Characterisation and improvement of the structure function estimation for application in PSI

Numerical Weather Prediction (NWP) is becoming state of the art for the compensation of atmospheric effects in InSAR and especially in PSI. The structure function (variogram) estimated on the InSAR data and on the NWP data is a required statistical characteristic and very useful in the processing of the data. For example, the structure function (typically the semivariogram) estimated on InSAR data is fundamental for the Kriging of the atmospheric phase screen (APS) based on irregular PSI estimates. The required parameters are the nugget, the range and the slope of the structure function. The practical implementation has shown that the NWP predicted structure function and the InSAR estimated structure function based on the conventional semivariogram equation do not match. Straightforward explanations are wrongly estimated APSs due to an insufficient number of interferograms or hindcasts that fail to capture the turbulent water vapour signal. However, in this paper we explain the effects of noise in interferograms and coarse resolution in the NWP on the conventional structure function estimation resulting in the observed mismatch. In order to avoid the mismatch, an alternative implementation based on wavelets is suggested and demonstrated using real Sentinel-1 data. We show that the wavelet based structure function estimation outperforms the conventional structure function estimation based on a variogram. Application of the proposed structure function alternative based on NWP data are the master selection, the estimation of the effective NWP data resolution, and a statistical consistency check of the estimated InSAR APS. To support the understanding, we demonstrate the effects of noise and resolution using simulated fractals. An application of the proposed wavelet based structure function estimation is the estimation of the effective NWP data resolution, which is demonstrated using the same Sentinel-1 data test case.