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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 σ0 2 (thus the condition W 0=0 at ground level is statistically verified). The optimum choice of σ0 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 σ0 2 (thus the condition W 0=0 at ground level is statistically verified). The optimum choice of σ0 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
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
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.
A real-time and automated multiple-Doppler analysis method for ground-based radar data, with an emphasis on observations conducted over complex terrain, is presented. It is the result of a joint effort of the radar groups of Centre National de Recherches Météorologiques and Laboratoire d'Aérologie with a view to converging toward a common optimized procedure to retrieve mass-conserved three-dimensional wind fields in the presence of complex topography. The multiple-Doppler synthesis and continuity adjustment technique initially proposed for airborne Doppler radar data, then extended to ground-based Doppler radars and nonflat orography, is combined with a variational approach aimed at improving the vertical velocity calculation over mountainous regions. This procedure was successfully applied in real time during the Mesoscale Alpine Programme Special Observing Period. The real-time processing and display of Doppler radar data were intended to assist nowcast and aircraft missions, and involved efforts of the United Sates, France, and Switzerland.
A real-time and automated multiple-Doppler analysis method for ground-based radar data, with an emphasis on observations conducted over complex terrain, is presented. It is the result of a joint effort of the radar groups of Centre National de Recherches Météorologiques and Laboratoire d'Aérologie with a view to converging toward a common optimized procedure to retrieve mass-conserved three-dimensional wind fields in the presence of complex topography. The multiple-Doppler synthesis and continuity adjustment technique initially proposed for airborne Doppler radar data, then extended to ground-based Doppler radars and nonflat orography, is combined with a variational approach aimed at improving the vertical velocity calculation over mountainous regions. This procedure was successfully applied in real time during the Mesoscale Alpine Programme Special Observing Period. The real-time processing and display of Doppler radar data were intended to assist nowcast and aircraft missions, and involved efforts of the United Sates, France, and Switzerland.
