A discrete view of the Indian monsoon to identify spatial patterns of rainfall

Abstract

We propose a representation of the Indian summer monsoon rainfall in terms of a probabilistic model based on a Markov Random Field, consisting of discrete state variables representing low and high rainfall at grid-scale and daily rainfall patterns across space and in time. These discrete states are conditioned on observed daily gridded rainfall data from the period 2000-2007. The model gives us a set of 10 spatial patterns of daily monsoon rainfall over India, which are robust over a range of user-chosen parameters as well as coherent in space and time. Each day in the monsoon season is assigned precisely one of the spatial patterns, that approximates the spatial distribution of rainfall on that day. Such approximations are quite accurate for nearly 95% of the days. Remarkably, these patterns are representative (with similar accuracy) of the monsoon seasons from 1901 to 2000 as well. Finally, we compare the proposed model with alternative approaches to extract spatial patterns of rainfall, using empirical orthogonal functions as well as clustering algorithms such as K-means and spectral clustering.

Spatio-temporal patterns of daily Indian summer monsoon rainfall

“…However for the model to be robust, it should be possible to approximate x(·, t) and Z(·, t) from outside this period using the same set of patterns. In the companion paper [9] (Section 3.2), we have already established that our patterns achieve this, using the 2 distance to compare daily rainfall data for all spatial locations to CRPs {φ u } K u=1 , and Hamming distance to compare Z(·, t) with the CDPs {φ d u } K u=1 . In this part, we examine how such fitting of patterns varies from year to year.…”

“…A day belonging to an active cluster may be considered an active day, and a day belonging to a break cluster may be called a break day. This is realistic due to the observation in the companion paper [9] (Table 3) that these clusters exhibit homogeneity with respect to daily aggregate rainfall, as they have low intra-cluster variation. Let us denote this new set of active/break days as ACT 1 and BRK 1 , respectively, and the corresponding runs of 3 or more such days as the active and break spells, respectively.…”

“…The last two lines account for the relationship between the variables Z, U , V and X, via the edge potentials ψ ST , ψ SS , ψ DZ and ψ DU , over the edges appearing in the graphical representation in Figure 1. For complete details about the functional form of these edge potentials, we refer the reader to the accompanying article [9]. We note that the spatio-temporal patterns we analyze here are obtained as a function of the latent variables (U, V, Z) given the data x.…”

“…and in Figure 3 we plot µ k for each of the 24 clusters obtained by setting ζ = 9 (see Table 1 of [9]), and also for the 10 prominent clusters among them. Notably, these patterns have been sorted in ascending order of daily aggregate rainfall.…”

“…As in the case of daily clusters (Section 3.1), we evaluate the temporal clustering using intracluster similarity. We use the same baselines as those used to evaluate the clusters in the companion paper [9]: K-Means [6] where the real-valued rainfall time-series x(s, ·) are clustered; Spectral Clustering [11] based on Euclidean distance between the rainfall time-series (Spect1); and Spectral Clustering based on Hamming Distance (Spect2) between discretized time-series Z(s, ·). In case of Spect2, this discretization is done by comparing local daily rainfall X(s, t) with local daily mean µ s .…”