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## Abstract

This paper is the first of three dealing with the three-dimensional wind field analysis from dual-Doppler radar data. Here we deal with the first step of the analysis which consists in interpolating and filtering the raw radial velocity fields within each coplane (or common plane simultaneously scanned by the two radars). To carry out such interpolation and filtering, a new method is proposed based on the principles of numerical variational analysis described by Sasaki (1970): the “filtered” representation of the observed field should be both “close” to the data points (in a least-squares sense) and verify some imperative of mathematical regularity. Any method for interpolating and smoothing data is inherently a filtering process. The proposed variational method enables this filtering to be controlled. The presented method is developed for any function of two variables but could be extended to the case of three or more variables.

Numerical simulations substantiate the theoretically predicted filtering characteristics and show an improvement on other filtering schemes. It is found, compared to the classical filtering using the Cressman weighting function, that the variational method brings a substantial improvement of the gain curve (in the sense of a steeper cut-off), when the “regularity” of the second-order derivatives is imposed. It is worth noting that this improvement is achieved without increasing the computing time. It is also emphasized that an elaborate numerical differentiation scheme should be used to estimate the divergence, otherwise the gain curve for this parameter may be different from that for the Cartesian coplane velocities (which may induce distortion in the final three-dimensional wind field).

## Abstract

This paper is the first of three dealing with the three-dimensional wind field analysis from dual-Doppler radar data. Here we deal with the first step of the analysis which consists in interpolating and filtering the raw radial velocity fields within each coplane (or common plane simultaneously scanned by the two radars). To carry out such interpolation and filtering, a new method is proposed based on the principles of numerical variational analysis described by Sasaki (1970): the “filtered” representation of the observed field should be both “close” to the data points (in a least-squares sense) and verify some imperative of mathematical regularity. Any method for interpolating and smoothing data is inherently a filtering process. The proposed variational method enables this filtering to be controlled. The presented method is developed for any function of two variables but could be extended to the case of three or more variables.

Numerical simulations substantiate the theoretically predicted filtering characteristics and show an improvement on other filtering schemes. It is found, compared to the classical filtering using the Cressman weighting function, that the variational method brings a substantial improvement of the gain curve (in the sense of a steeper cut-off), when the “regularity” of the second-order derivatives is imposed. It is worth noting that this improvement is achieved without increasing the computing time. It is also emphasized that an elaborate numerical differentiation scheme should be used to estimate the divergence, otherwise the gain curve for this parameter may be different from that for the Cartesian coplane velocities (which may induce distortion in the final three-dimensional wind field).

## Abstract

The choice of the boundary condition when integrating the air mass continuity equation, is a major problem of the 3D wind field analysis from dual (or multiple) Doppler radar data. A zero vertical velocity at ground level seems the most natural boundary condition. Unfortunately, it is known that the integration processes is unstable with respect to this condition: it leads to errors amplifying exponentially with height. In order to overcome this difficulty various solutions have been proposed, the most recent ones using the variational analysis: (i) integrating from storm top level, (ii) integrating from storm top level while constraining the height integrated divergence to be as small as possible (Ziegier, 1978), and (iii) constraining the direct estimates of the 3D wind field to satisfy the continuity equation (Ray *et al*., 1980). The analysis proposed in this paper is also based upon a variational concept, but it differs in its principle from those previously cited. It consists in adjusting the boundary condition field at ground level in order to optimize the “mathematical regularity” of the vertical velocity field, followed by upward integration of the continuity equation. In such a formulation, the boundary condition at ground level is “floating” (i.e., not specified). However it is possible to require. as a subsidiary condition of the variational problem, that the vertical velocity at ground level fluctuate about zero with a specified variance σ_{0}
^{2} (thus the condition *W*
_{0}=0 at ground level is statistically verified). The optimum choice of σ_{0} is established from considerations of statistical theory. It should be noted that the horizontal divergence (or coplane divergence) profile is unadjusted and that the equation of continuity is integrated upward from the optimum lower boundary condition to obtain *W*.

