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Uncertainty Quantification In Geosciences with Computationally Expensive Simulation Models (with last minute modification to Include Sustainability/Energy analysis. Christine A. Shoemaker School of Civil and Environmental Engineering and School of Operations Research and Information Engineering Cornell University

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**Uncertainty Quantification with Computationally Expensive Simulation Models**

Many problems in geoscience and environmental engineering are described by computationally expensive simulation models. Probably the main obstacle to doing rigorous statistical analysis of uncertainty for a given data set and model is computational time because: Simulation models typically need to be run many (e.g. thousands of ) times for uncertainty quantification Multiple types of uncertainty need to be incorporated including data error, model error, parameter error, randomness in model input (static and dynamic) I’m interested in developing new algorithms for this problem.

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**Examples of Static Geoscience Problems Requiring Uncertainty Quantification**

Examples of Goals: Determine the spatial distribution of different types of geologic materials in the subsurface Determine location of oil reservoirs or underground water Data sources: based on sound waves or radar (many spatial points but not highly accurate). Based on drilling into the subsurface (more accurate but very few spatial points because because of high cost of drilling each well.

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**Examples of Uncertainty Quantification Needs for Dynamic Geoscience Problems**

Examples of problems include: Forecasting man-made changes in fluid flow in geologic formations with effects on floods and contaminant transport Predicting the impact of Greenhouse gases on climate change over time. Forecasting dynamic response of multiple fluids in the subsurface to waste disposal (e.g. carbon sequestration)

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**Quantifying Uncertainty from Dynamic Model Predictions with Model Parameters based on Data**

Measured (Weather) Dynamic Input Predicted (Weather) Input-perhaps many scenarios to represent randomness Calibration Time Periods With Measured Static Input and Dynamic Output (flow, contamination, etc.) Forecasting Time Period Using Estimates of Static Input TIME Data is taken during Calibration time period and it can be used to establish parameter values (as deterministic values or parameters as random variables with pdf’s estimated based on data). We then want to forecast model output in the future using a probabilistic representation of model parameters and input. This forecast then has uncertainty including static and dynamic inputs, model error, and parameter error

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**Excess Phosphorous from Watershed would result in cost of **

How Do We Protect This Water From Pollution? This is New York City (NYC) water supply. Excess Phosphorous from Watershed would result in cost of $US 8 Billion Water Treatment Plant for NYC!

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**Study Area Cannonsville Reservoir Basin – agricultural basin**

New York City water supply P ‘restriction’ impedes economic growth of county Model incorporates over 20,000 data values available for this watershed. note eutrophication problem no filtration of drinking water here means that eutrophic reservoir causes taps to be shut off Filtration plant for this Basin is on the order of 8 billion dollars so avoidance is a priority!

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**Cow Waste (Manure) Is the Primary Source of Phosphorous Transported By Water to Lake**

Cow Waste (Manure) is Primary Source of Phosphorous Reaching River and Lake

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**SWAT2000: A Spatially Distributed Simulation Model with Subbasins, Land Use and Soil Type in “HRUs”**

Using a spatially distributed model helps us evaluate management options. Different than GWLF 43 subbasins 758 HRUs Avg HRU = 1.6 km2 SWAT model developed by USDA and is used worldwide.

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Simulation Model Predictions of Annual Phosphorous Contamination: Repeating Climates for 72 years By repeating 12 year blocks of climate, the trends in climate inputs are eliminated P load to reservoir simulated to increase over time! Increase is measured relative to first climate block = ( )/34297*100% = 11.4% increase in P load to reservoir over 72 yrs Data points in chart are the average of 12 years of total P loading. This is how future P loading trends will be assessed for other P management scenarios. *** In the absence of trends in climate, model predicts P loading to increase over time! Sept 12, 2003 10

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**Significance of Results on Phosphorous in Watershed**

The forecast of increase in phosphorous loading to the water (with no increase in human or animal activity) is very serious because: NY City’s drinking water quality will decline (and there might not be a replacement) Future cost could be Billions of dollars Steps should be taken now to stop the increase in phosphorous pollution Hence the rate of increase (11%/72 years) is important and we would like to know the uncertainty associated with it.

