AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-9-3145-20164-D-VAR assimilation of disdrometer data and radar spectral reflectivities for raindrop size distribution and vertical wind retrievalsMercierFrançoisfrancois.mercier@latmos.ipsl.frChazottesAymericBarthèsLaurentMalletCécilehttps://orcid.org/0000-0002-0943-3510SPACE (Statistiques, Processus, Aérosols, Cycle de
l'Eau)/LATMOS-IPSL,
CNRS-INSU, Laboratoire Atmosphères, Milieux, Observations
Spatiales, Université de Versailles-Saint-Quentin/Université Paris-Saclay, FranceFrançois Mercier (francois.mercier@latmos.ipsl.fr)20July2016973145316312October201525November20151April201631May2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/9/3145/2016/amt-9-3145-2016.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/9/3145/2016/amt-9-3145-2016.pdf
This paper presents a novel framework for retrieving the vertical raindrop
size distribution (DSD) and vertical wind profiles during light rain events.
This is also intended as a tool to better characterize rainfall microphysical
processes. It consists in coupling K band Doppler spectra and ground
disdrometer measurements (raindrop fluxes) in a 2-D numerical model
propagating the DSD from the clouds to the ground level. The coupling is done
via a 4-D-VAR data assimilation algorithm. As a first step, in this paper,
the dynamical model and the geometry of the problem are quite simple. They do
not allow the complexity implied by all rain microphysical
processes to be encompassed (evaporation, coalescence breakup and horizontal air motion are
not taken into account). In the end, the model is limited to the fall of
droplets under gravity, modulated by the effects of vertical winds. The
framework is thus illustrated with light, stratiform rain events.
We firstly use simulated data sets (data assimilation twin experiment)
to show that the algorithm is able to retrieve the DSD profiles and
vertical winds. It also demonstrates the ability of the algorithm to
deal with the atmospheric turbulence (broadening of the Doppler
spectra) and the instrumental noise. The method is then applied to
a real case study which was conducted in the southwest of France during
the autumn 2013. The data set collected during a long, quiet event
(6h duration, rain rate between 2 and 7mmh-1) comes from an optical disdrometer and
a 24GHz vertically pointing Doppler radar. We show that the
algorithm is able to reproduce the observations and retrieve realistic DSD and
vertical wind profiles, when compared to what could be expected
for such a rain event.
A goal for this study is to apply it to extended data sets for
a validation with independent data, which could not be done with our
limited 2013 data. Other data sets would also help to
parameterize more processes needed in the model (evaporation,
coalescence/breakup) to apply the algorithm to convective rain and to
evaluate the adequacy of the model's parameterization.
Introduction
Knowledge of the raindrop size distribution (DSD), both at the ground
and throughout the vertical atmospheric profile, is an important
topic. In remote sensing, parameters of the Z–R relationship,
allowing a radar reflectivity to be converted into a rain rate, depend on
rain microphysics. These parameters are generally based on a particular DSD
distribution and are supposed to be constant through the
entire atmospheric column. In practice, rain rates estimated from
reflectivity through the Z–R relationship can vary at least by
a factor of 2 depending on the DSD (). In
telecommunications, and especially for ground to space
telecommunication, rain-induced attenuation also depends on rain
microphysics, especially in the Ku band and beyond. More generally the
observation of vertical DSD profiles is fundamental to address many
questions regarding the numerous processes involved during a rain
event. A large number of investigations have been conducted in order
to model the vertical evolution of the DSD during rain for different
meteorological situations (; ; ;
; ; ; ). The
parameterization of the different phenomena occurring during droplet
fall is a complex task due to the great number of processes
involved. Among them, some are considered as predominant, notably
a raindrop can break up, coalesce or evaporate. It is also subject to
sorting by gravity along with updrafts, downdrafts and horizontal
wind. All these processes, playing an important role in the nature of
the DSD, are the subject of numerous studies. More specifically,
raindrop sorting induced by vertical drafts can play an important role
in the vertical DSD variability. For instance,
show that updraft structures can cause horizontal and vertical
sorting of raindrops with a lack of large raindrops (>3mm)
in the updraft core and an increase at the periphery. Moreover,
concerning evaporation, drizzle can never reach the ground in specific
meteorological situations (). On the other
hand, summarizes the main results on models concerning
the coalescence and breakup processes. However, he explains that
these models, built on laboratory or simulation experiments, generally
suffer from a lack of validation under real conditions. Nevertheless,
a very large number of direct or indirect observations of DSD are
available either on the ground, from disdrometers (among others,
), or at various
heights, from the use of Doppler radars
(). Several studies have been carried
out to compare these different observations
(). However, the
combination of these sources of information about the DSD is
a difficult task given their very different natures. It is even more
demanding when you add the different spatiotemporal scales and the
various locations of the measurements.
In order to improve vertical DSD profile retrievals, this study
establishes a framework allowing the assimilation at a fine scale
(<5min, 100m) of heterogeneous observations in
a dynamic model. The developed method allows the use of observations
from different instruments, each with its own spatiotemporal
resolution. Compared to direct observation retrievals, merging data in
a dynamic model includes spatiotemporal consistency to the DSD
retrievals. Moreover, it could help to assess and improve the model
parameterization.
In this paper, we will focus on disdrometer and
vertical radar reflectivity collocated observations. The data are
merged through the use of a 4-D-VAR assimilation algorithm. The
evolution in time and space of the DSD is based on a very simplified version
of the
propagation model (see Eq. ). It is actually limited to
the fall of droplets under gravity, solely modulated by the effect of
vertical wind. The goal of this study is to develop a new mathematical
framework in a simplified context. Once introduced, it will be tested
by evaluating its performance, sensitivity and limits on both simulated
data and real data (a calm stratiform real event of light rain). Note that,
even if it is simple, this model ensures a DSD which matches the measurements of all
instruments (and their different scales). This aspect is not negligible when,
as mentioned previously, you consider all the work done to establish
consistent Z–R relationships.
Improvements of this framework (encompassing evaporation, horizontal air motion,
coalescence/breakup) as well as validation, both requiring more measurements,
are deferred to future work allowed by extended data sets.
The paper is organized as follows. In Sect. , we present
the simple model used and discuss the terms of the complete propagation model, namely wind,
collision and evaporation terms. Then, model simplifications are
retained considering different hypotheses. General principles behind 4-D-VAR
data assimilation are presented and a focus is made on the cost
function and regularization terms. Disdrometer and radar observation
operators are then given. Finally we describe the algorithm used to
retrieve the DSD and vertical wind fields from radar and disdrometer
data. Then, in Sect. , the instruments available and the
case study are described. In Sect. , through a twin
experiment (i.e., on simulated data), we explore the impact of
observation errors in addition to non-modeled phenomena on the
algorithm performances as well as the quality of the
retrievals. Section gives some results obtained in
a stratiform real case study. Finally, Sect.
concludes the study. Perspectives are also drawn concerning the
improvements needed to extend the application of the method to a more
general context.
Assimilation method
The aim of this work consists in retrieving information about the drop
size distribution (DSD) vertical profiles by linking the measurements
made by different instruments at different scales and heights. To
carry out this relationship, we will use a dynamic model which
propagates the DSD through space and time (denoted below as
“propagation model”). In this section, firstly, we describe the
corresponding partial differential equation (PDE) model used. We also
introduce the associated discretization scheme, and the unknowns of
our model. Then, we explain how the 4-D-VAR data assimilation algorithm
combines the data available through the model to retrieve its unknowns.
The second part of this section details this algorithm.
