  The Upper Indus Basin, Himalayas provides water for 270 million people and
countless ecosystems. However, precipitation, a key component to hydrological
modelling, is poorly understood in this area. A key challenge surrounding this
uncertainty comes from the complex spatial-temporal distribution of
precipitation across the basin. In this work we propose Gaussian processes with
structured non-stationary kernels to model precipitation patterns in the UIB.
Previous attempts to quantify or model precipitation in the Hindu Kush
Karakoram Himalayan region have often been qualitative or include crude
assumptions and simplifications which cannot be resolved at lower resolutions.
This body of research also provides little to no error propagation. We account
for the spatial variation in precipitation with a non-stationary Gibbs kernel
parameterised with an input dependent lengthscale. This allows the posterior
function samples to adapt to the varying precipitation patterns inherent in the
distinct underlying topography of the Indus region. The input dependent
lengthscale is governed by a latent Gaussian process with a stationary
squared-exponential kernel to allow the function level hyperparameters to vary
smoothly. In ablation experiments we motivate each component of the proposed
kernel by demonstrating its ability to model the spatial covariance, temporal
structure and joint spatio-temporal reconstruction. We benchmark our model with
a stationary Gaussian process and a Deep Gaussian processes.