An application to simulated or real data helps us to appreciate the improvements brought by the present variational approach with respect to standard methods of integration: 1) the random errors are as small as in the case of an integration from storm top level, but here the boundary condition *W*
_{0}=0 at ground level is statistically preserved; and 2) for the integration paths where no cannot be specified (lack of data at low level), the analysis automatically generates a boundary condition which realizes the best regularity of *W* with respect to the neighbouring paths.

This variational analysis can be easily implemented on a computer from the program prepared for the standard integration, and it requires a short additional computation time.

## Abstract

The choice of the boundary condition when integrating the air mass continuity equation, is a major problem of the 3D wind field analysis from dual (or multiple) Doppler radar data. A zero vertical velocity at ground level seems the most natural boundary condition. Unfortunately, it is known that the integration processes is unstable with respect to this condition: it leads to errors amplifying exponentially with height. In order to overcome this difficulty various solutions have been proposed, the most recent ones using the variational analysis: (i) integrating from storm top level, (ii) integrating from storm top level while constraining the height integrated divergence to be as small as possible (Ziegier, 1978), and (iii) constraining the direct estimates of the 3D wind field to satisfy the continuity equation (Ray *et al*., 1980). The analysis proposed in this paper is also based upon a variational concept, but it differs in its principle from those previously cited. It consists in adjusting the boundary condition field at ground level in order to optimize the “mathematical regularity” of the vertical velocity field, followed by upward integration of the continuity equation. In such a formulation, the boundary condition at ground level is “floating” (i.e., not specified). However it is possible to require. as a subsidiary condition of the variational problem, that the vertical velocity at ground level fluctuate about zero with a specified variance σ_{0}
^{2} (thus the condition *W*
_{0}=0 at ground level is statistically verified). The optimum choice of σ_{0} is established from considerations of statistical theory. It should be noted that the horizontal divergence (or coplane divergence) profile is unadjusted and that the equation of continuity is integrated upward from the optimum lower boundary condition to obtain *W*.

An application to simulated or real data helps us to appreciate the improvements brought by the present variational approach with respect to standard methods of integration: 1) the random errors are as small as in the case of an integration from storm top level, but here the boundary condition *W*
_{0}=0 at ground level is statistically preserved; and 2) for the integration paths where no cannot be specified (lack of data at low level), the analysis automatically generates a boundary condition which realizes the best regularity of *W* with respect to the neighbouring paths.

This variational analysis can be easily implemented on a computer from the program prepared for the standard integration, and it requires a short additional computation time.

## Abstract

Recently, 95-GHz Doppler radars have been developed to document microphysical and dynamical properties of nonprecipitating clouds. RALI (airborne radar and lidar) plans to associate a dual-beam 95-GHz Doppler radar and a backscattering lidar on the same airborne platform. In optically thin clouds, data from both instruments (looking at nadir) will be available and can be combined to retrieve the microphysical characteristics of the cloud particles size distribution. But in dense clouds the lidar signal is extinguished over a few tenths of meters, and reflectivity may be biased by along-path attenuation. Thus the sampling strategy anticipated by RALI is the dual beam; that is, the radar operates along two viewing angles: nadir and 40° fore. The fact that the along-path attenuation acts in a different way along the two viewing angles is exploited by means of two algorithms (“stereoradar” and “dual beam”) allowing to retrieve the true (nonattenuated) reflectivity *Z* and the specific attenuation *K.*

In this paper, after recalling the principle of the two algorithms, the authors simulate the sampling of various typical clouds with RALI. The two algorithms are then applied to the simulations in order to evaluate their performances in the retrieval of true reflectivity and attenuation. The stereoradar algorithm retrieves *Z* fields very well and *K* fields less accurately. This is true even when a drizzle cell is embedded in the cloud. The dual-beam algorithm provides very good retrievals in the absence of drizzle, but its results are severely biased when there is drizzle. However, in such a situation the algorithm provides a diagnostic that two types of particles are present in the cloud.

It is argued that the retrieval of the true reflectivity and specific attenuation allows a determination of the liquid water content, the total droplet concentration, and the effective radius of the cloud.

