparable to those provided by budget studies\
(4-10 K day^-1^). On the other hand, several differences are evident in
the retrieved vertical profiles and no evidence is yet available to
determine which algorithm (if any) is performing best. Among the
possible causes for this differences three factors seem to dominate: 1)
uncertainties in the estimation of surface rainfall rate, 2)
classification of convective vs. stratiform areas, 3) characterization
of convection in different portions of the globe and different seasons,
4) impact of different spatial resolution. In general, each of the three
algorithms would benefit of measurements of vertical velocity. In fact,
such measurements would a) expand the available databases that
characterize convection in different portions of the globe (CSH), b)
provide a very selective criterion to select a CRM-simulated hydrometeor
profile (GPROFH), c) provide direct estimates of vertical velocity (HH),
d) improve the reliability of convective/stratiform classification for
any given profile (all). Furthermore, estimates of *w* would allow to
calculate the wind divergence which would, in turn, provide estimates of
the eddy flux contribution in Eq. (1) (which is otherwise ignored), and
of the horizontally advected mass fluxes.

# SPACEBORNE MEASUREMENTS OF MEAN DOPPLER VELOCITY

Recent studies ^10,11,12^ have focused on the system requirements for a
Doppler Precipitation Radar mounted on a Low Earth Orbiting (LEO)
spacecraft. It was found that the high linear speed of a LEO satellite
causes a significant Doppler shift to the radar echo originated from
different portions of the antenna mainlobe. For narrow beam,
nadir-looking (or cross-track scanning) radars, such Doppler shift (in
m/s) is roughly proportional to *x v~s~ / h~s~* where *x* is the
along-track displacement of the target with respect to the spacecraft,
*v~s~* is the satellite velocity and *h~s~* is the local satellite
altitude. If the radar volume of resolution is uniformly filled with
hydrometeors, the result is that the natural rainfall Doppler spectrum
(determined by the different vertical velocity of the hydrometeors) is
broadened by the additional Doppler shift. The observed Doppler spectrum
is approximately Gaussian (provided that the pattern of the antenna
mainlobe is approximately Gaussian) with normalized spectral width:

$w_{N} = \frac{2}{\lambda\text{PRF}}\sqrt{\sigma_{R}^{2} + \frac{\theta_{3\text{dB}}^{2}v_{S}^{2}}{\text{16}\text{ln}(2)}}$
()\
where λ is the operating wavelength, *PRF* is the Pulse Repetition
Frequency, *θ~3dB~* is the antenna 3-dB width, and *σ~R~* is the spread
in hydrometeor vertical velocity due to different size, turbulence and
wind shear.

The normalized Doppler spectral widths that correspond to several system
configurations and regimes of turbulence are described in Table 1. The
category of moderately broad spectra (*i.e.*, 0.1 \< *w~N~* \< 0.3) that
can be obtained with antenna sizes *D~a~* \< 10 m is suitable only for
estimation of the first spectral moment ^13^, (i.e, the mean,
backscattering weighted, Doppler velocity) and the variance of such
estimate can be expressed as:

$\text{var}({\hat{v}}_{P}) = \frac{\lambda^{2}\text{PRF}^{2}}{4M_{S}}\left\lbrack \frac{w_{N}}{4\sqrt{\pi}} + 2w_{N}^{2}\frac{S_{N}}{S_{S}} + \frac{1}{\text{12}}\left( \frac{S_{N}}{S_{S}} \right)^{2} \right\rbrack$
()

where M~S~ is the number of samples, S~S~ is the signal power and N~S~
is the noise power. Several possibilities are available, in principle,
to remove the effect of the Doppler shift (*e.g.*, operation from a
geostationary platform, or use of a dual antenna on a LEO satellite, use
of a very large antenna on a LEO satellite), however they all present
further technological challenges and will not be addressed in this
paper.

