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\title{Stochastic space-time modeling for agricultural decision support in the Argentine Pampas}

\author{Doctoral Dissertation Defense \\~\\ by \\
	Andrew Verdin\\~\\
		{\it Dept of Civil, Environmental, and Architectural Engineering\\University of Colorado at Boulder}\\~\\
	Advised by Prof. Balaji Rajagopalan}


\date{\today}

\titlepage

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{\footnotesize \tableofcontents }

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\section{\scshape Overview}
\subsection{Motivation}
\begin{frame}{Motivation}
``It is arguable that the artificiality of agricultural production systems make them less flexible, and therefore more vulnerable to climatic change than the naturally occurring species of the ecosystem within which they fit, and that the more unstable the climate the greater this vulnerability is likely to be.'' \\ \emph{~~~ - Oram (1985)}%.. In an undisturbed state there is no pressure on an ecosystem to deliver a product within a given period; in farming it may literally be `produce or perish'.'' \\ \emph{~~~ - Oram (1985)}
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\begin{frame}{The Pampas}
\begin{columns}[t]
\begin{column}[T]{5cm}
\begin{center}
\includegraphics[width=0.69\textwidth]{figures/PAMPAS2.png} \\
%\includegraphics[width=0.95\textwidth]{figures/santa-rosa.pdf}
\includegraphics[width=0.999\textwidth]{figures/Arg-gauchos-1-629x418.jpg}
\end{center}
\end{column}
\begin{column}[T]{5cm}
\begin{center}
\includegraphics[width=0.95\textwidth]{figures/IMG_4074.JPG} \\
\includegraphics[width=0.95\textwidth]{figures/water-table.png}
\end{center}
\end{column}
\end{columns}

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\begin{frame}{Pampas climate \& land use change}
Alternating wet and dry epochs:
\begin{columns}[t]
\begin{column}[T]{6cm}
	
	{\small \begin{itemize} 
		\item Early 1900s floods.
		\item 1930-50 drought.
%		\begin{itemize}
%			\item ``Dust bowl.'' 
%		\end{itemize}
		\item Increasing precipitation 1960-2000.
		%\begin{itemize}
			\item[] $\rightarrow$ Extremes \& frequency.
		%\end{itemize}
		\end{itemize} }
\begin{center}
\includegraphics[width=0.99\textwidth]{figures/trend-pcp.pdf} 
\end{center}

\end{column}
\begin{column}[T]{6cm}
	
	{\small 
			\begin{itemize}
			\item Agricultural expansion.
			\item[] $\rightarrow$ Reduced annual ET.
%				\begin{itemize}	
%					\item Fallow land after harvest.
%				\end{itemize}
			\item[] $\rightarrow$ Rising water table.
%				\begin{itemize}
%					\item Rainfall increases \& land use change.
%				\end{itemize}		
			\item 1997-2003 flooding.
			\item 2008 drought.
		\end{itemize} }
\begin{center}
\includegraphics[width=0.99\textwidth]{figures/dry-salado.jpg}
\end{center}
\end{column}
\end{columns}

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\begin{frame}{Land use change}
\begin{center}
\includegraphics[width=0.99\textwidth]{figures/landUse-change-pasturas-soja1.pdf} \\
$\Delta$ = (1981-2013) -- (1961-1980)
\end{center}
\end{frame}

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\begin{frame}{Crop yield mapping}
\begin{center}
\includegraphics[width=0.80\textwidth]{figures/WT-Crops.jpg} \\
WTD has been shown to affect crop yields.
\end{center}
\end{frame}

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\begin{frame}{Crop yield mapping}

{\tiny\it Nosetto et al. [2009]}
\begin{center}
\includegraphics[width=0.8\textwidth]{figures/Nosetto-et-al-raw2.pdf} %\hspace{0.05\textwidth}
%\includegraphics[width=0.32\textwidth]{figures/Nosetto-et-al-relative.pdf}
\end{center}
\end{frame}