Propagation model
As explained before, this study develops a simple framework and so
uses a very simple model presented in Sect. . This simple model
has the merit to add a spatiotemporal coherence to the DSD field. Since
it only incorporates the effects of gravity and vertical wind, it lacks the
complexity required to fully model rain.
In Sect. , we introduce a complete PDE governing the evolution of the DSD
(). This propagation model is not within the purview of this article. It is solely used
to discuss the different terms of the complete PDE (Eq. ) and see how their
inclusion or exclusion could affect the results. Finally, in Sect. ,
we present the effective ”simplistic” numerical model and the discretization underlying
hypotheses.
Simple propagation model
The DSD, noted N, is the number of raindrops by unit of volume and
diameter (unit: m-4). It is a function of time, position and
diameter. In the framework presented here, we will work in an air column,
and so limit the study to a time–height–diameter space, so that N=N(t,z,D).
Then, the PDE modeling the vertical evolution of DSD has the form:
∂N∂t(t,z,D)+∂∂z(v(D)+w(t,z)N(t,z,D))=0.
The first term of Eq. () represents the instantaneous
variation of N. The second term is the vertical advection of droplets:
w is the vertical wind (component of the 3-D-wind W along
the vertical axis uz); v is the terminal
velocity of droplets under gravity. In this study, the parameter v is
assumed to depend only on the diameter D, according to the
relation:
v(D)=9.65-10.3e-600D,
with D in m and v in ms-1. This relation was fitted to
be in good agreement with measurements (which are often
used as a reference) in the diameter range [0.6–5.8 mm]
().
Equation () can be considered as a simplified version of the model
proposed for instance in . We will now present this complete model and
discuss its different terms.
Complete propagation model (Hu and Srivastava, 1995) and discussion of its different terms
In the study, N depends on time, three coordinates of space and
diameter (N=N(t,x,y,z,D)) and obey the PDE:
∂N∂t+divvuz+WN+∂∂DτdDdmN=I.
We recognize the two first terms of Eq. () with the difference
that we have a 3-D wind W advection here. The third term of Eq. () represents
the evaporation. τ is the net rate of change of droplets' masses
(unit kgs-1) and dD/dm is the
derivative of the diameter–mass relation for spheric drops (unit
mkg-1). The last term I represents the collision effects
(drop mass changes not due to evaporation).
3-D wind W
The vertical wind is considered to add an offset to the terminal
velocity of drops. Additionally, the knowledge of the real droplet vertical
speed is critical to have information about the DSD from the return
power spectra of a vertically pointing Doppler radar. This question
has been widely investigated in the literature
() and it shows that we
have to take the vertical wind into account to properly deal with
Doppler radar data.
The horizontal wind is the main force which makes the droplets move in
the horizontal plane. However, with only one vertically pointing radar
and co-localized ground point measurements of the DSD, it is not
possible to record the horizontal variability of the
DSD. Consequently, as mentioned above, we limit the study to
a (z,t) plane.
This simplification is consistent if the horizontal air motion is weak or if the DSD is
homogeneous in the horizontal plane. (Of course the DSD can be
inhomogeneous with height.)
To limit the errors caused by this simplification, we will only treat quiet,
stratiform light rain events (see the case study in Sect. ) and we will
study the ability of the algorithm to reproduce the observations.
Evaporation (third term of Eq. )
The evaporation is mathematically modeled as a term of advection along
the mass of droplets coordinate. The advection speed (parameter τ
of Eq. ) depends on the drop diameter, and on several
meteorological variables, including pressure, temperature and mainly
humidity. proposes a parameterization of this term.
According to this parameterization and for long cold stratiform rain events
similar to the one investigated in this study (see Sect. ), we verified
(not shown) that evaporation can be neglected.
Collisions (fourth term of Eq. )
Collisions between drops can lead to coalescence (two drops producing
one drop) or breakup (two drops producing at least two
drops). Coalescence/breakup phenomena are generally assumed to be
an important factor governing the DSD temporal evolution
(), even if some studies reconsider this
assumption (). The phenomena have been
widely investigated. Numerous parameterizations have been
proposed. Among them, some characterize the ability of two drops to
coalesce, depending on the energy involved in the collision
( or ). Others are more
interested in the distribution of resulting drops in the case of
breakup. The resulting distributions can be based on laboratory
experiments, , or on numerical fluid dynamics models,
.
Since our main objective is to retrieve DSD profiles using data
assimilation, we restrict ourselves to a simple framework, and we do
not take into account the collisions between drops. Moreover, among
the wide literature mentioned above, many authors show that the
coalescence/breakup processes are less critical for low rain rates
(). Thus, this paper
will only focus on a light rain event in order to limit the errors caused
by this simplification. Additionally, as for the horizontal wind exclusion, we will
study the ability of the algorithm to reproduce the observations despite
this simplification.
Numerical model (discretization)
This simplified PDE used in this study (Eq. ) is discretized
on an (time/height/diameter) Arakawa C-grid (), meaning that
N is evaluated at the grid-box centers, while w is evaluated at the
grid-box faces. For this discretization of the PDE, we use the finite
difference scheme of , developed for advective problems in
atmospheric modeling requiring a large number of variables to be dealt with. For
this scheme, the Courant–Friedrichs–Lewy stability condition is
Δtv+w2Δz2<12,
with Δt and Δz the time and height steps,
respectively. In order to be consistent with radar observations (see
Sect. ), we choose Δz=100m. Then,
according to the usual range of drop terminal velocities
(v∈[0;10]ms-1) and of stratiform vertical wind
speeds (w∈±2ms-1), we choose Δt=5s. The number of time and height discretization steps are
denoted NT and NZ, respectively. In order to have a good coverage
of the range of diameters, they are discretized from
Dmin=0.2mm to Dmax=7.5mm,
with a diameter step ΔD=0.1mm. The resulting number
of diameter bins (ND) is thus equal to 73.
We note Ñ=(Ñin)n∈1:NT,i∈1:NZ, the
discretized DSD field. Ñin indicates a specific
discretized DSD at grid point (n,i), namely, at time
tn=t0+nΔt and height zi=z0-iΔz. Similarly, the
notations w̃ and ṽ stand respectively
for the discretized wind field and terminal velocity vector.
The unknowns of such a model are as follows.
The initial (t0) DSD field (initial condition),
(Ñi0)i∈1:NZ, is unknown. However,
if we suppose that the model starts running before the beginning of
the rain event, we can suppose that this initial field is 0
everywhere. This is the assumption made hereafter.
The top (z0) DSD field (boundary condition),
(Ñ0n)n∈1:NT, is unknown. This
corresponds to NT DSD bins, leading to NT×ND
unknowns.
The vertical wind time–height field, w̃ (which is,
in our case, a parameter of the model), represents
NT×NZ unknowns.
In order to reduce the dimension of the problem, and thus to
substantially reduce the number of degrees of freedom, we have to add
some a priori information. Since the density of raindrops in the
atmosphere is well approximated by a gamma distribution , we parameterize the top DSD
(Ñ0n)n∈1:NT
under a gamma form:
N(D)=α⋅f(D;θ,k),
with α the total volumetric number of droplets (unit
m-3), and f the gamma probability density function (unit
m-1):
f(D;θ,k)=Dk-1Γ(k)θke-Dθ,
with k the shape coefficient and θ the scale coefficient.
Thanks to this parameterization, for each time step n, we limit the
top DSD
(Ñ0n)n∈1:NT
determination to the choice of 3 parameters and so divide the
resulting number of unknowns by almost 25 (from 73 to 3). We
note x̃, the corresponding three parameters' field
(dimension: 3×NT):
x̃=(x̃n)n∈1:NT.
Concerning the implementation of the algorithm, at each time step n,
x̃n is directly converted into a top DSD
Ñ0n using Eqs. () and (); therefore
the propagation model can use it as input data.
