Adaptive K_dp Moment Estimation

The specific differential phase, K_dp, is the range derivative of the Angular PhiDP, described in Cubic spline fit. This is written as:

Angular PhiDP has no folding, but is still noisy due to backscattering effects and measurement errors. When performing differentiation of noisy data in the more traditional LSQ method of calculating K_dp, the errors are magnified in the K_dp output.

With the continuous complex form of Angular PhiDP data, a cubic spline methodology is chosen, because it has the greatest smoothness over all functions evaluating derivatives. Any smoothing function can be overly aggressive causing loss in data fidelity; the over smoothed situation. Therefore, an adaptive process is created to adjust the smoothing function.

The weather classification algorithm described in Weather classification algorithm is re-used to calculate K_dp. The weather classification algorithms defines segments having liquid rain versus other segments. In areas where there is no liquid rain, K_dp should be 0. This assumption provides a convenient end condition to determine all the coefficients needed in the cubic spline equation.

The cubic spline processing ¹ of the Angular PhiDP is performed in 2 steps. The first pass uses a standard fixed smoothing function, which evaluates the mean and dispersion of Angular PhiDP. The dispersion results from the first pass is then used as a weighting function to adapt or scale the smoothing function to match the properties of the Angular PhiDP data. This adaptive function ensures that the trade-off between smoothness and data fidelity is optimized.