Rain gauges are considered the most accurate method to estimate rainfall and are used as the “ground truth” for a wide variety of applications. The spatial density of rain gauges varies substantially and hence influences the accuracy of gridded gauge-based rainfall products. The temporal changes in rain gauge density over a region introduce considerable biases in the historical trends in mean rainfall and its extremes. An estimate of uncertainty in gauge-based rainfall estimates associated with the nonuniform layout and placement pattern of the rain gauge network is vital for national decisions and policy planning in India, which considers a rather tight threshold of rainfall anomaly. This study examines uncertainty in the estimation of monthly mean monsoon rainfall due to variations in gauge density across India. Since not all rain gauges provide measurements perpetually, we consider the ensemble uncertainty in spatial average estimation owing to randomly leaving out rain gauges from the estimate. A recently developed theoretical model shows that the uncertainty in the spatially averaged rainfall is directly proportional to the spatial standard deviation and inversely proportional to the square root of the total number of available gauges. On this basis, a new parameter called the “averaging error factor” has been proposed that identifies the regions with large ensemble uncertainties. Comparison of the theoretical model with Monte Carlo simulations at a monthly time scale using rain gauge observations shows good agreement with each other at all-India and subregional scales. The uncertainty in monthly mean rainfall estimates due to omission of rain gauges is largest for northeast India (~4% uncertainty for omission of 10% gauges) and smallest for central India. Estimates of spatial average rainfall should always be accompanied by a measure of uncertainty, and this paper provides such a measure for gauge-based monthly rainfall estimates. This study can be further extended to determine the minimum number of rain gauges necessary for any given region to estimate rainfall at a certain level of uncertainty.
monsoon over India. The default model with mean parameter estimates does not contain a bifurcation, but the model admits bifurcation as parameters are varied.
We consider statistics of spatial averages estimated by weighting observations over an arbitrary spatial domain using identical and independent measuring devices, and derive an account of bias and variance in the presence of missing observations. We test the model relative to simulations, and the approximations for bias and variance with missing data are shown to compare well even when the probability of missing data is large. Previous authors have examined optimal averaging strategies for minimizing bias, variance and mean squared error of the spatial average, and we extend the analysis to the case of missing observations. Minimizing variance mainly requires higher weights where local variance and covariance is small, whereas minimizing bias requires higher weights where the field is closer to the true spatial average. Missing data increases variance and contributes to bias, and reducing both effects involves emphasizing locations with mean value nearer to the spatial average. The framework is applied to study spatially averaged rainfall over India. We use our model to estimate standard error in all-India rainfall as the combined effect of measurement uncertainty and bias, when weights are chosen so as to yield minimum mean squared error.
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