## Philip White and Shane Reese, Statistics

### Introduction

Antarctica’s significance to the global climate is due to the vast amounts of water stored in its ice sheet. Indeed, its ice sheet stores enough water to increase the global sea level by about 200 feet if it were to melt. Even though radical climate change could not melt the entire Antarctic ice sheet for thousands of years, smaller, more realistic changes would still make a significant impact in the global climate, sea level, and growing seasons. For this reason, climatologists and geologists model water accumulation and loss over the Antarctic ice sheet. Accumulation, as defined here, is the net precipitation, sublimation, melt, and wind redistribution of snow.

### Methodology

We construct a Markov random field expressed by Gaussian spatial process (GaSP) models to provide an accurate representation of accumulation as a function of spatial arrangement of data collection sites. Moreover, accurate prediction and associated uncertainties in accumulation over the Antarctic ice sheet can be estimated using GaSP. A GaSP model is a stochastic process for which any finite collection of random variables from the process jointly have a multivariate Gaussian distribution.

In the case of Antarctic accumulation modeling we build the covariance function on latitude, longitude, and elevation. In particular, we will model our covariance with an isotropic covariance function so that our covariance function *k* is a function of *d* where *d = *|*x - xʹ*| and *x, xʹ* are two locations of interest. We choose the Mat´ern covariance function,

due to computational stability and because it provides a general covariance function.

To account for variability between nearby measurements, we will include a spatial nugget δ_{s} in our model. The spatial nugget accounts for the base (minimum) autocorrelation of the spatial process. Since we make no assumption that σ^{2}, ρ, δ_{s}, ω_{d}, and ω_{e} are known, we will set prior distributions on these parameters and use Markov chain Monte Carlo (MCMC) to solve for them. Using the solutions for σ^{2}, ρ, δ_{s}, ω_{d}, and ω_{e}, we predict accumulation at all prediction locations using the posterior predictive distribution

To propose new measurement locations, we have used an iterative selection process where we considered two criteria: the scaled model uncertainty *V* at grid locations and the minimum scaled distance of prediction locations to proposals *d _{i}*. Specifically, for constants

*c*and

_{1}*c*, we select the prediction locations with maximum values for an

_{2}This design uncertainty metric

*U*accounts for the distance between other proposals and the model uncertainty. We will solve for

*c*and

_{1}*c*by constraining

_{2}*c*+

_{1}*c*= 1 and selecting the combination of

_{2}*c*and

_{1}*c*which minimize

_{2}*U*.

### Results

Using the posterior draws from the Metropolis-Hastings algorithm, we use the posterior predictive distribution to predict SMB over the Antarctic ice sheet (see Table 1 and Figure 1). For measurement proposals, we find that when *c _{1}* = 0:5 and

*c*= 0:5, we minimize

_{2}*U*. The 15 highest priority proposals are plotted in Figure 2.

Average 95% Credible Interval Margin for SMB predictions: 114.66 mm ∙ w.e. ∙ yr

^{-1}

Table 1: Total and average SMB estimates and 95% credible interval results. Note that C.I. signifies a 95 % credible interval.

### Discussion

Our predicted average SMB, 158 mm ∙ w.e. ∙ yr^{-1}, is lower than previously estimated values by Vaughan et al., 169 mm ∙ w.e. ∙ yr^{-1}, and Van de Berg et al., 186 mm ∙ w.e. ∙ yr^{-1}, suggesting less water accumulation than previously thought. Specifically, Vaughan et al.’s prediction is 7% higher than our estimate and Van de Berg et al.’s estimate is 18% larger (Van de Berg et al. 2006, Vaughan et al. 1999). However, Vaughan et al.’s estimate is within our prediction error bounds; thus, we cannot conclude that our prediction varies significantly from this estimate.

### Conclusion

Because our results are lower than others, it provides scientists with important areas for glaciological and climatological research. We will continue to incorporate new data to improve this model. In the future, we will also analyze how SMB is changing over time. In analyzing how SMB is changing over time, we will need to develop a new model that integrates time into our spatial model. This spatio-temporal model will enable us to analyze how water accumulation has changed over time.