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\begin{frame}{Agricultural planning in the Pampas}

\begin{itemize}
	\item Crop simulation models (DSSAT).
	\item Analyze different management strategies.
	\item Correct crop yields with WTD  (MIKE-SHE).
	\item[] {\small $\rightarrow$ MIKE-SHE has great computational costs. (5km x 5km, daily)}
	\item[]
	\item[] \ldots the need for data.
	\begin{itemize}
		\item Observed record limited.
		\item Variety of scenarios.
		\item Specific trends; \underline{traits of seasonal forecast.}
		\item At \underline{unobserved locations.}
		\item[]
	\end{itemize}
	\item[$\rightarrow$] {\bf Stochastic weather generation!}
\end{itemize}

\begin{center}
\includegraphics[width=0.75\textwidth]{figures/first-flowchart.png}
\end{center}
\end{frame}

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\subsection{Background}
\begin{frame}{Single site weather generators}

Parametric:
\begin{itemize}
	\item[] {\footnotesize {\it [Richardson, 1981; Richardson \& Wright, 1984; Semenov \& Barrow, 1997]} }
	\item {\small Precipitation occurrence modeled as Markov chain. {\footnotesize\it [Katz, 1977]}}
	\item {\small Precipitation amount modeled as Gamma random variable.}
	\item {\small Temperatures modeled using autoregressive time  series.}
	\item {\small\it Fit separate models for each month (or finer).}
	\item {\small\it Captures climatology and linear relationships; fails to capture extremes.}
\end{itemize}

~\\ Nonparametric: \\

\begin{itemize}
	\item[] {\footnotesize {\it [Lall \& Sharma, 1996; Rajagopalan \& Lall, 1999; Harrold et al., 2003]} }
	\item {\small Kernel density estimators of precipitation.}
	\item {\small K-nearest neighbor time series bootstrap.}
	\item {\it\small Non-normal features captured.}
	\item {\it\small Resamples historic record, not simulating unobserved values.}
	
\end{itemize}
\end{frame}

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\begin{frame}{Multiple site weather generators}

Parametric:
\begin{itemize}
	\item[] {\footnotesize {\it [Wilks, 1998, 1999; Qian et al., 2002; Baigorria \& Jones, 2010; Khalili et al., 2009]} }
	\item {\small Climate variables modeled consistent with single site.}
	\item {\small Estimate parameters for each variable at each location.}
	\item {\it\small Unwieldy to preserve temporal statistics and spatial dependence.}
	\item {\it\small Difficult to simulate at \underline{unobserved} locations, or large network of stations.}
\end{itemize}

~\\ Nonparametric: \\

\begin{itemize}
	\item[] {\footnotesize {\it [Yates et al., 2003; Apipattanavis et al., 2007; Sharif \& Burn, 2007]} }
	\item {\small Resample observed vector of variables.}
	\item {\small Preserve spatial dependence and climatological statistics.}
	\item {\it\small Due to resampling, not simulating unobserved values.}
	\item {\it\small Difficult to simulate at \underline{unobserved} locations, or large network of stations.}
\end{itemize}
\end{frame}

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\begin{frame}{Generalized linear models}

Generalized linear models:
\begin{itemize}
	\item[] {\footnotesize {\it [Stern \& Coe, 1984; Chandler, 2005; Furrer \& Katz, 2007; Kleiber et al., 2012, 2013]} }
	\item[] Model and simulate skewed or discrete variables.
	\item[] $Y \sim f(\mu, \sigma)$
	\item[] {\small $f(.) \rightarrow$ binomial, Gamma, poisson, gaussian.}
	\item[] $\eta(\mu) = \bX'\bbeta$
	\item[] {\small $\eta(\mu) \rightarrow$ logit, probit, inverse, log, identity.}
%	\item[] $\rightarrow \bY = \bX'\bbeta + \bepsilon$
	\item[]
	\item Time series models at each location.
%	\item[] $\rightarrow$  No need for Markov chains (each location, each month).
	\item Covariates capture seasonality, temporal dependence.
	\item[] $\rightarrow$ e.g., sine, cosine, autoregressive.%wet/dry, hot/cold.
%	\item Spatial models on GLM coefficients (i.e., $\bbeta$).
%	\item Residuals as spatial processes. 
%	\item[] $\rightarrow$ e.g., monthly.
	\item[]
	\item GLM coefficients (i.e., $\beta_i$) are fixed.
	\item[] $\rightarrow$ Uncertainty is neglected.
	\item[] $\rightarrow$ Variability from residuals process.
\end{itemize}
\end{frame}

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%\begin{frame}{Coupled framework}