## Abstract

Recently, 95-GHz Doppler radars have been developed to document microphysical and dynamical properties of nonprecipitating clouds. RALI (airborne radar and lidar) plans to associate a dual-beam 95-GHz Doppler radar and a backscattering lidar on the same airborne platform. In optically thin clouds, data from both instruments (looking at nadir) will be available and can be combined to retrieve the microphysical characteristics of the cloud particles size distribution. But in dense clouds the lidar signal is extinguished over a few tenths of meters, and reflectivity may be biased by along-path attenuation. Thus the sampling strategy anticipated by RALI is the dual beam; that is, the radar operates along two viewing angles: nadir and 40° fore. The fact that the along-path attenuation acts in a different way along the two viewing angles is exploited by means of two algorithms (“stereoradar” and “dual beam”) allowing to retrieve the true (nonattenuated) reflectivity *Z* and the specific attenuation *K.*

In this paper, after recalling the principle of the two algorithms, the authors simulate the sampling of various typical clouds with RALI. The two algorithms are then applied to the simulations in order to evaluate their performances in the retrieval of true reflectivity and attenuation. The stereoradar algorithm retrieves *Z* fields very well and *K* fields less accurately. This is true even when a drizzle cell is embedded in the cloud. The dual-beam algorithm provides very good retrievals in the absence of drizzle, but its results are severely biased when there is drizzle. However, in such a situation the algorithm provides a diagnostic that two types of particles are present in the cloud.

It is argued that the retrieval of the true reflectivity and specific attenuation allows a determination of the liquid water content, the total droplet concentration, and the effective radius of the cloud.

## Abstract

This paper presents a new concept in the measurement of precipitation by radar. The principle consists of stereoscopic observations: the same precipitation cell is observed by two radars operating at the same attenuated frequency but following two different angles of view. For instance, a system of two around based 20 km-spaced radars operating in *X*-band may provide stereoscopic observations. But the present paper considers more particularly an airborne radar configuration where a dual beam antenna system, combined with the aircraft displacement, views a precipitation cell under two incidences. In the method developed, the signal of interest is the difference *z*
_{1} – *z*
_{2} between the apparent radar reflectivities (in dBZ) observed store the two viewing angles. It is shown that a second-order differentiation of *z*
_{1} – *z*
_{2} allows one to estimate the along track gradient in the attenuation coefficient ∂*K*/∂*X*. However, because this estimate is too noisy, a variational method is proposed to retrieve the *K*-field from the *z*
_{1} and *z*
_{2} observations.

The validation of such a concept is then investigated on the basis of a numerical simulation. In consideration of a simple raincell model and realistic sampling of the apparent reflectivity fields *z*
_{1} and *z*
_{2}, it is demonstrated that the *K*-field can be reliably retrieved in the range of values 0.5 to 7 dB km^{−1}. The obtained *K*-field may then be used to correct the apparent reflectivities for attenuation, and achieve an estimate of the “true” reflectivity *Z*.

Another variational approach is subsequently proposed which draws benefit of the simultaneous knowledge of *Z* and *K* to achieve an improved estimate of the rainfall rate *R*. A numerical simulation is again used as a tool to test it.

The last part of the paper discusses possible applications of the new concept developed. Some emphasis is placed on the stereoradar from space in competition with the techniques already proposed for measurement of rain from a spaceborne radar.

## Abstract

This paper presents a new concept in the measurement of precipitation by radar. The principle consists of stereoscopic observations: the same precipitation cell is observed by two radars operating at the same attenuated frequency but following two different angles of view. For instance, a system of two around based 20 km-spaced radars operating in *X*-band may provide stereoscopic observations. But the present paper considers more particularly an airborne radar configuration where a dual beam antenna system, combined with the aircraft displacement, views a precipitation cell under two incidences. In the method developed, the signal of interest is the difference *z*
_{1} – *z*
_{2} between the apparent radar reflectivities (in dBZ) observed store the two viewing angles. It is shown that a second-order differentiation of *z*
_{1} – *z*
_{2} allows one to estimate the along track gradient in the attenuation coefficient ∂*K*/∂*X*. However, because this estimate is too noisy, a variational method is proposed to retrieve the *K*-field from the *z*
_{1} and *z*
_{2} observations.