Finally, the problem is reduced to the determination of the parameter
field (x̃) along with the vertical wind field
(w̃), resulting in a total number of unknowns of
(NZ+3)×NT.
Data assimilation algorithmGeneral principles
In the previous section, we have seen that the model propagates the
top DSD
(Ñ0n)n∈1:NT
through space and time according to the vertical wind
w̃. Besides the numerical model, we have a full set of
observations at various heights and times (from radar and disdrometer;
see Sects. and ).
Diagram of the assimilation algorithm. The unknowns of the problem (top DSD
parameters and vertical winds) are circled in red. The top DSD parameters are
converted into bin DSD and propagated through space and time, under gravity
and vertical wind, by the propagation model. Radar and disdrometer
observations (circled in green) are available at different places and times.
Observation operators are used to convert the model DSD into observations-like data (circled in purple). Then, the gap between these data and the
observations is the value of the observation term of the cost function.
Regularization and penalization terms are added in the cost function. Then,
we use a numerical minimizer to estimate a set of unknowns which minimize the
cost function. The notations are the ones defined in Sect. .
The retrieval of the top DSD and of the vertical wind is thus carried
out through the use of variational data assimilation which basically
consists in minimizing the distance between the model and the
observations (see Fig. which details our
contextualized assimilation process).
In this study, the well-known 4-D-VAR data assimilation algorithm is
used to allow the fusion of observations thanks to our propagation
model. Detailed descriptions of its theoretical principles can be
found in , or online on the European
Centre for Medium-Range Weather Forecasts (ECMWF) website:
http://www.ecmwf.int/sites/default/files/Data assimilation concepts and methods.pdf.
To perform a 4-D-VAR assimilation, we still have to build a 4-D-VAR cost
function, which is crucial since it is used to evaluate the gap between
the observations and the top DSD
(Ñ0n)n∈1:NT
propagated by the model to different times and heights. The numerical
minimization of the cost function allows a set of unknowns to be achieved
(namely, x̃ and w̃) which minimize the
cost function.
In the data assimilation context, the cost function usually takes the
form:
J(Ñ)=Jb(Ñ)+Jo(Ñ),
with Ñ the discretized DSD as previously defined.
Jo is the observation term of the cost function,
presented later. Jb is the background term (also called
first guess) of the cost function. Jb keeps the solution
in the vicinity of a given a priori state. In our case, we suppose
that there is no background available; therefore Jb will be dropped from the cost function. Moreover, we add regularization
and penalization terms (see Sect. ).
The cost function finally used takes the form:
J(x̃,w̃)=Jo(x̃,w̃)+Jr(x̃,w̃)+Jx(x̃)+Jw(w̃),
where the cost function is the sum of a regularization term
(Jr) along with two penalization terms (Jx), dedicated to
(x̃) the DSD parameter field, and Jw, dedicated to
(w̃) the wind field. These three terms will be discussed
and explained in Sect. .
Location of the studied area in the southeast of France, on a topographic
map. The two instruments used in the assimilation algorithm (MRR and DBS, see
Sect. and ) are located at Le
Pradel, 278m height. The three automatic Météo France weather
stations of Le Pradel (same place), Aubenas (7.5km westward,
180m height) and Berzème (7.5km in the northeast,
650m height) are also shown.
In the end, the cost function J(x̃,w̃) is a
function of both the top DSD parameter field (x̃) and the
wind field (w̃) that allows the overall distance
between a propagated set of parameters
(x̃,w̃) and the observations to be computed. By minimizing
the cost function, the algorithm will estimate an optimal set of
(x̃,w̃), minimizing this distance. Once the
assimilation process is completed, for each time step, we will have a top DSD
parameter and wind values for all heights.
Observation term of the cost function Jo
Jo, the observation term of the cost function, takes the form:
Jo(Ñ)=12∑n∈1:NT∑i∈1:NZyin-Hi,nÑin2.
For a given grid point (n,i), yin is the vector composed
of the observations from radar and disdrometer. Hi,n is
the operator projecting the DSD on the observation space. Note that,
here, the covariance matrix of the observation error is the identity.
At time (tn), using the vertical wind field (w̃), the
model has propagated the top DSD Ñ0n′
from times tn′<tn. We note Min the numerical model
propagating all the necessary top DSD
Ñ0n′ to compute
Ñin, the DSD at time (tn) and height
(zi). Then, we have
Ñin=Min(x̃,w̃),
with x̃ the three DSD parameters field as defined above
(see Sect. ). Consequently, the cost function
(Jo) can be expressed as a function of the unknowns
x̃ and w̃ only:
Jo(x̃,w̃)=12∑n∈1:NT∑i∈1:NZyin-HinMin(x̃,w̃)2.
Observation operator H
The observation operator, H, can be considered as an
aggregation of 2 sub-operators Hdis and
Hdop, modeling the disdrometer and radar
observations, respectively.
For disdrometer measurements, we build an operator
Hdis able to convert the propagation model ground
(i=NZ) discretized DSD
ÑNZnn∈1:NT
into disdrometer-like measurements (i.e., drop histograms in our
case). A disdrometer records a flux of droplets crossing a given area
(A, in m2). Then, at time tn and for drops in the diameter
class p, this operator has the form:
Hdis(ÑNZ,pn)=ÑNZ,pn⋅A⋅vp+w̃NZnΔtΔD.
For radar measurements, we build an operator Hdop
converting model DSD into Doppler radar-like measurements. Radars
operating in a vertically pointing mode provide the distribution of
the radar return power as a function of the Doppler velocity, the so-called
Doppler spectrum (). The retrieval of the DSD directly from
these Doppler spectra is an inverse problem ()
mixing various phenomena. Here we build a “direct” operator which
acts in reverse order (from DSD to Doppler spectra). It takes vertical air motion and rain attenuation into
account. At a given time and location, this operator can be computed according to the three following
steps.
Calculate the theoretical non attenuated and non-noisy spectrum F(v) (sm-2) from a DSD N(D):F(v)=σ(D)N(D)dDdv(v),where, given a drop diameter D, σ(D) is the backscattering
cross section (unit m2), calculated for the radar frequency
according to the Mie scattering theory
(). dD/dv is the derivative
of the diameter-velocity relation.
Then derive the final attenuated spectrum Fa(v), encompassing both the effects of the attenuation and of the vertical air motion w.Fa(v)=F(v+w)e-2∫0zK(r)dr.The exponential part of Eq. () adds the effect of rain
attenuation (see for instance ), with K(z) the
attenuation coefficient at height z (unit m-1), calculated
through K(z)=∫Dσext(D)N(D,z)dD,
with σext(D) the extinction cross section (Mie
theory).
Finally, for minimization purposes, Fa is converted into a logarithmic
scale, resulting in a logarithmic bounded attenuated spectrum Z(v):Z(v)=ln1010Fa(v)+1.
We recall that the operator run on the computer is necessarily a discretized version of what is presented here.
Regularization and penalization terms
In this section, we present a brief description of the penalization
term Jr as well as the two regularization terms Jx
and Jw of the cost function (see Eq. ).
Jr: we have seen in Sect. that
the time step of the model is set to 5s. It means that
x̃ and w̃ have to be retrieved at this
resolution. However, the minimal resolution of the instruments is
generally greater than this value (10s in our case; see
Sects. and ). Then, using
Jr avoids the assimilation algorithm producing high-frequency fluctuations on both of the unknowns x̃ and
w̃ by giving lower cost to smooth fields.