%Weather generator coupled with DSSAT, MIKE-SHE. \\~\\
%DSSAT offers lumped modeling (fast running). \\~\\
%MIKE-SHE offers distributed modeling (computationally intensive). \\~\\
%Generally only seek the WTD for season of interest. \\~\\
%{\it Ways to reduce computational intensity?}
	
%\includegraphics[width=0.95\textwidth]{figures/first-flowchart.png}
%\end{frame}

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%\begin{frame}{Limitations}
%	\begin{itemize}
%		\fail Simulating at unobserved locations.\\~\\ \pause
%		\fail Seasonal forecasts uninformative to decision makers.\\~\\ \pause
%		\fail GLM weather generators neglect uncertainty.\\~\\ \pause
		%$\rightarrow$ Not exploring full range of future conditions.\\~\\ \pause
%		\fail Distributed hydrologic models have great costs.
%	\end{itemize}
%\end{frame}

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\begin{frame}{Research objectives}
	\begin{itemize}
		\item[1.] { Develop a weather generator for simulating at unobserved locations.}
			\begin{itemize}
				\item Space-time GLM weather generator.\\~\\
			\end{itemize}
		\item[2.] { Generate space-time weather scenarios consistent with seasonal forecasts and multidecadal trends, for resources planning and management.}
			\begin{itemize}
				\item Modification of GLM weather generator.\\~\\
			\end{itemize}
		\item[3.] { A generalized weather generator that can model and propagate uncertainty.}
			\begin{itemize}
				\item Bayesian space-time weather generator.\\~\\
			\end{itemize}
		\item[4.] { An efficient water table depth model for seasonal agricultural planning.}
			\begin{itemize}
				\item Hydrologic model emulator (i.e., ``w'').\\~\\
			\end{itemize}
	\end{itemize}
	
\end{frame}

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\section{\scshape Chapter 2}
\subsection{Coupled stochastic weather generation using spatial and generalized linear models}
\begin{frame}{Chapter 2}
\begin{itemize}
{\bf \item[1.] { Develop a weather generator for simulating at unobserved locations.}}\\~\\
\end{itemize}
%GLM and spatial models for stochastic weather generation.\\~\\
%Address the issue of simulating daily weather at {\it unobserved locations}.\\~\\
Fit GLMs at each location, obtain $\bbeta(\bs)$. \\~\\
$\rightarrow$ {\small Captures temporal variability.}
\\~\\
Spatial models to estimate $\hat{\bbeta}(\bs)$ coefficients at unobserved locations.\\~\\
$\rightarrow$ {\small Captures spatial structure.}
\\~\\
Exponential covariance functions defined by residuals.

\end{frame}

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\begin{frame}{Precipitation models}
%Occurrence and amounts modeled separately. \\~\\  
Occurrence $O(\bs,t) = \indicator_{[W_O(\bs,t)\ge 0]}$ modeled using probit regression. \\~\\
Latent Gaussian process $W_O(\bs,t)=\bX_O(\bs,t)'\bbeta_O(\bs)$ is linear regression. \\
By definition, probit residual variance $\sigma_O^2=1$\\~\\
If $W_O(\bs,t) \geq 0$, $O(\bs,t) = 1$ \\
Else, ~~~~~~~~~~~~~~~~~~$O(\bs,t) = 0$ \\
$\bX_O(\bs,t) = (1, O(\bs,t-1), \text{cos}(2\pi t/365), \text{sin}(2\pi t/365))$\\~\\


Amounts $A(\bs,t)$ modeled as Gamma$(\alpha_A(\bs), \alpha_A(\bs)/\mu_A(\bs,t))$.\\~\\
%$A(\bs,t) \sim Gamma(\alpha_A(\bs), \alpha_A(\bs)/\mu_A(\bs,t))$\\~\\
Gamma shape $\alpha_A(\bs)$ varies in space.\\
Gamma scale $\alpha_A(\bs)/\mu_A(\bs,t)$ varies in space and time.\\~\\
$\mu_A(\bs,t) = \text{exp}(\bX_A(\bs,t)'\bbeta_A(\bs))$\\
$\bX_A(\bs,t) = (1, \text{cos}(2\pi t/365), \text{sin}(2\pi t/365))$



