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
(

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

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

In order to improve vertical DSD profile retrievals, this study
establishes a framework allowing the assimilation at a fine scale
(

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

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.

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.

As explained before, this study develops a simple framework and so
uses a very simple model presented in Sect.

In Sect.

The DSD, noted

Equation (

In the

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
(

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

To limit the errors caused by this simplification, we will only treat quiet,
stratiform light rain events (see the case study in Sect.

The evaporation is mathematically modeled as a term of advection along
the mass of droplets coordinate. The advection speed (parameter

According to this parameterization and for long cold stratiform rain events
similar to the one investigated in this study (see Sect.

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
(

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
(

This simplified PDE used in this study (Eq.

We note

The unknowns of such a model are as follows.

The initial (

The top (

The vertical wind time–height field,

Thanks to this parameterization, for each time step

Finally, the problem is reduced to the determination of the parameter
field (

In the previous section, we have seen that the model propagates the
top DSD

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.

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

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

In the data assimilation context, the cost function usually takes the
form:

The cost function finally used takes the form:

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.

In the end, the cost function

For a given grid point

At time (

with

The observation operator,

For disdrometer measurements, we build an operator

For radar measurements, we build an operator

Calculate the theoretical non attenuated and non-noisy spectrum

Then derive the final attenuated spectrum

Finally, for minimization purposes,

We recall that the operator run on the computer is necessarily a discretized version of what is presented here.

In this section, we present a brief description of the penalization
term

The numerical minimization of the cost function

We use the YAO software (

We finally note that we have to initialize the unknowns of the problem

In this section, we firstly give the characteristics of the two
instruments used in this study, namely a Doppler

The first instrument used in this study is a micro-rain radar (MRR,

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

The radar range gate size is

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 (

For the real case study of the 12 October 2013,

The two instruments described above were deployed in Ardèche
(southwest of France), during autumn

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

The temperature was around

For this case study, the number of time and height
discretization steps (NT and NZ; see
Sect.

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.

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.

Figure

Then, from these fields, we simulate realistic MRR and DBS
observations by adding model and instrumental errors (instrumental
noise and turbulence, see Sect.

Finally, the assimilation algorithm estimates an optimal set of
unknowns explaining these observations (step e). Assimilated

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

First, we have to simulate realistic time series of top DSD
parameters (

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

We also need a time–height field of vertical winds

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.

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.

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

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

At the end of the process, we check which drops, among the

In this section, we firstly assess the ability of the assimilation
process to reproduce the observations. Then, we compare the true
and assimilated DSD

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

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

For the twin experiment: comparisons between truth, observations and
assimilation for the moments of order

In the following analysis, we use the denominations true,
observed and assimilated (see Sect.

We recall (Sect.

The assimilated spectra are also broader than the true
ones, but to a lesser extent (spectral width

On the contrary, the characteristic velocity is not highly
affected by the addition of turbulence impact on observed fields
(bias

For the twin experiment: MAE (Eq.

To reproduce the turbulence broadening, the algorithm can act on
the two unknown fields,

An example of Doppler spectra at a

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

The results are slightly less satisfactory for liquid water content (LWC)
(10.0 vs. 142

We have seen that the

The rain rates are also satisfactory for resolutions over

In this last subsection, we briefly assess the influence of the
integration time of the observations (observation window, see
Sect.

Because the model fits, at best, the

The results are globally the same for the other parameters of
Table

Finally, we conclude that an observation window of

In this section, we apply the assimilation algorithm to the real
data case study described in Sect.

For the real case study: comparison between observations and assimilation for
three moments of the MRR Doppler spectra: order

We use the indicators introduced in Sect.

Table

The three columns on the right of Table

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

We can also note that the very small drops (

For large drops (

Figure

We remind that there are no Doppler spectra available for the two
lowest gates (0–

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

We also have an overall mean wind of

We have seen (Sect.

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.

Now we will also assess whether these structures are retrieved for some
important moments of the DSD. Figure

Because the DSD for this event is globally exponential (see
Fig.

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

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).

The data sets used in this paper (for instance micro-rain radar measurements) are available on the HyMeX website:

This work was supported by the French Programme National de
Télédétection Spatiale 759 (PNTS,