Jx allows us to proceed to the minimization under constraints
for x̃, the three DSD parameters penalization
term. Because the instruments used are not able to record the complete
drops diameter range, it is used to avoid the assimilation algorithm
producing unphysical DSD. To this end, we add this penalization term
which avoids the components of the x̃ DSD parameters
field to go out of predefined ranges: [0; 8000] m-3 for
α; [0; 3] for k; [0; 10-3] m for θ.
Jw: the vertical wind field w̃ modifies the
advection velocities and shifts the Doppler spectra. Then, the
vertical wind is mainly controlled (in the assimilation algorithm) by
the Doppler spectra. If no spectra are available for a given vertical
layer, the assimilation algorithm could produce unphysical temporal
fluctuations of the vertical wind which would not imply significant
extra cost without the term Jw. Since there may be no Doppler
spectra available at given heights (see Sect. ),
we use Jw, the wind penalization term, to force the vertical wind
to stay close to 0 by adding extra cost to strong vertical winds.
Minimization of the cost function and parameter estimation
The numerical minimization of the cost function J(x̃,w̃) goes through the calculation of its
gradient. Note that all the regularization terms are easily
differentiable. The gradient of the observation term (Jo) of the
cost function is
∇Jo(x̃,w̃)=∑n∈1:NT∑i∈1:NZtMintHinyin-HinMin(x̃,w̃),
with tMin the adjoint of the linearized version of the operator Min.
We use the YAO software (), developed by the LOCEAN
(Laboratoire d'Océanographie et du Climat). It provides a simple
method for deriving the adjoint of the model. For the numerical
minimization itself, it is coupled with M1QN3 (),
a quasi-Newton algorithm to solve unconstrained optimization
problems. Once the propagation model is implemented, it allows the use of
observations to compute realistic estimations for the unknowns.
We finally note that we have to initialize the unknowns of the problem (x̃,w̃)
at the very beginning of the assimilation process. We found that this initialization is
not critical. Consequently, we will initialize the wind field (w̃) at 0 and the 3 gamma parameters
time series (x̃) at constant very low values, namely α=1, k=0.8, θ=2.10-4.
Measuring instruments and case study
In this section, we firstly give the characteristics of the two
instruments used in this study, namely a Doppler 24GHz
radar, and a disdrometer. Then, we present the characteristics of the
case study examined.
Micro-rain radar (MRR)
The first instrument used in this study is a micro-rain radar (MRR,
), developed by METEK GmbH, Germany, and belonging
to Météo France. The MRR is a 24GHz frequency
modulated, continuous wave (FM-CW), vertically pointing Doppler radar,
with a small transmit power (50mW) and a beamwidth of
2∘. The raw return power spectrum is processed by the MRR (see ),
which supplies the spectral reflectivities. In our assimilation process, only the spectral reflectivities will be used as
observations (noted FMRR) .
The MRR software also supplies DSD estimations (obtained through an
inversion algorithm of the Doppler spectra, including attenuation
correction) and estimations of different moments of the DSD (liquid
water content, rain rate, reflectivity factor). Those quantities,
mentioned as “MRR products”, will be used as a comparison framework
for the DSD Ñ retrieved by our algorithm.
The radar range gate size is 100m. The MRR provides
data up to a 3100m height. The temporal resolution
is 10s. This fine temporal resolution will allow us to
integrate the observations over various time periods. This integration
time will be referred to later as an “observation window”. The Doppler
velocities used a range from 0.56 up to 9.54ms-1, with a step of
0.19ms-1. According to the manufacturer recommendations
(), the MRR records for the two first radar gates
(<200m) will not be used in the assimilation process.
Dual-beam spectropluviometer (DBS)
The DBS is an optical disdrometer developed by the LATMOS, whose main
advantage is its ability to cover a very large range of raindrop
diameters (). Its collecting area is
0.01 m2. For each recorded drop, the DBS supplies a triplet (time of
arrival/spherical equivalent diameter/fall velocity), allowing the
measurement of drop flux time series, noted NDBS, used as
observations in the assimilation algorithm. We apply a minimum diameter
threshold of 0.4mm. 0.4mm corresponds to a value
over which we are sure that the instrument avoids false detections
() and can be compared to other instruments ().We will only conserve the drops
above this diameter.
For the real case study of the 12 October 2013, 2min rain rates as
measured by the DBS (red) and as reproduced by the 4-D-VAR assimilation
algorithm (blue).
Case study data set
The two instruments described above were deployed in Ardèche
(southwest of France), during autumn 2013, in the context of
the HyMeX campaign (HYdrological cycle in the Mediterranean
EXperiment, see for instance
). Figure presents
a topographic map of the area. The two instruments (DBS and MRR) were
co-localized at Le Pradel (44.6∘ N, 4.5∘ E), in a mountainous
area called the Cévennes-Vivarais, subject to so-called Cévenol flash flood
events (). However, for the purpose described in
Sect. , we will not work on extreme rain events, but
rather on a stratiform event of light rain which occurred on
12 October 2013.
For the real case study: rain rate (up), reflectivity factor (middle) and
characteristic velocity (bottom) for MRR products (left) and assimilated
fields (right).
This event occurred at Le Pradel in the afternoon, from 16:30
to 21:30 UTC. It is a long rain event characterized by low but
sustained rain rates at ground level (the disdrometer was
used to record 2min rain rates always between 1
and 6mmh-1; see Fig. , red
line). The event is also quite homogeneous. All the Météo
France automatic weather stations located around Le Pradel
(see Fig. ) record consistent cumulative
rainfall: 11.5mm at Le Pradel, 12mm at
Aubenas, and 12.9mm at Berzème. From the MRR
(Fig. , left) and DBS
(Fig. , red) measurements, we can distinguish
two distinct phases in the event. Until 18:15, there is
very light rain, with only small drops (rain rates around
2mmh-1 and very low reflectivities and
characteristic velocities). After 18:15, there are short
periods with higher rain rates (4–5 mmh-1) and,
mainly, higher reflectivities and falling velocities (up to
35 dBZ and 8ms-1).
The temperature was around 8.5∘C
at ground level (Le Pradel station, 278m
height). Assuming a gradient of
-1∘C/150m, we found a freezing
level at 1550m, which is consistent with the MRR
records (on Fig. , left, we see the so-called radar bright
band, indicative of melting ice crystals, at about 1500m height). As a consequence, the top of the retrieval domain
in our assimilation algorithm will be set
to 1400m in order to avoid bright band problems
(the propagation model is not able to treat the melting
layer). Note that the blank strip on the right of
Fig. is explained by this choice. The
10m horizontal southerly wind recorded by Météo
France at Berzème station is light during the event, ranging
from 1.0 to 2.3ms-1. The humidity rates
are always close to 100% (97% at
Berzème station, 99% at Aubenas; see
Fig. ). This rules in favor of the choice
to neglect horizontal wind and evaporation effect in the
propagation model (Sect. ).
For this case study, the number of time and height
discretization steps (NT and NZ; see
Sect. ) is set to 4900 and 15,
respectively. We remind the reader that, for numerical purposes, the time and
height steps were chosen to be Δt=10 s and Δz=100 m.
Simulated data (twin experiment)
We have seen in the last section that the case study chosen to
apply our retrieval algorithm is consistent with the propagation
model underlying hypothesis. As a reminder, it assumes DSD to be driven by gravity and
vertical wind alone. Thus the effects of evaporation, collisions
and horizontal wind are discarded. However, other phenomena remain
unaccounted for in the model. Since this could impair the ability
of the model to satisfactorily manage our real case study, we have
to assess the impact of two of these phenomena (instrumental noise
and turbulence) on the assimilation retrievals. This is done, in
this section, by performing a twin experiment, whose principle is
described below.