%\begin{flalign*}
%&A(\bs,t) \sim Gamma(\alpha_A(\bs), \alpha_A(\bs)/\mu_A(\bs,t))& \\~ \nonumber \\
%&\mu_A(\bs,t) = \text{exp}(\bX_A(\bs,t)'\bbeta_A)& \\~ \nonumber \\
%&\bX_A = (1, \text{cos}(2\pi t/365), \text{sin}(2\pi t/365))&
%\end{flalign*}

\end{frame}

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\begin{frame}{Model structure}
Temperatures modeled using linear regression:
\begin{flalign*}
&Z_N(\bs,t) = \bX_N(\bs,t)'\bbeta_N(\bs) + W_N(\bs,t)& \\~ \nonumber \\
&Z_X(\bs,t) = \bX_X(\bs,t)'\bbeta_X(\bs) + W_X(\bs,t)& \\~ \nonumber \\ 
&\bX_i(\bs,t) = (1, Z_N(\bs,t-1), Z_X(\bs,t-1),\text{cos}(2\pi t/365),& \nonumber \\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{sin}(2\pi t/365), r(t), O(\bs,t))~~~~~~~~~~~~~\text{for } i=N,X& \\
%&W_i(\bX,t) \sim GP(0, C_i(t))~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{for } i=N,X&
\end{flalign*}

The mean $\bX_i(\bs,t)'\bbeta_i(\bs)$ is climate.\\~\\
Residuals $W_i(\bs,t)$ is like daily weather.

\end{frame}

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\begin{frame}{Parameter estimation \& simulation}
Model fitting:
\begin{itemize}
	\item[1.] Four GLMs are fitted at each location. 
	\item[2.] Maximum likelihood estimates (MLE) of $\hat{\bbeta}_i(\bs)$ at each location.
	\item[3.] Spatial models on $\hat{\bbeta}_i(\bs)$ to simulate at any location.
	\item[4.] Exponential covariance functions defined from monthly residuals.\\~\\ \pause
\end{itemize}
Simulation:
\begin{itemize}
	\item[1.] Simulate occurrence. 
	\item[2.] Where occurrence is positive, simulate amounts.
	\item[3.] Update temperature covariates using simulated occurrence.
	\item[4.] Simulate temperatures.
	\item[5.] Repeat for all days to simulate.
\end{itemize}


\end{frame}

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\begin{frame}{Study region \& data}
\begin{center}
\includegraphics[width=0.995\textwidth]{figures/ch2-studyregion.pdf} \\
1908--2010 daily precipitation, maximum and minimum temperature.
\end{center}
\end{frame}

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\begin{frame}{Simulation \& validation}

\begin{center}
100 simulations: 1 Jan 1908 - 31 Dec 2010. \hfill Cross-validation\\
\end{center}

\includegraphics[width=0.74\textwidth]{figures/climatology-Junin-GLMGEN.pdf}
\includegraphics[width=0.25\textwidth]{figures/Figure3-take6.jpg}
%\begin{columns}[t]
%\begin{column}[T]{5cm}
%\begin{center}
%{\small Santiago del Estero }
%\includegraphics[width=0.85\textwidth]{figures/wet-dry-hot-cold2.png} 
%\end{center}
%\end{column}
%\begin{column}[T]{5cm}
%\begin{center}

%{\small Pergamino} \\ {\tiny Prolonged ($>$ 10 day) dry spells, {\bf Cross-validated} }
%\includegraphics[width=0.25\textwidth]{figures/Figure3-take3.jpg}

%{\small All stations} \\ {\tiny Pairwise correlations for min and max temperatures }
%\includegraphics[width=5cm]{figures/pairwise-corr-min-max-closeup2.png} 
%\end{center}
%\end{column}
%\end{columns}

\end{frame}

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\begin{frame}{Gridded simulation}
	\begin{center}
		\includegraphics[width=0.95\textwidth]{figures/studyregion-with-grid.pdf}\\
		1961--2013 daily precipitation, maximum and minimum temperature.
	\end{center}

\end{frame}

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\begin{frame}{Gridded simulation}

{\small Ensemble mean OND seasonal total precip, mean max temp, mean min temp.}

\begin{center}
\includegraphics[width=0.95\textwidth]{figures/CH2-OND.pdf}
\end{center}
\end{frame}