If the radar volume of resolution is not homogeneously filled with
hydrometeors (*i.e.*, in Nonuniform Beam Filling conditions, NUBF) the
uneven weighting of the Doppler shift in different portions of the
mainlobe causes the observed Doppler spectrum^14^ to be not symmetric
around the mean Doppler velocity and introduces a bias of several m/s in
vertical velocity measurements. The proposed Combined Frequency Time
(CFT) spectral processing technique is able to remove such NUBF-induced
bias^15^ provided that several spectra are measured on volumes of
resolution partially overlapping in the along-track direction. A further
advantage of adopting CFT is that the along-track horizontal resolution
is improved to the distance covered by the spacecraft between
consecutive spectra (in the notional configuration considered in that
study this corresponds to approximately 75 m). On the other hand, the
price for such high quality information at one specific scan angle is
that limited resources are left for cross-track scanning. Preliminary
trade-off studies show that configurations with 3 or 5 beams per
cross-track scan could be profitably used together with CFT, but
performances degrade rapidly if that number is further increased. In
general, it was proved that an accuracy of \<1 m/s can be achieved with
a Doppler Precipitation Radar configuration with a 3 to 6 m antenna
size. Such accuracy can be obtained over homogeneous rainfall fields
through standard spectral moment estimators (such as Pulse Pair
processing or Fourier-based Spectral Analysis), or trough CFT
(independently of NUBF conditions).

For a near nadir pointing angle, the measured Doppler velocity is the
reflectivity weighted velocity of the hydrometeors:

$v_{P} = \frac{1}{\eta}\int_{}^{}{v_{p}(D)\sigma_{b}(D)N(D)\text{dD}}$
()

where *σ~b~(D)* is the backscattering cross-section of an hydrometeor of
diameter *D*, *v~p~(D)* is its vertical velocity (*v~p~(D) = v~t~(D) -
w*), and *η* is the radar reflectivity. The velocity *v~P~* is related
to the two more physically meaningful quantities discussed in Section 2:
the vertical wind speed *w* and the mass-weighted mean Doppler velocity
*v~M~=v~T~-w*. The problem of estimating *w* and *v~M~* from *v~P~*
measurements is related to the estimation *N(D)*. Assuming for example
that N(D) is Gamma distribution with mean (mass weighted) drop diameter
D~m~ and normalized standard deviation s~m~, the relation between *v~P~*
and *v~T~* is described in Figure 1. Assuming an uncertainty of 0.2 mm
on *D~m~* and no information on *s~m~* we have that the uncertainty in
estimating *v~T~* from *v~P~* is less than \<0.5 m/s.

Several techniques have been developed to estimate the first-order
moments of *N(D)* from radar and radiometric measurements. While it is
possible to envisage the combined use of a DDPR with radiometers (as it
has been done with non-Doppler systems on TRMM and in the planning of
GPM), we focus on the use of Dual-frequency radar measurements in order
to simplify the problem to simultaneous, multiparametric measurements
with identical spatial resolution. In general, dual-frequency radar
measurements rely on the fact that Mie scattering must be accounted for
in modeling the e.m. characteristics of hydrometeors at operating
frequencies above 5 GHz. It follows that the shape of *σ~b~(D)/η*
changes depending on the operating frequency and *D~m~* can be estimated
from dual-frequency radar measurements.

# ESTIMATION OF HEATING PROFILES WITH DOPPLER RADAR MEASUREMENTS

In this section we describe three contributions that simultaneous
measurements of dual-frequency reflectivity (*Z~m1~,Z~m2~*) and mean
Doppler velocity (*v~P~*) can bring to the goal of estimating the
heating associated with cloud systems.

## CONVECTIVE/STRATIFORM CLASSIFICATION

As aforementioned, the distribution of heating in the stratiform
precipitation areas is considerably different from the vertical profile
of heating in the convective regions. As a matter of fact, it is
suggested ^9^ that inconsistencies in the estimation of the convective
portion of a rainfall system might be one of the leading sources of
error in large scale estimates of vertical profiles of heating.