The validation of such a concept is then investigated on the basis of a numerical simulation. In consideration of a simple raincell model and realistic sampling of the apparent reflectivity fields *z*
_{1} and *z*
_{2}, it is demonstrated that the *K*-field can be reliably retrieved in the range of values 0.5 to 7 dB km^{−1}. The obtained *K*-field may then be used to correct the apparent reflectivities for attenuation, and achieve an estimate of the “true” reflectivity *Z*.

Another variational approach is subsequently proposed which draws benefit of the simultaneous knowledge of *Z* and *K* to achieve an improved estimate of the rainfall rate *R*. A numerical simulation is again used as a tool to test it.

The last part of the paper discusses possible applications of the new concept developed. Some emphasis is placed on the stereoradar from space in competition with the techniques already proposed for measurement of rain from a spaceborne radar.

## Abstract

Airborne Doppler radar is an important meteorological research tool for sampling the flow field of large convective systems (hurricanes, squall lines, fronts, etc.). In order to improve space resolution, the next generation of airborne Doppler radars must operate at fast scanning rates, i.e., use a very short dwell time that requires coding transmission. In this paper, three types of coding are envisioned: phase coding (unweighted or weighted Barker codes), linear frequency modulation (unweighted or weighted chirp), and a comb of four stepped frequencies. The radar transmission, reception, and processing were simulated; and the statistical accuracy of the mean Doppler velocity estimates and the mean power estimates obtained in the various cases were compared. The comb of four stepped frequencies emerges as the best technique among those considered for this application.

## Abstract

Airborne Doppler radar is an important meteorological research tool for sampling the flow field of large convective systems (hurricanes, squall lines, fronts, etc.). In order to improve space resolution, the next generation of airborne Doppler radars must operate at fast scanning rates, i.e., use a very short dwell time that requires coding transmission. In this paper, three types of coding are envisioned: phase coding (unweighted or weighted Barker codes), linear frequency modulation (unweighted or weighted chirp), and a comb of four stepped frequencies. The radar transmission, reception, and processing were simulated; and the statistical accuracy of the mean Doppler velocity estimates and the mean power estimates obtained in the various cases were compared. The comb of four stepped frequencies emerges as the best technique among those considered for this application.

During May and June 1981 several French research organizations, the University of Abidjan (Ivory Coast) and the Agency for Security of Aeronautical Navigation (ASECNA), participated in the observational field program called “Convection Profonde Tropicale 1981” (COPT 81). COPT 81 was directed toward developing a better understanding of the dynamical and electrical features of precipitating convection in continental tropical regions.

The observational network was designed to study the development and evolution of diurnal convection and squall lines over the northern part of the Ivory Coast, which is an example of a tropical savanna region at the southern edge of the Sahel. It consisted of two Doppler radars, a central meteorological station equipped for the reception of satellite data, rawin sounding and interrogation of remote targets, an acoustic sounder, a central electrical and electromagnetical station, and a set of remote ground meteorological and electrical stations.

Some experimental results are presented to characterize the main features of a tropical continental squall line. The evolution of the boundary layer during its passage, its precipitation pattern and associated dynamical field, its surface trace and the modification it produces on the thermodynamical state of the atmosphere, as well as some of its associated electrical features, are given.

During May and June 1981 several French research organizations, the University of Abidjan (Ivory Coast) and the Agency for Security of Aeronautical Navigation (ASECNA), participated in the observational field program called “Convection Profonde Tropicale 1981” (COPT 81). COPT 81 was directed toward developing a better understanding of the dynamical and electrical features of precipitating convection in continental tropical regions.

The observational network was designed to study the development and evolution of diurnal convection and squall lines over the northern part of the Ivory Coast, which is an example of a tropical savanna region at the southern edge of the Sahel. It consisted of two Doppler radars, a central meteorological station equipped for the reception of satellite data, rawin sounding and interrogation of remote targets, an acoustic sounder, a central electrical and electromagnetical station, and a set of remote ground meteorological and electrical stations.