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\begin{frame}{Contributions}

\begin{itemize}
	\tick Developed GLM based stochastic weather generator. \\$\rightarrow$ Spatial models on GLM coefficients.\\ $\rightarrow$ Residuals as monthly Gaussian processes. \\~\\ \pause
	\tick Address problem of simulating daily weather at \underline{unobserved} locations. \\~\\ \pause
	\tick Ease modeling of skewed \& discrete variables.
\end{itemize}
\vfill
{\tiny Verdin et al., (2015). Coupled stochastic weather generation using spatial and generalized linear models. {\it Stochastic Environmental Research and Risk Assessment}, 29(2), 347--356. doi:10.1007/s00477-014-0911-6}

\end{frame}

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\section{\scshape Chapter 3}
\subsection{A conditional stochastic weather generator for seasonal to multi-decadal simulations}
\begin{frame}{Chapter 3}
\begin{itemize}
{\bf \item[2.] { Generate space-time weather scenarios consistent with seasonal forecasts and multidecadal trends, for resources planning and management.}} \\~\\
\end{itemize}
%Conditional GLM weather generator:\\~\\

Propose the use of GLM weather generator for planning purposes. \\~\\
%$\rightarrow$ Seasonal to multi-decadal scales.\\~\\
Additional covariates, i.e., seasonal climate information. \\~\\
%$\rightarrow$ Seasonal climate conditions. \\~\\
Enables translation of seasonal forecasts or climate model projections. \\~\\
$\rightarrow$ \underline{Downscaling} coarse scale information.\\~\\
$\rightarrow$ Address the disconnect between data \& decision.

\end{frame}

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\begin{frame}{Study region \& data}
	\begin{center}
		\includegraphics[width=0.95\textwidth]{figures/Figure1.pdf}\\
		1961--2013 daily precipitation, maximum and minimum temperature.
	\end{center}

\end{frame}

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\begin{frame}{Principal component analysis}
Domain average OND (lines) and PC1 of all stations.
	\begin{center}
		\includegraphics[width=0.9\textwidth]{figures/Figure2.pdf}\\
	\end{center}

\end{frame}

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\begin{frame}{Additional covariates}
Precipitation models: \\
Domain average seasonal total precipitation.
\begin{flalign*} 
&\bX_O(\bs,t) = (1, \ldots, ST1(t), ST2(t), ST3(t), ST4(t))& \\~ \nonumber \\
&\bX_A(\bs,t) = (1, \ldots , ST1(t), ST2(t), ST3(t), ST4(t))&
\end{flalign*}
Temperature models: \\
Domain average seasonal mean maximum and minimum temperature.
\begin{flalign*}
&\bX_i(\bs,t) = (1, \ldots, SMN1(t), SMN2(t), SMN3(t), SMN4(t),& \nonumber \\ 
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~SMX1(t), SMX2(t), SMX3(t), SMX4(t))&\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{for } i=N,X&\nonumber
\end{flalign*}

{\bf Validate by comparison to unconditional GLM weather generator.}
\end{frame}

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\begin{frame}{Temporal validation}
13 years: JFM 2001 -- OND 2013
\begin{columns}[t]
\begin{column}[T]{1.5cm}
{\small Unconditional:\\~ \\~ \\ Precip \\ ~ \\ ~ \\ ~ \\ ~ \\ Max Temp \\ ~ \\ ~ \\~ \\ ~ \\ Min Temp }
\end{column}
\begin{column}[T]{8cm}
\begin{center}
	\includegraphics[width=0.999\textwidth]{figures/Figure4.pdf}
\end{center}
\end{column}
\begin{column}[T]{1.5cm}
{\small Conditional:\\~ \\~ \\ Precip \\ ~ \\ ~ \\ ~ \\ ~ \\ Max Temp \\ ~ \\ ~ \\~ \\ ~ \\ Min Temp }
\end{column}
\end{columns}
\end{frame}

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\begin{frame}{Seasonal forecast}
\begin{columns}[t]
\begin{column}[T]{7cm}
	\includegraphics[width=0.99\textwidth]{figures/IRI-OND2010.pdf}