Stratiform precipitation is generally characterized by little or no
vertical air motion (*i.e.*, *\|w\|* \<\< 1 m/s) and by the presence of
a well defined melting layer of 0.3 to 1 km thickness which is
associated to a brightband signature in the radar reflectivity.
Spaceborne precipitation radars such as TRMM/PR are already recognized
as the optimal tool to discriminate stratiform systems from convection
and other precipitation types. This is mainly due to their higher
spatial (horizontal and vertical) resolution as compared to passive
sensors. While the performances of the classification are satisfactory
^17^, some inconsistencies are observed. Furthermore, the extension of
the observed latitudinal region from TRMM to GPM will preclude the usage
of some a priori conditions on the altitude of the brightband used in
the current TRMM algorithm for the detection of the brightband. In fact,
the TRMM classification algorithm ^16^ assumes that the brightband is
within a predetermined altitude range, typical of the tropical
latitudes, and that presence of reflectivities of 40 dBZ or higher below
the freezing level are always associated with convection, which has been
demonstrated to be not always true over the globe, see for example a
study performed in Montreal that showed that nearly half of the events
with radar reflectivity in the 40-45 dBZ range was resulting from non
convective rain ^18^.

We have recently developed a multiparametric radar brightband detection
algorithm ^19^ that makes use of dual-frequency reflectivity (14 and 35
GHz) and mean Doppler velocity measurements. The algorithm first
determines an approximate altitude of the melting layer by averaging
vertical velocity profiles over an area of several km (in order to
reduce the relative contribution of convective profiles). In stratiform
areas, the averaged velocity profile has a typical step-like shape
determined by the higher falling speed of liquid hydrometeors vs. low
density frozen hydrometeors (i.e., *v~P~* \~ 1 m/s above the melting
layer and *v~P~* \> 4 m/s below it). The first-guess of freezing level
altitude is set at 500 meters above the point where the average velocity
profile reaches its maximum. The algorithm proceeds on a
profile-by-profile basis, by finding the typical sharp increase in
reflectivity measurements that occurs when the frozen hydrometeors begin
their melting process. Such sharp increase is searched for at altitudes
within a 1.5 km range from the first-guess of freezing level and it is
associated with the top of the melting layer (*h~top~*). The reliability
of the detection of *h~top~* is significantly improved when both
reflectivity profiles are analyzed together. This is due to the fact
that any sharp increase in reflectivity due to the beginning of the
melting process must show a well defined and predictable correlation
between the two frequencies. Finally the negatively correlated
variations of reflectivity at 14 GHz and *v~P~* are analyzed starting
from the top of the melting layer to determine the bottom of the melting
layer (*h~low~*). In fact, in the lower part of the melting layer the
reflectivity at 14 GHz drops and the mean Doppler velocity increases
because the hydrometeors increase their density (hence reducing their
size and number density -- the latter because of the acceleration to a
faster terminal velocity). The reflectivity profile at 35 GHz is not
used here because the deeper Mie scattering regime at 35 GHz is such
that the reduction in average particle size does not always result in
correspondingly lower reflectivities. The algorithm was first developed
for the measurements obtained by the NASA/JPL airborne, dual-frequency,
dual-polarization, Doppler radar APR-2 ^20^ and therefore also the
vertical profiles of Linear Depolarization Ratio can be used to refine
the estimation of *h~low~*. It was found that, if no cross-polarimetric
measurements are available, the algorithm simply underestimates by \<10%
the melting layer thickness if measurements are obtained at the vertical
resolution of 30 m (APR-2 radar resolution). A spaceborne atmospheric
radar is in general designed to have a vertical resolution of 250m or
more for increased sensitivity, that is, significantly larger than this
underestimation which can be therefore ignored in first approximation.
Overall the algorithm was found to provide extremely reliable estimates
of the presence, altitude and thickness of the melting layer without the
a priori information currently used in non-Doppler
algorithms.![](media/image6.jpeg){width="2.902083333333333in"
height="2.1770833333333335in"}![](media/image7.jpeg){width="2.902083333333333in"
height="2.1770833333333335in"}

Profiles that do not show a brightband and whose vertical velocity
profile does not correlate with the vertical velocity profile of the
surrounding stratiform rain are classified as convective. Profiles with
no brightband but with vertical velocity consistent with the stratiform
ones are classified as other.

The convective vs. stratiform classification is therefore obtained with
no need for analyzing also the horizontal patterns of reflectivity (as
in TRMM classification algorithm).