Description of the twin experiment process
Twin experiments consist in applying the assimilation process to
simulated data with properties analogous to real data.
Diagram summarizing the twin experiment process (encompassing the observation
simulation). See Sect. for the description of the scheme. The
sections of the paper in which the different simulation processes are
detailed are indicated in brackets below the arrows.
Figure schematically summarizes all the successive
steps of our twin experiment (i.e., to simulate realistic
observations and to run the assimilation). Firstly, we simulate
realistic series of top DSD parameters and vertical winds (step a
on Fig. ). These parameters are what we would like to
retrieve with the algorithm. In Sect. , we will
explain how these fields are simulated. Then, these series are
propagated through space and time using the propagation model (step b) and “true” (namely, by supposing the model to be perfect) DBS and
MRR data are produced through the use of the observation operators
(step c). All the fields obtained through this process (DSD and
vertical wind, as well as observations, for instance Doppler spectra) are
mentioned below as true.
Then, from these fields, we simulate realistic MRR and DBS
observations by adding model and instrumental errors (instrumental
noise and turbulence, see Sect. ), which cannot be
directly produced by the propagation model (step d) since they
are not present in the model despite their effects on the
measurements. These data are mentioned below as “observed”. The
process used to simulate these observed fields is detailed in
Sect. .
Finally, the assimilation algorithm estimates an optimal set of
unknowns explaining these observations (step e). Assimilated
x̃ and w̃ fields are then propagated
by the model (step f) and the observation operators (step g). All these retrieved fields are mentioned below as
“assimilated”. They will be compared to true and observed
fields. The results of this data assimilation twin experiment are
detailed in Sect. .
This twin experiment demonstrates the ability of the assimilation
algorithm to produce coherent DSD fields when the assumptions of our propagation model are satisfied. It also shows how it
indirectly handles the turbulence, underlining its efficiency
despite the presence of non-modeled phenomena. Note that in this
section, all the observations (MRR spectra FMRR and DBS
DSD NDBS) are integrated over a 2min
observation window (if not, it will be explicitly specified).
Simulation of top DSD parameters (x̃) and vertical wind field (w̃)
First, we have to simulate realistic time series of top DSD
parameters (x̃). To do this, we use the rain rates
recorded on the ground by the DBS during the case study presented
above (Sect. ). From each of the successive disdrometer
rain rates, we calculate the two parameters of an exponential DSD
according to . This time series of Marshall–Palmer DSD
(special case of gamma DSD) is our simulated top DSD
(Ñ0n)n∈1:NT,
which is used as input in the propagation model.
For the twin experiment: vertical wind and Doppler spectra vertical profiles
for a given time. Left: vertical wind (positive downward) according to the
height. The dashed line is the true wind; the solid line is the assimilated wind.
Right: Doppler spectra for the successive radar gates from gate 4
(bottom) up to gate 15 (up). Marked lines are the true spectra, dashed
lines the observed spectra (with turbulence and instrumental noise) and solid
lines the assimilated spectra.
We also need a time–height field of vertical winds
w̃, (resolution 5s, 100m) to
run the propagation model and simulate observations. We have seen
that our algorithm is, up to now, limited to stratiform quiet rain
events. For this kind of event, vertical winds range in
±2ms-1, with temporal correlations around
2–5min and spatial correlations around
500–1000m (). The
process used to simulate such a field is as follows. (1) We create
a 5s/100m field of independent, normally
distributed, random variables. (2) We average this field over
a mobile (2min/500m) window. (3) We scale the
field to get winds within ±2ms-1. In the
end, we obtain a wind field with triangular correlations
corresponding to the expected values. Figure
(top, on the left) shows a part of the vertical wind field used in
the twin experiment (i.e., on simulated data). Note that the winds
are positive downward.
For the twin experiment: ability of the assimilation algorithm to retrieve
the wind field characteristics. Top left: true wind field; top right:
assimilated wind field. Bottom left: mean values of these two fields
according to the height: blue: truth; red: assimilation. Bottom middle:
the same for the standard deviation; bottom right: temporal normalized
autocorrelation function of the two wind fields. The wind is positive
downward.
Simulation of realistic observations
We detail the process used to simulate realistic
observations in this section (addition of turbulence effects and instrumental
noise). The purpose of this step is to build observations
consistent with what is expected from real measurements.
Radar Doppler spectra
We want to simulate Doppler spectra as they would be recorded by
an MRR in realistic conditions. For this purpose, we run the
propagation model and save the attenuated reflectivities calculated
by the algorithm (observation operator) from the DSD (variable
Fa(v), see Eq. ). Then, for each
spectrum, we apply the following process. (1) The
atmospheric turbulence impact is added. Turbulence can be seen as a fine-scale modulation of the vertical wind. For a time and location, the
effective vertical air motion is the sum of the mean vertical wind
(defined above and taken into account in the propagation model) and
of a random variable, modeling the turbulence (not taken into
account). Mathematically, the impact of turbulence on the Doppler
spectra can be modeled as the convolution of the Doppler spectra
with a Gaussian function (). So, the turbulence
is added to our spectra by applying a discretized version of the
convolution operator:
Ft(v)=Fa(v)∗12πσte-v22σt2,
with ∗ the convolution operator and σt the width of the
turbulence spectrum. We use σt=0.7ms-1,
considered a realistic but large value
() implying the use of a large
spectral broadening (). (2) Gaussian white
noise (intensity 1dB) is added to Ft. On
Fig. , on the right, the dotted lines show observed
Doppler spectra at different heights.
Disdrometer
We simulate the records of a ground disdrometer. For this purpose,
we run the propagation model and save the discretized DSD in the
grid box just above the ground. For a given time and diameter
class p, we note np, the corresponding number of drops in
a given box. Then, for each of these np drops, we choose
a diameter (following a uniform law in the interval [p;p+1]ΔD+D0) and a velocity. This velocity is the theoretical Atlas
velocity for the diameter modulated by Gaussian white noise (as
presented previously), representing the atmospheric turbulence. The
height of the drop is finally uniformly drawn in the interval
[0; 100] m (height range of the grid box).
At the end of the process, we check which drops, among the np, will touch the
ground during the time interval Δt (and so will be seen by
the disdrometer). The drops touching the ground are recorded as
observed by the disdrometer. This process is a way to mimic the
natural variability of rain and to run the
observation operator defined by Eq. () in a stochastic way.
Results – robustness of the algorithm
In this section, we firstly assess the ability of the assimilation
process to reproduce the observations. Then, we compare the true
and assimilated DSD Ñ and wind fields
w̃. Finally, we investigate how the model response
is impacted by the size of the observations window. First of all,
we have to introduce the numerical indicators needed to
perform this investigation.
Error indicators
In this section, we define the indicators used to quantitatively
evaluate the performance of the algorithm. Choosing error
indicators is rather sensitive to the range of values under
study. For fields with magnitude far from 0 and ranging in small
intervals, we use relative indicators without any risk of giving
too much importance to very low (close to 0) data. We define the
two following indicators: the mean absolute percentage error
(MAPE), and the relative bias (rbias), defined by
MAPE(%)=100⋅1P∑i=1PAi-BiAi,rbias(%)=100⋅∑i=1PAi-Bi∑i=1PAi,
where A and B are the respective reference and evaluated
fields rearranged in vector forms. P stands for the dimension of the vectors.
Fields with very low or negative values render the two previous indicators
inappropriate. For these fields, we use the mean absolute error (MAE),
defined by
MAE=1P∑i=1PAi-Bi.
