# Bayesian Methods for Discontinuity Detection in Climate Model Predictions

Khachik Sargsyan, Cosmin Safta, Bert Debusschere, Habib Najm

### Abstract

Discontinuity detection is an important component of many fields, e.g. climate change research, image recognition, digital signal processing. However, current methods of discontinuity or edge detection are often (a) restricted to one- or two-dimensional setting, (b) require a uniformly spaced, and typically quite dense data collection, and (c) lead to a deterministic, fixed answer without quantifying the confidence in the result. However, in predictive model analysis, response surface methods for surrogate modeling are strongly challenged by nonlinear or discontinuous output data structure, since global, smooth-basis methods could require prohibitively high order expansions as well as exhibit Gibbs phenomena. While domain refinement methods can reduce the impact of nonlinearities and jumps, they often require prohibitively expensive sampling in each subdomain.

We propose a probabilistic, Bayesian framework of discontinuity detection that parameterizes and infers the discontinuity location together with the associated uncertainties. Namely, adaptive Markov Chain Monte Carlo sampling is used to obtain joint distributions for the discontinuity location parameters, effectively leading to a distribution over all possible discontinuity curves. The methodology can be generalized to multiple dimensions and is robust with respect to limited and arbitrarily distributed data. This technique then allows uncertainty propagation from input parameters to the output distributions using spectral methods for each side of the discontinuity. Moreover, since the Bayesian approach leads to the full posterior distribution of the discontinuity location, the expectation of spectral expansions with respect to this distribution leads to a compact representation of the output of the interest, thus effectively accomplishing the uncertainty propagation.

We will illustrate our methodology on the example of the Meridional Overturning Circulation (MOC) in the Atlantic Ocean. It is known that the maximum stream function of the MOC exhibits discontinuity across a curve in the space of two uncertain parameters: the climate sensitivity and the CO2 forcing rate. The proposed methods prove very efficient in this context compared to other discontinuity detection algorithms, since uncertainty quantification in climate models is challenged by the sparsity of the available climate data due to the high computational cost of the model runs.