# Adaptive measure-theoretic parameter estimation for coastal ocean modeling

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## Abstract

Since Hurricane Katrina (2005), there has been a marked increase in the quantity of field observations gathered during and after hurricanes. There has also been an increased effort to improve our ability to model hurricanes and other coastal ocean phenomena. The majority of death and destruction due to a hurricane is from storm surge. The primary controlling factor in storm surge is the balance between the surface stress due to the wind and bottom stress. Manning's formula can be used to model the bottom stress; the formula includes the Manning's n coefficient which accounts for momentum loss due to bottom roughness and is a spatially dependent field. It is impractical to measure Manning's n over large physical domains. Instead, given a computational storm surge model and a set of model observations, one may formulate and solve an inverse problem to determine probable Manning's n fields using observational data, which in turn can be used for predictive simulations. On land, Manning's n may be inferred from land cover classification maps. We leverage existing land cover classification data to determine the spatial distribution of land cover classifications which we consider certain. These classifications can be used to obtain a parameterized mesoscale representation of the Manning's n field. We seek to estimate the Manning's n coefficients for this parameterized field.

The inverse problem we solve is formulated using a measure-theoretic approach; using the ADvanced CIRCulation model for coastal and estuarine waters as the forward model of storm surge. The uncertainty in observational data is described as a probability measure on the data space. The solution to the inverse problem is a non-parametric probability measure on the parameter space. The goal is to use this solution in order to measure the probability of arbitrary events in the parameter space. In the cases studied here the dimension of the data space is smaller than the dimension of the parameter space. Thus, the inverse of a fixed datum is generally a set of values in parameter space. The advantage of using the measure-theoretic approach is that it preserves the geometric relation between the data space and the parameter space within the probability measure. Solving an inverse problem often involves the exploration of a high-dimensional parameter space requiring numerous expensive forward model solves. We use adaptive algorithms for solving the stochastic inverse problem to reduce error in the estimated probability of implicitly defined parameter events while minimizing the number of forward model solves.