For the twin experiment: comparisons between truth, observations and
assimilation for the moments of order 1 and 2 of the Doppler spectra
(characteristic velocity Wc and spectral width SW,
respectively) of the MRR and of the total number of drops seen at successive
2min observation windows by the DBS. For the MRR data, the MAPE
(Eq. ) and the relative bias (Eq. ) are presented, as
well as the mean value of the true fields. For DBS data, the MAE
(Eq. ) are presented, as well as the mean value of the true field.
In the following analysis, we use the denominations true,
observed and assimilated (see Sect. and
Fig. ). Table gives the MAPE and the
relative biases between true, observed and assimilated
characteristic velocities and spectral widths.
We recall (Sect. ) that the atmospheric
turbulence, which broadens the Doppler spectra, is included in the
observations but not in the true data. Consequently, for the
observed spectra, we obtain spectral widths larger
(+12 %) than the true ones.
The assimilated spectra are also broader than the true
ones, but to a lesser extent (spectral width 9.5%
larger). They are slightly less broad than the observed spectra
(-2.3 %). We also note that the MAPE and rbias have
almost the same absolute values (Table , third and fourth
rows). This means that the sign of the differences between the two
compared fields is the same for almost all the grid points. This
implies that all spectra are broadened by turbulence independently
of time and height.
On the contrary, the characteristic velocity is not highly
affected by the addition of turbulence impact on observed fields
(bias -0.36 %, MAPE 0.40%). Consequently,
the characteristic velocities are correctly reproduced by the
algorithm (biases and MAPE inferior to 2%). Obtaining
the result that turbulent broadening only impacts the spectral width and
not the reflectivity-weighted mean downward velocity in the simulations
confirms the robustness of the assimilation procedure and expected results
from prior work ().
For the twin experiment: MAE (Eq. ) between true and assimilated
fields for different parameters: vertical wind; rain rate, RR;
reflectivity factor, Ze; mean volume diameter, Dm;
liquid water content, LWC; number of drops, N0. The indicators are
presented for two observation windows (10s, in italics, and 2min) and
for different temporal field resolutions (from 10s to 8min).
The absolute bias is also presented, as well as the absolute values of the means of
the true fields. In each case, the best result (lowest MAE) has been made bold if it
happened in the case of the 2min observations window.
FieldObs.windRRZeDmLWCN0resolutionwindow(ms-1)(mmh-1)(dBZ)(µm)(µgm-3)(m-3)10s10s0.210.270.9628.514.53372min0.460.511.0543.019.225730s10s0.170.240.9427.414.23342min0.400.340.9233.414.82511min10s0.170.240.9126.413.83302min0.290.170.8326.210.72462min10s0.160.230.8725.013.53232min0.140.140.8024.710.02414min10s0.150.220.8123.313.13112min0.120.130.7723.59.542348min10s0.140.220.7421.812.82962min0.0980.120.7222.49.13225Abs. bias10s0.100.200.6922.613.03052min0.0540.0870.7122.78.67212True absolute mean 0.342.1827.611201421070
To reproduce the turbulence broadening, the algorithm can act on
the two unknown fields, x̃ and
w̃. Firstly, it can affect the DSD shape, by adding
small and large droplets and removing drops of intermediate
diameters; but because the DSD is directly controlled on the
ground by the disdrometer (observation NDBS), it forces
the algorithm to reproduce the ground DSD
(ÑNZn)n∈1:NT well. Between
true and assimilated disdrometer observations, we find an MAE (see
Table , last row) of 27 mm-1, while it is
42 mm-1 between truth and observations (the mean number of
drops being 689 mm-1). Secondly, the algorithm can use another way,
which consists in using the fast modulations of the vertical wind
w̃ at fine resolution.
We have seen that all the
observations are integrated over a 2min observation
window (Sect. ), while the wind is retrieved at
the 5s model resolution. Consequently, if the 5s winds
oscillate inside the 2min observation window, it will result in shifting the 5s
corresponding spectra alternatively towards small and large Doppler
velocities. Once averaged over the 2min time period, these shifted spectra
will result in a final 2min spectrum with a spectral width larger than all the
5s spectral widths, whatever the original DSDs. This is the way used by the
algorithm to mimic the turbulence.
This phenomenon can be seen in the third column of Table ,
which shows the MAE between true and assimilated wind fields for
an observation window of 2min and for various field
resolutions (spreading from 10s to 8min). We
can see that the MAE of the 10s resolution wind fields
is large (0.46ms-1) compared to the true absolute
mean value (0.34ms-1; see last row of
Table ), but it sharply decreases for
resolutions greater than 2min. This means that there
are fluctuations at the 5s model resolution which do not
correspond to the physics. These are numerical artifacts produced
by the model which uses any degree of freedom available to
reproduce the broadening observed in the measurements.
An example of Doppler spectra at a 2min resolution is
given in the right part of Fig. . In this plot,
examples of vertical profiles of Doppler spectra are shown. On the
observed and assimilated spectra, we can clearly see the turbulence
broadening, absent in true ones. Since this broadening
on assimilated spectra is due to a turbulence reproduction (using
the wind resolution) and not to a modification of the DSD, the good
behavior of the assimilation algorithm for realistic stratiform
rain is thus assessed.
Wind and DSD reproduction
Even if we confirmed the ability of the assimilation algorithm to
handle the turbulence, we still have to assess some other important rain
properties.
As in the previous section, we compare the
MAE between true and assimilated fields for an observation window
of 2min and for various resolutions in Table . The quantities
Ze and Dm (moments of the DSD, see caption of
Table ) are very well reproduced by the assimilation
algorithm, with very small MAE relative to the true absolute mean
values (for instance, at 2min field resolution, MAE of
0.80dB and 24.7 µm compared to true absolute mean
values of 27.6 dBZ and 1.12mm, respectively).
The results are slightly less satisfactory for liquid water content (LWC)
(10.0 vs. 142 µgm-3) and for N0 (241 vs. 1070). This is due to the fact that
the number of drops is mainly driven by the number of very small
drops (<0.4mm), which are not seen by the
disdrometer. Moreover, given their very low backscattering
cross section, they have almost no weight in the cost
function. They are consequently not effectively
controlled. However, we can note that these drops represent a very
low mass of water and that their non-control is not
critical. Moreover, since the top DSD is parameterized, we cannot
obtain unreasonable values.
We have seen that the 2min MAE for the wind field is
quite satisfactory, meaning that the global trends of the wind
field have been reproduced by the algorithm; but we would also
like to know whether the wind patterns are well
reproduced. Figure shows extracts of the true and
assimilated wind fields, as well as the mean and standard
deviations of these fields according to the height (radar gate) and
the temporal autocorrelation functions. The top images confirm
a satisfactory reproduction of the wind fields. We also see that
the correlation time, the absolute mean values and the standard
deviations are well reproduced by the assimilation, even if the
standard deviation results are more dubious for the two lowest
radar gates, where no Doppler spectra are available (as seen in
Sect. ).
The rain rates are also satisfactory for resolutions over
2min (Table , column 4), with an MAE of
0.14mmh-1 with a mean value of
2.18mmh-1.
Impact of the observations window
In this last subsection, we briefly assess the influence of the
integration time of the observations (observation window, see
Sect. ). Table presents the MAE
between different fields, depending on the observations window,
10s or 2min. We have seen that
a 2min observations window allows the algorithm to use
the 5s model resolution to reproduce the turbulence
spectral broadening by adding artificial synthetic fluctuations. If
the observation window is reduced to 10s, no room is
left for fluctuation. Consequently, the MAE for the wind is
smaller with 10s observations window for fine
resolutions (for instance, 0.21
vs. 0.46ms-1 at 10s).
Because the model fits, at best, the 10s resolution, it
is natural that the algorithm performs less accurately at the 2min resolution
and hence beyond.
