copulas for integrating weather and land information in space and time
Fakhereh Alidoost is a PhD student in the department of Earth Observation Science (EOS). Her supervisors are prof.dr.ir. A. Stein and prof.dr.ir. Z. Su from the Faculty of Geo-information Science and Earth Observation.
Environmental processes are driven by weather, land, and water variables and their interactions that change continuously in space and time. A complete process description considers both spatio-temporal dependencies and associations between those variables. Describing the dependencies is challenging because natural phenomena are often observed at a discrete set of locations and times. In this thesis I focus on reanalysis data of ECMWF (ERA-I) that are being used increasingly for those process descriptions. Major dilemmas locally are that observations are sparse, and the use of reanalysis data is prone to uncertainty because of the coarse spatial resolution and systematic bias. The complete study of dependencies will also lead to an increase in the number of involved variables. To address these problems, this research demonstrates the potentials of copulas. It uses two datasets: daily mean air temperature collected from weather stations and reanalysis data in the Qazvin Plain, Iran, and daily air temperature and precipitation retrieved from weather stations and reanalysis data in the Netherlands.
First, copulas described the dependencies between measurements and reanalysis data in the absence of ancillary data in Iran. The conditional distribution of air temperature given the reanalysis data was estimated with copulas. This thesis illustrated a systematic bias in the reanalysis air temperature data as compared to weather station measurements. I predicted bias-corrected air temperatures using two new predictors based upon Conditional Probabilities (CP): CP-I offers a single conditional probability as a predictor, while CP-II is a pixel-wise version of CP-I and offers spatially varying predictors. The CPs reduced the bias with 44 – 68% as compared to commonly applied predictors. I concluded that CPs locally improved existing bias correction methods.
Second, copulas took care of the spatial dependencies between weather variables and associations between land variables. Ancillary information was obtained from remote sensing images. The classical and common method for bias correction, i.e. a univariate Quantile Mapping (QM) produced smooth maps. To locally rectify for smoothness, the conditional distribution of air temperature given reanalysis data and elevation was estimated with copulas. Three Multivariate Copula Quantile Mappings (MCQMs) were proposed to predict bias-corrected air temperature. MCQMs reduced bias with 16-63% as compared to QM. The study showed that MCQMs were well able to represent spatial and temporal variations of air temperature and its associations with elevation.
Third, in this thesis I exploited copulas to improve the spatial resolution of air temperature data. Two new interpolators were investigated embedding remote sensing products, in particular land surface temperature, leaf area index and surface elevation: a spatial copula interpolator including covariates, and a mixed copula interpolator. The spatial copula interpolator including covariates improved the spatial predictions with 46-58% as compared to the spatial copula interpolator, the ordinary kriging predictor and the co-kriging predictor. The copula-based interpolators well represented spatial variability of air temperature and its associations with land variables at spatial resolution of 1 km. The methods are potentially useful for other sparsely and irregularly distributed weather data.
Fourth, copulas helped me to describe the multivariate dependencies of the weather extremes and yield, production, and price of potatoes in the Netherlands. In this thesis, a procedure was proposed to select the dominant driving climate indices of air temperature and precipitation in space. The conditional distributions of the non-climatic variables given the indices were estimated. The non-climatic variables were predicted with relative mean absolute errors equal to 5.4%, 3.6%, and 27.9%, respectively. I showed in this study that the proposed copula-based method optimally quantified the impact of climate extremes including their uncertainties.
The main conclusion drawn from this research is that copula-based methods can well represent the spatial variability and associations between air temperature and precipitation and other variables. They are also able to improve existing methods locally. Findings illustrate the practical advantages of copulas to describe multivariate dependencies, to define several predictors and to assess uncertainties.
 ECMWF: the European Centre for Medium-range Weather Forecasts