## ESTIMATION OF SURFACE RAINFALL

A mass conservation approach^4^ applied to a vertical profile shows that
the total heating released by a portion of a precipitating system is
roughly equivalent to the liquid water mass flux exiting the column
multiplied by the specific latent heat of vaporization of water
(*L~v~*). Assuming that no horizontal advection is present, such mass
flux corresponds to the surface rainrate. Furthermore, at low altitudes
(*i.e.*, within the boundary layer) the average vertical wind over an
area of few km^2^ is negligible. Therefore, the 'still-air' rainfall
rate *R* typically obtained by rain retrieval algorithms is a good
estimator of the total heating.

Several groups of algorithms are used to estimate R from non-Doppler
spaceborne systems (single and dual frequency radar, multifrequency
radiometric or radar/radiometer combined algorithms). In principle, one
can modify any of them to exploit the information provided by Doppler
measurements of average vertical velocity.

Figure 3 shows the isopleths of three radar observables: radar
reflectivity (*Z*), specific attenuation (*k*) and mean Doppler velocity
(*v~P~*) as a function of liquid water content (*M*), mean (mass
weighted) drop diameter (*D~m~*) and normalized drop standard deviation
(*s~m~*). These values were calculated for a modified Gamma distribution
for the DSD at 14 GHz.

While each of the three observable parameters is only moderately
affected by the variability of *s~m~* (small spread of dashed and dotted
curves around the solid curves), it is clear that the measurements of
*v~P~* are more sensitive to *D~m~* than *Z* or *k*. Furthermore, for
radar operating at attenuating frequencies such as DAR, *Z* and *k* are
both derived from measurements of the apparent reflectivity factor
*Z~m~* and are therefore not truly independent (errors in estimating one
reflect to the other). On the other hand *v~p~* measurements are not
proportional to the returned power and are affected by different error
sources, they can therefore be considered independent from the pair
(*Z,k*). One should also note that even if *v~P~* measurements are
obtained with a 1 m/s accuracy, they provide valuable information on
*D~m~* and they do reduce the uncertainty on the estimation of DSD
parameters.

This rationale can be extended directly to dual-frequency algorithms. In
fact, the use of algorithms exploiting the information present in
differential measurements (*e.g.*, using the pairs (*Z~14~-Z~35~,
Z~14~*), (*k~35~-k~14~, Z~14~*) or even triplets obtained by similar
combiations) allows in principle to reduce significantly the range of
ambiguity. However, the aforementioned problems arising from the fact
that these parameters are obtained from the same measurements (of
returned power) are only partially mitigated by the fact that two power
measurements are now available (since the returned power at the two
frequencies is not completely independent in a statistical sense, one
reason being that all scatterers that fall in the Rayleigh regime at
both frequencies contribute equally to the observed reflectivity factor
at the two frequencies). Since all these algorithms aim at retrieving
*R* and *D~m~* , the availability of *v~p~* would help in increasing the
constraining of such retrieval (*i.e.*, three observables for two
unknowns) and therefore they increase our confidence in the retrieved
parameters.

## HYDROMETEOR HEATING METHOD

Estimates of vertical heating profiles through the HH algorithm are
sensisitive to the correct classification of the phase of hydrometeors.
As described in Section 4.1, such discrimination is easily obtained by a
DDAR on stratiform areas, or more generally, when no significant
vertical wind is present. Furthermore, the improved estimates of
rainrate and *D~m~* described in section 4.2 result in general in
accurate profiles of latent heating. On the other hand, in convective
areas (*i.e.*, where *w* is not negligible), frozen and liquid particles
can be discriminated only by dual-frequency reflectivity measurements
(which rely on the fact that the specific attenuation of frozen
particles is markedly smaller than that of liquid hydrometeors) and,
similarly, their bulk quantities and mean diameters cannot be improved
through Doppler measurements. Even assuming that the hydrometeor type,
bulk quantity, and mean diameters are well estimated by the
dual-frequency measurements, further uncertainty in estimating the
mass-flux *F(z)* is introduced in Eq. (5) by convective updrafts and
downdrafts if measurements of vertical velocity are not available. On
the other hand, the average (mass weighted) vertical velocity *v~M~* can
be estimated from measurements of *v~P~* with an error smaller than the
threshold that defines whether convection is present or not*.*