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**Sustainability/Energy Issues**

Development of renewable energy devices –fuel cells, photovoltaics, biofuels, etc. Wind Energy also has some of the same issues as the watershed analysis—you have a calibration period, but the real uncertainty is during the forecast period and is driven by variability in weather inputs (and fear of disasters). My group has started to work on Carbon Sequestration which is important in waste disposal and has some similarity to the problems that arise in nuclear waste disposal.

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**Carbon Sequestration Another Application where Uncertainty Quantification is Important**

Carbon sequestration: storage of super critical carbon dioxide in geological formations about 1 kilometer below the surface.

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**Uncertainty Quantification Importance for Carbon Sequestration**

The issue is that there are serious public health and environmental risks associated with the unexpected movement of CO2 upward to groundwater or to the surface. We would like to use a model plus monitoring data to generate an estimate of the CO2 plume in the ground to determine if the system is functioning OK and where the plume is going to move in the future.

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Related Issues How do we eﬃciently quantify tail probabilities/rare events of high-impact? Can we augment the predictive ability by using experiments? How? Can we derive predictive reduced-order models and/or response surfaces? Likelihoods can be multimodal and/or rough.

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Most Nonlinear Simulation Models will Lead to Likelihood Models with Multiple Local Minima Multi-Modal Problems have Multiple local minima F(x) Local minimum Global minimum X (parameter value)

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Likelihood can be a Rough Surface because of Numerical Simulation Objective (likelihood) Function Versus Parameter Value

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**Our Approach to Uncertainty Quantification**

I have been working with statistician David Ruppert and others on quantification of uncertainty for computationally expensive simulation models. My approach has been to imbed both optimization search and response surfaces into the algorithms for uncertainty quantification to significantly reduce computational effort (by orders of magnitude). Our methods are designed to work with multi modal functions and rough surfaces.

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Paper 1 on SOARS Blizniouk, N., D. Ruppert, C.A. Shoemaker, R. G. Regis, S. Wild, P. Mugunthan, “Bayesian Calibration of Computationally Expensive Models Using Optimization and Radial Basis Function Approximation.” Journal of Computational and Graphical Statistics, July 2008.

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**Notation Y0 - vector of observed data;**

η - parameters in the (joint) statistical model [list1|list2] - conditional density of random variables in list1 given list2, e.g. Is the conditional density of η given the data Y0 η= b in this example

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Kernel estimates of the marginal posterior densities by a) (solid line) exact joint posterior obtained from conventional MCMC .Analysis with 10,000 function evaluations & b) (dashed lines) with our function approximation method with 150 function evaluations. One graph for each parameter.

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**OUTPUT UNCERTAINTY: Again we got excellent agreement between **

our approach with 150 evaluations and the conventional approach with 10,000 evaluations. Output Comparison

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**SWAT2000: A Spatially Distributed Simulation Model with Subbasins, Land Use and Soil Type in “HRUs”**

Using a spatially distributed model helps us evaluate management options. Different than GWLF 43 subbasins 758 HRUs Avg HRU = 1.6 km2 SWAT model developed by USDA and is used worldwide.

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**Marginal posterior Densities of the 5 parameters (βi) for watershed model**

This contains all the statistical information . Everything else is computed on the basis of this.

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**Joint Posterior Density of Six Different Model Outputs**

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**Quantiles, Means, Standard Dev.**

These are based on statistically rigorous analysis with transformations to account for non normal data obtained with a small fraction of the number of simulations required by other methods including MCMC and GLUE

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Questions?

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**Groundwater Simulation Model for Beijing**

Figure 1. Miyun-Huai-Shun groundwater aquifer with hydraulic head observation wells of the study area. The main aquifer covers 456 km2 of the basin area. This is primary water supply for Beijing. Upper layers of the Aquifer are contaminated, thereby significantly reducing Beijing’s water supply. Parameter estimation goal in order to understand effect of extracting water and potential for contamination. Model involves solving PDE’s.

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Groundwater Aquifer is 3-dimensional. Below is a vertical cross section. Cross Hatched areas indicate different “conductivities”, which need to be estimated as parameters. Simulation model solves a systems of partial differential equations by finite difference method.

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