Uncertainty assessment of remotely sensed data using copulas

Quantification of rainfall and its spatial and temporal variability is extremely important for reliable hydrologic modeling. Recent technological advances in the field of remote sensing have led to an increase in available rainfall data on a regional and global scale. However, the advantages of remotely sensed data are limited by complications related to the indirect nature of remotely sensed estimates. Previous studies confirm that remotely sensed rainfall estimates are subject to various errors, and for future use in hydrologic and climate studies, efforts are required to determine the accuracy of data and their associated uncertainties. Despite extensive research, however, uncertainties associated with remotely sensed rainfall estimates are not yet well quantified.

So far, a multitude of approaches and extensive research efforts have been undertaken to develop an uncertainty model for remotely sensed rainfall estimates. In order to assess rainfall uncertainties, one can simulate an ensemble of precipitation fields that consists of a large number of realizations, each of which represents a possible rainfall event that can occur. Subsequent runs of a hydrological model using simulated ensembles would then allow an assessment of uncertainty propagation due to the precipitation input. One way to generate an ensemble of rainfall estimates is to stochastically simulate random error fields and impose them on remotely sensed estimates. This study intends to develop different stochastic techniques for simulation of rainfall fields through simulating random error fields and imposing them over remotely sensed rainfall estimates. Three different models are developed and discussed in this work. In the first and second models, two elliptical copulas, Gaussian and t-copula, are used to describe the dependence structure of rainfall error and to simulate multivariate rainfall error fields. In the third model, an asymmetrical non-Gaussian v-transformed copula is employed for error simulations. In the the models, having described the dependencies using copulas, the empirical distribution function of observed rainfall error is numerically approximated and applied to the simulated error fields so that the simulated realizations are similar to those of the observed in terms of the distribution function. Furthermore, available observations of rainfall error (the differences between remotely sensed estimates and rain gauge measurements) are used to condition the simulated fields on observations.

In order to examine reliability and performance of the developed models, several case studies are presented over two watersheds. The simulated rainfall ensembles obtained from different models are compared with original rainfall estimates with respect to statistical properties, extreme values and spatio-temporal dependencies. The results indicate that the models generate rainfall uncertainty ensembles with similar spatio-temporal characteristics and stochastic properties to those of observed rainfall data. However, using the t-copula model may have significant advantages over the other models particularly with respect to extremes. It is hoped that the results of this research can be used to assess the uncertainties associated with remotely sensed rainfall estimates; as it is believed, that with accurate information about surface rainfall and its associated uncertainties, hydrologists have the potential to improve hydrologic predictions and global climate studies.