The performances of DDPR in estimating the latent heating of a
precipitating system have been analyzed through simulations. The case
study shown in Figure 4 is obtained from a CRM generated tropical storm
showing a very strong convective cell at km 20 and a weaker one at km 30
(see panel e for bulk quantities of rain and graupel, panel f for the
corresponding mass-fluxes and panel c for the hydrometeor vertical
velocity). The area between km 10 and km 20 is a stratiform area 'fed'
mainly by the first convective cell. The strong updraft of cell 1 occurs
between 3 and 15 km altitude and is connected with a high-level
downdraft around km 13 altitude and a low-level downdraft below 4 km
altitude. The DDPR measurements were simulated through a
three-dimensional Doppler radar simulator^14^ and the CFT technique^15^
was applied to retrieve *v~P~* . The DDPR was configured as follows:
not-scanning (nadir-looking only), PRF = 7kHz, antenna size 5m with
matching beams (Ka band using only 40% of the antenna size), pulse
duration 1.6 μs and 64 samples per spectrum (corresponding to a 240 m
vertical resolution and along-track sampling rate of 1/64 m^-1^). The
simulated reflectivities Z~14~ and Z~35~ are shown in panels a and b,
respectively, and the CFT retrieved *v~P~* is shown in panel d.

Panels g and h show latent heating estimates obtained through an adapted
version of the HH algorithm: in this version only the heating from
precipitating hydrometeors is calculated. In panel g, the mean terminal
velocities are obtained through hydrometeor-specific *M-v~T~* relations
and no vertical wind is assumed (HH0). In panel h, the mass flux is
obtained by estimating *v~M~* from the *v~P~* measurements as described
in section 2 (HHV).

![](media/image10.wmf){width="6.394444444444445in"
height="7.436805555555556in"}

The retrieved latent heating fields were compared to the latent heating
field generated by the CRM. While both retrievals are in good agreement
(i.e., within 20%) with the model on the column integrated latent
heating, significant differences occur in the vertical profiling. The
along-track averaged vertical profile generated by the CRM has a maximum
peak of 30 K/hr at 5 km altitude which is retrieved correctly by HHV
while HH0 has the maximum of 23 K/hr at 4.5 km. Also, the HHV profile
shows a high-level cooling of --10 K/hr at 12 km altitude in agreement
with the CRM-generated profile, while HH0 does not detect it. Both
algorithms fail to detect a second peak of 25 K/hr heating at 8 km
altitude and overestimate the low level cooling (-12 K/hr instead of --4
K/hr). These differences can be understood by accounting for the
features in the heating field indicated in panels g and h. The cooling
areas C3 and C4, and the heating areas H3 and H4 visible in panel h are
in good agreement with the CRM-generated latent heating field. Note that
C4 is erroneously replaced by H1 in the HH0 retrieval, H3 is
significantly underestimated and partially replaced by the cooling C2
due to a non-existing melting layer (updraft region corresponding to
re-freezing instead, see panel e), and C3 and H4 are also lost in the
HH0 retrieval. The weak high-level cooling C1 is overestimated by both
algorithms, and the low level heating H2 below the melting layer of the
stratiform region corresponds to a region of weak cooling in the model,
also the low level cooling C5 in the lower part of both convective cells
is not present in the model. The cause for the errors relative to C5 and
H2 has been clearly identified as the contribution of horizontal
advection.

# CONCLUSIONS

The results of a preliminary study on the performances of a
Dual-Frequency Doppler Precipitation radar in estimating the vertical
profiles of latent heat in precipitating system are presented in this
paper. The results of simulations show that the availability of
measurements of vertical velocity of the hydrometeors with the 1 m/s
accuracy achievable with the current technology by a spaceborne Doppler
radar would allow to directly estimate the vertical profiles of latent
heating

# ACKNOWLEDGMENTS {#acknowledgments .unnumbered}

The research described in this paper was based on the work performed for
the Tropical Rainfall Measuring Mission (TRMM) and the Global
Precipitation Measurements (GPM) programs at the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration, and at the University of
Florence (Italy) through a program funded by the Italian Space Agency.

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