Some experimental results are presented to characterize the main features of a tropical continental squall line. The evolution of the boundary layer during its passage, its precipitation pattern and associated dynamical field, its surface trace and the modification it produces on the thermodynamical state of the atmosphere, as well as some of its associated electrical features, are given.

## Abstract

Several space agencies are presently considering missions with active instruments (radar, lidar), which are able to document cloud stratification and cloud microphysical properties on the global scale. The objective of this paper is to develop an algorithm to derive as much information as possible from a single-frequency, nadir-looking cloud radar operating from an airborne or spaceborne platform. It is impossible to derive all parameters of interest in the radiative budget of a cloud from only the radar reflectivity profile, unless some a priori knowledge of cloud processes is introduced in the formulation of the algorithm itself. The a priori knowledge considered here (total concentration of particles invariant with altitude, adiabatic liquid water content) does not apply to all cloud types but only to warm stratiform clouds where entrainment is weak.

The algorithm concept and inversion procedure, including a stable scheme for correcting the radar reflectivity for attenuation, are first described. A test of the algorithm is then performed using numerical simulations in order to investigate the sensitivity of the retrieval to measurement noise, degradation of the range resolution, shape of the cloud droplet distribution, and presence of entrainment. In the realistic conditions of an airborne experiment, the retrieval of cloud base *h*
_{b}, total number concentration of particles *N*
_{T}, profiles of the liquid water content, and effective radius *r*
_{e} can be performed with good accuracy (provided the entrainment coefficient is below 1 km^{−1}). With the sampling characteristics of a spaceborne radar, retrievals of the cloud base and liquid water content remain reasonably accurate, but the estimates of *N*
_{T} and *r*
_{e} are degraded to a level where they become meaningless.

A test of the algorithm is performed using a dataset from the zenith-pointing ground-based 94-GHz radar of The Pennsylvania State University, obtained during the Continental Stratus Experiment. The algorithm is found to be successful in 43% of cases. An attempt of evaluation of the retrieval is made by comparison with ceilometer data. Most failure cases are probably due to the presence of drizzle.

## Abstract

Several space agencies are presently considering missions with active instruments (radar, lidar), which are able to document cloud stratification and cloud microphysical properties on the global scale. The objective of this paper is to develop an algorithm to derive as much information as possible from a single-frequency, nadir-looking cloud radar operating from an airborne or spaceborne platform. It is impossible to derive all parameters of interest in the radiative budget of a cloud from only the radar reflectivity profile, unless some a priori knowledge of cloud processes is introduced in the formulation of the algorithm itself. The a priori knowledge considered here (total concentration of particles invariant with altitude, adiabatic liquid water content) does not apply to all cloud types but only to warm stratiform clouds where entrainment is weak.

The algorithm concept and inversion procedure, including a stable scheme for correcting the radar reflectivity for attenuation, are first described. A test of the algorithm is then performed using numerical simulations in order to investigate the sensitivity of the retrieval to measurement noise, degradation of the range resolution, shape of the cloud droplet distribution, and presence of entrainment. In the realistic conditions of an airborne experiment, the retrieval of cloud base *h*
_{b}, total number concentration of particles *N*
_{T}, profiles of the liquid water content, and effective radius *r*
_{e} can be performed with good accuracy (provided the entrainment coefficient is below 1 km^{−1}). With the sampling characteristics of a spaceborne radar, retrievals of the cloud base and liquid water content remain reasonably accurate, but the estimates of *N*
_{T} and *r*
_{e} are degraded to a level where they become meaningless.

A test of the algorithm is performed using a dataset from the zenith-pointing ground-based 94-GHz radar of The Pennsylvania State University, obtained during the Continental Stratus Experiment. The algorithm is found to be successful in 43% of cases. An attempt of evaluation of the retrieval is made by comparison with ceilometer data. Most failure cases are probably due to the presence of drizzle.

