Spear-man’s rank correlation incorporating different classes of inter-gauge distances and seasons is clearly seen in Figure 5.13. In general, there is a substantial increase in the magnitude of the spatial correlation with temporal aggregations. However, increasing this correlation trend is also followed by increasing its uncertainty on the whole. The number of outliers also increases substantially in accordance to the rise of temporal scales. Most outliers occur below the lower boundary and are more concentrated within the higher distance groups.
Effect of anisotropic assumption Figure 5.14 shows iso-correlation lines in the two-dimensional spaces of inter-gauge distances using the pairwise approach in Bavaria and Baden-W ¨urttemberg in order to detect the anisotropic spatial correlation. Within a radius of 60 km, the direction of anisotropic spatial correlations can be detected from South-West to North-East at the daily scales for all seasons and both regions except for the season of DJF in Bavaria where it seems to be the opposite way. Increasing time scales yield the pattern of iso-tropical spatial correlation for both regions within different seasons except for the season of DJF in Baden-W ¨urttemberg.
5.5 Summary
• Most precipitation amounts in Bavaria exhibit a greater spatial rank correlation than those occurring in Baden-W ¨urttemberg and the spatial rank correlation functions for Baden-W ¨urttemberg are higher than the ones for Singapore. The spatial correlation in Singapore is very low indicating that the high spatial variability exists caused by local convective precipitation mainly driven by vertical processes. In contrast, in South-ern Germany, there is a factor of the large scale frontal system from the atmospheric variables influencing the local variable of precipitation mainly driven by horizontal processes.
• The spatial rank correlations calculated using the pairwise approach with simultane-ous temporal occurrences exhibit a systematically higher value than using the geosta-tistical approach. This empirical evidence is consistent with different climate regions, in Singapore and the South of Germany. The reason for this is mainly caused by the driving processes with large spatial extent where the geostatistical approach assumes spatial stationary over the spatial domain of interest and therefore neglects the influ-ence of the mean value.
• Spatial correlations depend on aggregation. The longer the time interval is, the higher the correlations are.
• The concept of geostatistical approach has been widely applied to the precipitation fields for solving the spatial interpolation issues, especially used for the rainfall-runoff processes where both precipitation and runoff require the same time event. The pair-wise approach containing simultaneous temporal observations is useful for infilling missing precipitation observations.
Figure 5.12: The Spearman’s rank correlation functions over distances using the pairwise ap-proach in the regions of Bavaria (BY) and Baden-W ¨urttemberg (BW). The verti-cal axes represent the rank correlation. Panels from top to bottom represent the seasons (DJF, MAM, JJA, and SON) and regions (BY, BW). The horizontal axes represent distances (from 5 to 100 km). Panels from left to right represent the time scale (daily (1d) to monthly (1m).
74 5.5. SUMMARY
Figure 5.13: The Spearman’s rank correlation functions over time scales using the pairwise approach in the regions of Bavaria (BY) and Baden-W ¨urttemberg (BW). The ver-tical axes represent the rank correlation. Panels from top to bottom represent seasons (DJF, MAM, JJA, and SON) and regions (BY, BW). The horizontal axes represent a variety of time scales from daily to monthly. Panels from left to right represent distances from 5 km to 50 km.
Figure 5.14: Isocorrelation lines of daily precipitation in the two-dimensional spaces of dis-tances using the pairwise approach in Bavaria (BY) and Baden-W ¨urttemberg (BW). The vertical axes represent the lag distances in North and South direction in km unit. The horizontal axes stand for the lag distances in East-West direc-tion. Panels from left to right represent seasons (DJF, MAM, JJA, and SON).
Panels from top to bottom represent regions of BY and BW.
dependence using bivariate copulas
6.1 Introduction
Reliable spatial information on precipitation data sets is required for an accurate quantifi-cation of water resources planning and management since much empirical solid evidence shows that precipitation exhibits high spatial variability. Unfortunately, the spatial coverage of available precipitation data sets is frequently inadequate for representing the higher de-gree of spatial variability of precipitation. The low densities of precipitation gauge network due to a small number of stations installed in many parts of the world brings poor quality of spatial precipitation data sets. Hence, a decent spatial interpolation method is necessary in order to estimate ungauged areas.
There are a variety of spatial interpolation models that have been developed for precipi-tation applications over the last a half of century. Many of them are developed based on Gaussian assumption because they are derived from the concept of linear regression. One of the spatial models, which is popularly implemented in precipitation studies is Kriging.
Kriging type models can be classified into stochastic models, which are able to deliver un-certainty of the interpolation quality directly with its variances, as precipitation behavior exhibits a higher uncertainty of spatial distribution of precipitation.
However, Kriging models offer the quality of interpolation, which is just a function of the observation density and the variogram models (B´ardossy,2006). These functions can only describe the spatial dependence as an integral over the whole distribution of the parameter values. Anomalies in the observations and areas with high or low variabilities cannot, there-fore, be considered for estimating the variances (B´ardossy and Li,2008). These models also cannot detect properly the different percentile values, for example, extremes, which might have a different spatial dependence structure from the central values (B´ardossy,2006). This issue is relevant to precipitation fields which are characterized skewed distribution, and variogram functions which can only be dominated by a few anomalously differing pairs, might occur.
Copula based model is a promising tool to overcome those drawbacks. Copulas are also flexible to analysis the marginal distributions separately from the joint distributions and al-low them to be same or different from the distribution family. Furthermore, copulas are capable of expressing dependence structures on quantile scale, which is able to detect both non-linear and tail, or asymmetric dependence (Nelsen,2006). Recently, spatial interpola-tion copula models have been implemented in the precipitainterpola-tion studies among few scientists
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78 6.2. GEOSTATISTICAL APPROACH