{\footnotesize ~~~OND 2010 Precipitation: (15:35:50) (A:N:B) \\			~~~OND 2010 Temperature: (40:35:25) (A:N:B)} 
\end{column}
\begin{column}[T]{5cm}
\includegraphics[width=0.99\textwidth]{figures/terciles.pdf}\\~\\
{\footnotesize 1. Classify observed OND precipitation and temperatures as A:N:B. \\ ~\\ 2. Bootstrap ensemble of OND covariates, A:N:B as weights. \\~\\ 3. One simulation per ensemble member. \\~\\ 4. Ensemble of weather reflects seasonal forecast.}
\end{column}
\end{columns}
\end{frame}

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\begin{frame}{Seasonal forecast}

{\footnotesize Differences in ensemble mean (unconditional minus conditional):}
\begin{center}
	\includegraphics[width=0.85\textwidth]{figures/OND2010-med.pdf}
\end{center}
{\footnotesize Differences in 95\% ensemble spread (unconditional minus conditional):}
\begin{center}
	\includegraphics[width=0.85\textwidth]{figures/OND2010-spr.pdf}
\end{center}
\end{frame}

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\begin{frame}{Multi-decadal projection}
CORDEX-CMIP5 RCM (2015--2050)
\begin{columns}[t]
\begin{column}[T]{1.5cm}
{\small Unconditional:\\~ \\~ \\ Precip \\ ~ \\ ~ \\ ~ \\ ~ \\ Max Temp \\ ~ \\ ~ \\~ \\ ~ \\ Min Temp }
\end{column}
\begin{column}[T]{8cm}
\begin{center}
	\includegraphics[width=0.999\textwidth]{figures/Figure8.pdf}
\end{center}
\end{column}
\begin{column}[T]{1.5cm}
{\small Conditional:\\~ \\~ \\ Precip \\ ~ \\ ~ \\ ~ \\ ~ \\ Max Temp \\ ~ \\ ~ \\~ \\ ~ \\ Min Temp }
\end{column}
\end{columns}
\end{frame}

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\begin{frame}{Contributions}

\begin{itemize}
	\tick Condition GLM weather generator on seasonal forecasts, trends.\\~\\ \pause
	\tick Translate coarse seasonal forecasts.\\~\\ \pause
	\tick \underline {Downscale} regional climate model output.\\~\\ \pause
	\tick Addressed the disconnect between data \& decision.
\end{itemize}
\vfill
{\tiny Verdin et al., (2016). A conditional stochastic weather generator for seasonal to multi-decadal simulations. {\it Journal of Hydrology}. doi:10.1016/j.jhydrol.2015.12.036}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\scshape Chapter 4}
\subsection{BayGEN: A Bayesian space-time stochastic weather generator}
\begin{frame}{Chapter 4}
\begin{itemize}
{\bf \item[3.] { A generalized weather generator that can model and propagate uncertainty.}} \\~\\
\end{itemize}
Address non-treatment of uncertainty in weather generation.\\~\\

``Why do I care about uncertainty?'' \\ $\rightarrow$ {\small Variability between ensemble members. }\\ $\rightarrow$ {\small Better estimates of crop production risk.}\\~\\

%Treatment of uncertainty not trivial with traditional methods.\\~\\
Bayesian methods widely used to quantify uncertainty. \\ $\rightarrow$ {\small Bayesian model averaging {\it [Leamer, 1978]}.} \\ $\rightarrow$ {\small Coupling with weather generators {\it [Pezzulli et al., 2006; Hashmi et al., 2009]}.}\\~\\

Bayesian weather generator propagate uncertainty to simulations.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Methodology}
Consider the model from Chapter 2:\\
\begin{flalign}
&O(\bs,t) = \indicator_{[W_O(\bs,t) \geq 0]}& \\ ~\nonumber \\
&W_O(\bs,t) = \bX_O(\bs,t)'\bbeta_O(\bs)& \\ ~\nonumber \\
&A(\bs,t) \sim Gamma(\alpha_A(\bs),\alpha_A(\bs)/\mu_A(\bs,t)& \\ ~\nonumber \\
&\mu_A(\bs,t) = \text{exp}(\bX_A(\bs,t)'\bbeta_A)& \\ ~\nonumber \\
&Z_N(\bs,t) = \bX_N(\bs,t)'\bbeta_N(\bs) + W_N(\bs,t)& \\ ~\nonumber \\
&Z_X(\bs,t) = \bX_X(\bs,t)'\bbeta_X(\bs) + W_X(\bs,t)&
\end{flalign}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Model structure}

