Interactive comment on Retrieval of three-dimensional wind fields from Doppler radar data using an efficient two-step approach by C
This manuscript is concerned with the description and testing of a variational dualDoppler radar wind retrieval algorithm. The constraints imposed in the analysis procedure are: observational constraints, which incorporate data from two or more Doppler radars (and possibly other data sources, e.g., dropsonde or sounding observations), a mass conservation constraint (sometimes referred to in the literature as a mass continuity constraint), and spatial smoothness constraints. The procedure is general enough to incorporate data from two or more stationary Doppler radars or from a so-called "pseudo-dual-Doppler radar", that is, data from an airborne-radar that, when appropriately processed, are equivalent to data gathered from two stationary Doppler radars. The observational and mass conservation constraints are standard in dual-Doppler analysis procedures. The use of smoothness constraints is standard in variational (e.g., 3DVAR) analysis techniques. The new aspect of this study is a data gridding/reduction step designed to minimize loss of information and to generate estimates of the variances of the reduced velocities. Any improvements made in these directions would be of interest to radar and mesoscale meteorologists as it could potentially improve dualDoppler analyses of a variety of convective and non-convective weather phenomena. The new retrieval is tested with pseudo-dual-Doppler radar data from a tropical depression. Unfortunately, the results from these tests are inconclusive. A comparison of the retrieved horizontal wind vectors with the horizontal wind vectors from an independent data source (dropsondes) does indicate a qualitatively good agreement (Fig. 8). But there is relatively little challenge in obtaining qualitatively reasonable horizontal wind components from dual-Doppler or pseudo-dual Doppler data, especially on the coarse analysis grid indicated on that figure. A more challenging problem and one of prime interest to mesoscale meteorologists is the analysis of the vertical wind component. Apparently it was not possible to evaluate the accuracy of the retrieved vertical wind component in this test case. In the future it would probably be better to consider a more stringent test of the algorithm, perhaps
Stability of an Iterative Dual-Doppler Wind Synthesis in Cartesian Coordinates
Abstract A popular method for obtaining a wind field that satisfies mass continuity and that is consistent with dual-Doppler radial velocity observations is the iterative wind synthesis in Cartesian coordinates. However, in some instances, the iterative method does not converge. Herein, a stability condition is derived for the iterative technique that depends on the horizontal and vertical grid spacings, and the azimuth and elevation angles of the dual-Doppler observations. Alternately, the condition for stability can be expressed as a relationship between the grid box dimensions and the direction of the coplanar arc passing through the grid point. The results of experiments with simulated radar data are consistent with the predicted stability characteristics.
Decidability of some intuitionistic predicate theories
Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T. Question. Is T decidable? One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows: Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.