Errors in Fixed and Moving Frame of References: Applications for Conventional and Doppler Radar Analysis

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  • 1 Department of Meteorology, University of Maryland, College Park 20742 and NASA/Goddard Space Flight Center, Laboratory for Atmospheric Sciences, Greenbelt, MD 20771
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Abstract

Procedures for the estimation and correction of advection effects with single and multiple conventional and Doppler radars are developed. In the case of scalars or Cartesian vectors, the essence of the method is finding a moving frame of reference where (in the least-square sense) the observations are as stationary as possible. It is shown that this is quite different from the more traditional correlation techniques. In the case of non-Cartesian vectors (e.g., radial velocities from Doppler radars), very different criteria are derived. For a single Doppler radar, we look for a frame of reference where ∫[∂2/∂t2(vrr)]2 is a minimum. Here vr is the radial velocity, t is time and r is the distance of the observed point from the radar. In the multiple Doppler case, it is shown that in a frame of reference moving with the advection speed,
tvr(1)r(1)tvr(2)r(2)2
is a minimum. Here superscripts (1) and (2) refer to radars (1) and (2), respectively. For scalars (such as radar reflectivities) or Cartesian vectors, the correction for advection is trivial; one merely redefines the observations in the new frame. It is shown that in general, this correction, when applied to radial velocities, is wrong. The appropriate correction procedures are derived. Using scale analysis, error estimates with and without the correction procedures are derived. Experiments with dual-Doppler radar data of the optically clear boundary layer demonstrate that for that case it is possible to estimate advection velocities from the data. When a correction procedure is applied, the horizontal velocities, as well as the derived vertical motions, display improved temporal correlation between scans.

Abstract

Procedures for the estimation and correction of advection effects with single and multiple conventional and Doppler radars are developed. In the case of scalars or Cartesian vectors, the essence of the method is finding a moving frame of reference where (in the least-square sense) the observations are as stationary as possible. It is shown that this is quite different from the more traditional correlation techniques. In the case of non-Cartesian vectors (e.g., radial velocities from Doppler radars), very different criteria are derived. For a single Doppler radar, we look for a frame of reference where ∫[∂2/∂t2(vrr)]2 is a minimum. Here vr is the radial velocity, t is time and r is the distance of the observed point from the radar. In the multiple Doppler case, it is shown that in a frame of reference moving with the advection speed,
tvr(1)r(1)tvr(2)r(2)2
is a minimum. Here superscripts (1) and (2) refer to radars (1) and (2), respectively. For scalars (such as radar reflectivities) or Cartesian vectors, the correction for advection is trivial; one merely redefines the observations in the new frame. It is shown that in general, this correction, when applied to radial velocities, is wrong. The appropriate correction procedures are derived. Using scale analysis, error estimates with and without the correction procedures are derived. Experiments with dual-Doppler radar data of the optically clear boundary layer demonstrate that for that case it is possible to estimate advection velocities from the data. When a correction procedure is applied, the horizontal velocities, as well as the derived vertical motions, display improved temporal correlation between scans.
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