The results are globally the same for the other parameters of
Table . Because the algorithm cannot use the wind
w̃ to adjust the assimilated spectra to the observed
broadened ones, it has no choice but to fit the observations
through a change of the top DSD (excess of small and large drops)
in order to produce the best compromise at the cost level. This
effect is visible in the N0 column of Table : the
results are worse for a 10s observation window whatever
the resolution (excess of small droplets). We can also note that
the spectra are much noisier at 10s, implying
a tougher minimization.
Finally, we conclude that an observation window of 2min
is a better choice than 10s. It allows the algorithm to
handle the turbulence and limit the instrumental noise
influence. Thus the assimilation in the next section will be run
with a 2min observation window.
Results for real case study
In this section, we apply the assimilation algorithm to the real
data case study described in Sect. . Again, we show that
the assimilation algorithm is efficient (able to fuse different
observations) for this case study. We then show that it produces
spatiotemporally coherent wind and DSD fields.
For the real case study: comparison between observations and assimilation for
three moments of the MRR Doppler spectra: order 0 (reflectivity η);
order 1 (characteristic velocity Wc); order 2 (spectral
width SW). For η, MAE (Eq. ) and bias are presented. For
Wc and SW, MAPE (Eq. ), rbias (Eq. ) and
the mean value of the observed field are presented. The results are shown for
all heights, as well as for three different height ranges. Positive bias mean
overestimation by the assimilation is compared to the observations. Comparisons
between observed and assimilated DBS measurements are also presented.
All gatesTopMiddleBottom(1100–1400m)(700–1000m)(200–500m)ηMAE (dB)1.21.80.861.1bias (dB)-0.017+0.60-0.12-0.40WMAPE (%)4.45.73.54.4rbias (%)-0.39-1.5+0.16-0.15mean(obs) (ms-1)5.115.285.184.87SWMAPE (%)14141217rbias (%)+5.9+3.1+3.5+12mean(obs) (ms-1)1.201.251.221.11DBSMAE (number of drops: mm-1)102mean(obs) (number of drops: mm-1)1080Observation reproduction
We use the indicators introduced in Sect. again
(see Table ). Firstly, we note that the disdrometer
observations NDBS are well reproduced. The MAE between
observed and assimilated data is 102 mm-1, while the mean
observed number of drops is 1080 mm-1. These results are very
similar to the results of the previous simulated study, meaning that the
algorithm successfully reproduces the disdrometer
observations. Figure shows the ground rain rates as
calculated from the disdrometer observations (red) and as reproduced
by the assimilation algorithm (blue). It confirms that the algorithm
is efficient for ground data. We will later come back to the
remaining differences between the two rain rate time series.
Table also presents results concerning the MRR
observations. Even if the MAPEs for the characteristic velocity
Wc and spectral width SW are larger than the ones on simulated
data, they remain satisfactory (mean error of 4.4 and
14%, respectively).
The three columns on the right of Table present these
results at different heights and help us to analyze the vertical
structure of MRR observations and assimilated data. When considering
reflectivity η (zeroth order moment, second row), we see that
there is globally no bias between observed and assimilated
reflectivities (-0.017 dB). However, the assimilation
process overestimates the reflectivities at the top
(+0.60 dB) and underestimates the reflectivities close to
the ground (-0.40 dB), while there is a smaller bias
at middle heights (-0.12 dB). Since the model propagates
the drops from the top to the bottom, without changing their diameter,
the only process in our propagation model that can produce a decrease of the
mean reflectivity with height at the scale of the event is attenuation.
This means that the MRR observations display more
reflectivity decreasing with height than what can be explained by
the model attenuation. This feature is confirmed by the left part
of Fig. . We clearly see that the rain rate
(top) and reflectivity factor (middle) display a
decrease with height for the MRR products. This phenomenon cannot
be reproduced by our algorithm (right part of
Fig. ) which more or less (depending on the wind
field, see Sect. below) conserves these quantities
with height. We also see that the algorithm reaches a compromise
between the different heights; therefore the assimilated
reflectivities (and hence the rain rates and other DSD parameters
displayed in Fig. ) are overall (compared to
MRR products) slightly overestimated on the top and underestimated
on the ground. This would also explain why, in Fig. ,
assimilated rain rate peaks are generally underestimated compared
to disdrometer data. Figure shows the bottom DSD
(continuous lines) for the MRR products (200–300m, red),
for the DBS measurements (on the ground, in black), as well as
the assimilated ones (on the ground, in blue). For the intermediate
diameters ([0.7; 2.5] mm, corresponding to the range which mainly
impacts the rain rate and the reflectivity factor at low rain
rates), the assimilation underestimates the number of drops. This
could be explained by the MRR observed spectra at the top, which
induce a lower number of drops
(Fig. , red dotted marked line). The origin of
this variation of reflectivity with height for the MRR observations
is not clear. It could be a calibration problem (height-dependent
transfer function; see ), as well as the effects
of the horizontal wind or of some microphysical processes
(coalescence/breakup) assumed negligible in our propagation model. Anyway, the assimilation algorithm appears
able to merge the different data available to produce a solution,
making a compromise between the observations available (low biases and
MAE/MAPE for the three first moments of the spectra and for disdrometer data;
see Table ). It produces results embedding the
spatiotemporal coherence brought by the propagation model.
For the real case study: mean DSD between 20:00 and 20:30 evaluated on the
ground (continuous lines) from: disdrometer data (black) and assimilation
outputs (blue) and from MRR on the lowest radar gate available (the 3rd one)
(continuous, red). The same, evaluated on the top (dotted marked lines) from
MRR data (red) and assimilation outputs (blue).
For the real case study: time–height fields of different assimilated
parameters induced by DSD field Ñ or wind
w̃. Top: 2min vertical wind field (positive
downward). Second highest panel: mean volume diameter Dm. Second lowest panel:
number of drops N0 (moment of order 0 of the DSD). Bottom: slope of the
DSD, evaluated as the slope of the linear regression of the logarithm of the
DSD on the diameters.
We can also note that the very small drops (<0.4mm) are
loosely constrained and that the corresponding results are then
dubious (Fig. ). The disdrometer does not cover
these diameters and the MRR does not receive a lot of energy for
the corresponding Doppler velocities, so that the MRR observed
spectra seem less coherent. Consequently, the assimilation algorithm
produces a very large number of small drops (more than an
exponential law would) without any physical justification
associated.
For large drops (>2.5mm), we note
(Fig. ) that the disdrometer provides observed
ground DSD NDBS which are not reproduced by the
assimilation, probably because they are not consistent with the MRR
observed spectra. However, large drops are very rare for this light
rain event; therefore the presence or absence of a particular drop can
deeply affect the tail of the disdrometer DSD. Moreover, the
terminal velocity of these drops is >7.3ms-1; therefore
disdrometer measurements with 2min observation
window correspond to very large probed volumes (corresponding to
a height of around 850m, crossed in 2min for
a 7.3ms-1 drop) which absolutely do not match the
MRR probed volumes (100m height). Because large drops are
rare for this event, we nevertheless chose to work with
a 2min observation window. Naturally, for convective
events, this choice could be questioned, at least for the
disdrometer (we have seen in Sect. that a decrease
of the observation window width for the Doppler spectra decreases
the ability of this algorithm to handle the turbulence).
Wind
Figure (top panel) presents the 2min vertical
winds as retrieved by the algorithm from 19:30 to 20:30.
We remind that there are no Doppler spectra available for the two
lowest gates (0–200m); therefore the wind retrievals for
these two heights are not constrained by observations in the algorithm, since
the wind also acts on the vertical advection of drops and therefore also on the
disdrometer records.