## Abstract

One of the major problems in three-dimensional wind field analysis from dual (or multiple) Doppler radar data resides in the non-stationary of the observed air flow within the volume sampling time which ranges typically from 2 to 5 min. The present part II is focused on this problem. Most often, the storm moves horizontally at a speed of 5–25 m s^{−1}.Therefore, the temporal variation for a fixed observer at ground level results from the superposition of two effects: 1) the intrinsic temporal variation (or variation seen in a frame moving with the storm) and 2) the effect of horizontal advection.

The first contribution of the paper concerns the development of an algorithm for correcting for the advection effect in the case of a dual-Doppler radar observation. This algorithm, which provides a mathematically exact solution to the problem of correcting for advection, can be very easily implemented in a computer program.

The second contribution deals with the errors that may arise from an accurate (or lack of) evaluation of the advective velocity, or from an “Intrinsic” temporal variation in the moving frame. A spectral decomposition of the 3D wind field is considered, allowing us to study the dependence of the error on the scale of the motion. Specific conclusions are drawn about the requirements necessary to achieve a given accuracy in the vertical velocity field. i.e., admissible uncertainty in the advective velocity, and characteristic time of intrinsic temporal variation.

Finally an example of application to actual Doppler radar data is presented. The results obtained from non-advected analyses are compared and discussed.

## Abstract

One of the major problems in three-dimensional wind field analysis from dual (or multiple) Doppler radar data resides in the non-stationary of the observed air flow within the volume sampling time which ranges typically from 2 to 5 min. The present part II is focused on this problem. Most often, the storm moves horizontally at a speed of 5–25 m s^{−1}.Therefore, the temporal variation for a fixed observer at ground level results from the superposition of two effects: 1) the intrinsic temporal variation (or variation seen in a frame moving with the storm) and 2) the effect of horizontal advection.

The first contribution of the paper concerns the development of an algorithm for correcting for the advection effect in the case of a dual-Doppler radar observation. This algorithm, which provides a mathematically exact solution to the problem of correcting for advection, can be very easily implemented in a computer program.

The second contribution deals with the errors that may arise from an accurate (or lack of) evaluation of the advective velocity, or from an “Intrinsic” temporal variation in the moving frame. A spectral decomposition of the 3D wind field is considered, allowing us to study the dependence of the error on the scale of the motion. Specific conclusions are drawn about the requirements necessary to achieve a given accuracy in the vertical velocity field. i.e., admissible uncertainty in the advective velocity, and characteristic time of intrinsic temporal variation.

Finally an example of application to actual Doppler radar data is presented. The results obtained from non-advected analyses are compared and discussed.

## Abstract

This paper is based on the observation of a cold front using a C-band Doppler radar. The extent of the precipitation system associated with the front allowed collection of Doppler radar data during 12 consecutive hours. The methodology for data acquisition presently used is conical scanning. The data analysis has been extended to the case of a nonuniform distribution of tracers.

The air circulation is presented in a reference frame moving at the speed of the front. A pronounced cross-frontal circulation is found to be associated with significant cross-frontal acceleration. The thermal structure across the front is reconstructed by means of the equations of motion.

From the vertical velocity field an estimate of the height-integrated condensation rate is made. It is found to agree with the rainfall rate inferred from the radar reflectivity data.

Also, large-amplitude small-scale motions are detected and identified as a well-characterized atmospheric wave. Theoretical considerations support the explanation that it is the manifestation of a dynamical instability of the shear flow within the frontal zone.

## Abstract

This paper is based on the observation of a cold front using a C-band Doppler radar. The extent of the precipitation system associated with the front allowed collection of Doppler radar data during 12 consecutive hours. The methodology for data acquisition presently used is conical scanning. The data analysis has been extended to the case of a nonuniform distribution of tracers.

The air circulation is presented in a reference frame moving at the speed of the front. A pronounced cross-frontal circulation is found to be associated with significant cross-frontal acceleration. The thermal structure across the front is reconstructed by means of the equations of motion.

From the vertical velocity field an estimate of the height-integrated condensation rate is made. It is found to agree with the rainfall rate inferred from the radar reflectivity data.

Also, large-amplitude small-scale motions are detected and identified as a well-characterized atmospheric wave. Theoretical considerations support the explanation that it is the manifestation of a dynamical instability of the shear flow within the frontal zone.