The structure of the assimilated wind field is consistent with what
we could expect for such an event. The correlation characteristic
time and height are around 1min and 300m for
the 10s wind field (which can be used by the algorithm
to handle the turbulence as explained in Sect. ) and
around 3min and 500m for the 2min
field. These results, as well as the wind range (±1.5 ms-1), are consistent with the results of
or , for
instance. Even if we cannot validate the retrieved field using
independent measurements in this study, we could nevertheless
notice that our method produces the same wind field structures
whatever the algorithm initialization.
We also have an overall mean wind of +0.004 ms-1,
slight and not notable downward air motion. We get slight
downward air motion in the upper layers of the atmosphere (mean
wind around +0.15 ms-1 above 700m) and
slight upward air motion closer to the ground (around
-0.2 ms-1 between 200 and 500m). The
2min standard deviation is quite constant with height
and around 0.25ms-1.
DSD
We have seen (Sect. ) that, on average, the MRR
products (rain rate, reflectivity factor) are consistent with our
assimilated data. There is nevertheless a vertical decreasing
tendency on MRR products, which cannot be reproduced by the
algorithm, given that the propagation model produces vertically
coherent fields (Fig. ). However, in this
section, we focus on the structure of the assimilated DSD fields,
independently of the MRR products.
We have seen that the algorithm conserves the reflectivity
factor during a drop fall well. It results in diagonal structures on the
time–height plane (see Fig. , middle, right).
The slope of a diagonal structure reflects the fall velocity. We can
see that the reflectivity field is consistent with the
characteristic velocity field (bottom, right): the lower the
velocity, the lower the slope of the reflectivity diagonal
structure.
Now we will also assess whether these structures are retrieved for some
important moments of the DSD. Figure shows the
mean volume diameter Dm, and the number of drops N0, resulting
from the assimilated DSD Ñ. We note that the
retrieved values for Dm (0.5–1.5mm) are reasonable for
light stratiform rain (see Fig. A4 in ). Dm
is also quite well conserved with height, but the patterns seem
noisier than for the reflectivity, as it also does for the rain rate
field (Fig. ). This is a consequence of the
vertical wind, which modifies the vertical advection and so the
DSD. The vertical wind has a greater impact, relatively, on small
drops than on large ones (given their smaller terminal velocity); so
the low-order moments, mainly driven by the small drops, are more
affected than the high-order moments. This is particularly clear for
N0. We have seen that the algorithm, underconstrained for these
diameters, produces a very large number of very small drops
(<0.4mm, i.e., with terminal
velocities <1.5ms-1) (see
Fig. ). Consequently, (1) the vertical
consistency of N0 is low. (2) The slope of the diagonal
patterns of the N0 time–height field is low (because N0 is
driven by small drops, with low terminal velocities). (3) Patches
of raindrops are formed by upward winds (see
Fig. ), and these patches disappear as soon as the
wind in the lower layers turns downward. For instance, around
19:55, there is an upward wind between 100 and
400m, which results in a patch of drops between 300 and
500m. We can also note that even if the absolute values
of N0 are dubious, these patches are probably real, since they
are driven by the wind.
Because the DSD for this event is globally exponential (see
Fig. ), it is reasonable to estimate a DSD slope
(Fig. , bottom). In addition, because this slope is
calculated over the entire range of diameters, the retrieved slope
field is vertically consistent (less affected by the
wind). We also note that the retrieved slopes are consistent with
the results of (Fig. 8 in this paper) in
a similar case study.
Conclusions
We have built a 4-D-VAR data assimilation algorithm for retrieving
vertical DSD profiles and vertical wind under the bright band from
observations coming from a micro-rain radar (MRR) and from a co-located
disdrometer, associated with a vertical propagation model. In
this paper, we focused on the data assimilation algorithm. The
algorithm finely handles measurements of various natures collected
at different resolutions. We consequently chose to use a simple
propagation model, only taking gravity and vertical
wind into account. Because of the limitations of the model, the retrieval
algorithm is currently only suitable to study stratiform, light rain events.
The algorithm was firstly applied to simulated data in a twin
experiment context. In simulated observations, despite the addition
of instrumental noise and turbulence effects, we showed that the proposed technique was
able to retrieve DSD fields along with vertical wind patterns and
intensities thanks to radar Doppler spectra and disdrometer drop
fluxes. In particular, the algorithm appears able to handle the
turbulence impact on Doppler spectra even if the turbulence is not
explicitly implemented in the propagation model.
The behavior of the algorithm is assessed on a light rain
event during which co-localized MRR and optical disdrometer were
used in the south of France during the HyMeX campaign. According to
this work, the algorithm appears able to make the best use of the data from the two
instruments. In particular, we noticed (suspicious) vertical
trends on the MRR products which cannot be reproduced by the
propagation model due to its conservative nature. However, despite
this trend, the assimilation algorithm is able to produce a good
compromise between the observations at different heights. The
vertical retrieved wind field, although not independently
validated, is spatially and temporally coherent with
what can be expected for a rain event such as the one studied
here. The DSD fields also appear spatially coherent and
self-consistent. By combining ground and radar data, the presented
algorithm is therefore able to retrieve the vertical wind field and
to improve the DSD profile retrieval.
Independent information on the vertical wind and/or on the DSD are
nevertheless necessary to estimate the performances of the
algorithm. For this purpose, we plan to use dual-frequency wind profilers (see for
instance , which retrieves vertical wind in
rain from two profilers). Another possibility would be to use the
methods developed by or ,
which retrieve vertical winds and DSD from K band and/or W band
Doppler spectra. We could also study the impact on the quality of
the retrievals of the addition of other frequencies directly in the
assimilation process.
Some improvements are needed to provide an algorithm suitable for
various weather situations (tropical rain and convective events,
for instance). In our model, we discarded horizontal motions. This
hypothesis is valid only if there is no significant horizontal wind
or at least if the DSD is sufficiently homogeneous in the
horizontal plane. It will be necessary to evaluate
the impact of such an hypothesis more precisely. Another possibility would consist
in taking into account the horizontal wind by extending the model
with an additional spatial dimension. In our case, this can be done
with the use of the YAO assimilation tool that provides a simple
way to do it.
Other processes such as evaporation and coalescence/breakup have to
be implemented in the propagation model. This enhanced model would
allow the relative importance of the different
physical phenomena (wind, evaporation, collisions, …) to be investigated, and
therefore which are really essential for the
assimilation algorithm to be determined. It would also help to better parameterize
the algorithm through its ability to explain the observations.
Finally, the method is well suited for merging observations of
different types each with its own temporal and spatial
resolution. Indeed, when merging new observations, only the
observation operator has to be adapted. For instance we could use
more disdrometers located in the vicinity or the reflectivity at
another frequency from another radar. Any kind of sensor which
observes geophysical parameters playing a role during the fall of
raindrops could also be easily introduced in the assimilation
algorithm (ground wind, pressure, humidity).
Data availability
The data sets used in this paper (for instance micro-rain radar measurements) are available on the HyMeX website:
http://mistrals.sedoo.fr/HyMeX/.
Acknowledgements
This work was supported by the French Programme National de
Télédétection Spatiale 759 (PNTS,
http://www.insu.cnrs.fr/pnts), grant no. PNTS-2013-01, and by
the IPSL (Institut Pierre-Simon Laplace). The authors wish to thank
Jean-Marie Donier from the “Centre National de Recherches
Météorologiques – Groupe d'étude de l'Atmosphère
Météorologique” (CNRM-GAME) for supplying us with the MRR data. We
also thank the LOCEAN team for providing us with the YAO software.
Edited by: M. Portabella
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