{\small Now model parameters are treated as random variables. \\
Monte Carlo sampling produces distribution of model parameters.\\$\rightarrow${\it Posterior distribution.}}
\begin{columns}[t]
\begin{column}[T]{6cm}

{\small {\bf Data layer:}
\begin{flalign}
	&O(\bs,t) = \indicator_{[W_O(\boldsymbol{s},t) \geq 0]}& \\ ~\nonumber \\
	&A(\bs,t) \sim Gamma(\alpha_A(\bs), \alpha_A(\bs)/\mu_A(\bs,t))& \\ ~\nonumber \\
	&Z_N(\bs,t) \sim GP(\mu_N(\bs,t),C_N(t))& \\ ~\nonumber \\
	&Z_X(\bs,t) \sim GP(\mu_X(\bs,t),C_X(t))&\\~\nonumber
	\end{flalign}
*{\footnotesize Priors centered on MLE.}}
\end{column}
\begin{column}[T]{6cm}

{\small {\bf Process layer:}
\begin{flalign}
	&W_O(\bs,t) \sim N(\mu_O(\bs,t),\sigma_O^2)& \\ ~\nonumber \\
	&\mu_A(\bs,t) = \text{exp}(\bX_A(\bs,t)'\bbeta_A(\bs))& \\ ~\nonumber \\
	&\mu_i(\bs,t) = \bX_i(\bs,t)'\bbeta_i(\bs)& \nonumber \\ 
	&~~~~~~~~~~~~~~~~~~~~~~\text{for}~i = O,N,X& \\
	&\bbeta_j(\bs) \sim GP(\hat{\beta}_j(\bs),C_{\beta_j})& \nonumber \\
	&~~~~~~~~~~~~~~~~~~~~~~\text{for}~j = O,A,N,X& \\~\nonumber \\
	&\balpha_A(\bs) \sim GP(\hat{\balpha}_A(\bs),C_{\balpha_A})& 
	\end{flalign}
}
\end{column}
\end{columns}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}{Likelihood computation}
%Define $\btheta$ as a vector of all model parameters:
%\begin{flalign}
%&\btheta = (\bbeta_O, \bbeta_A, \bbeta_N, \bbeta_X, C_X(t), C_N(t), C_{\bbeta_O}, C_{\bbeta_A}, C_{\bbeta_X}, C_{\bbeta_N}, \balpha_A, C_{\balpha_A}),&
%\end{flalign}
%and $\bY$ as a vector of the data:
%\begin{flalign}
%&\bY = (O(\bs,t), A(\bs,t), Z_N(\bs,t), Z_X(\bs,t)).&
%\end{flalign}
%Seek the {\it posterior distribution}:
%\begin{flalign} \label{eqn:posterior}
%&p(\btheta | \bY) = \frac{p(\bY|\btheta)p(\btheta)}{p(\bY)} \propto p(\bY|\btheta)p(\btheta)&\\
%\end{flalign}
%$p(\bY|\btheta) =$ likelihood function. \\
%$p(\btheta) =$ prior joint distribution.
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Study region \& data}
	\begin{center}
		\includegraphics[width=0.95\textwidth]{figures/Figure1-junin.pdf}\\
		1961--2013 daily precipitation, maximum and minimum temperature.\\
		{\bf Validate BayGEN by comparison to GLMGEN.}
	\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Posterior distribution}
\begin{center}
	\includegraphics[width=0.95\textwidth]{figures/bocc-intercept.pdf}\\
{\small Posterior distribution and MLE of precipitation occurrence intercept: ($\beta_0$)}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Climatology \& variability}
\begin{center}
{\small Mean Precip ~~~~~~~~~~~~~~~~~~ Mean Max Temp ~~~~~~~~~~~~~~~~ Mean Min Temp}\\
	\includegraphics[width=0.99\textwidth]{figures/climatology-Junin-mn.pdf}\\
{\footnotesize BayGEN as grey boxplots; GLMGEN as white boxplots; observed as lines.}
\end{center} 

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Climatology \& variability}
\begin{center}
{\small S.D. Precip ~~~~~~~~~~~~~~~~~~ S.D. Max Temp ~~~~~~~~~~~~~~~~ S.D. Min Temp}\\
	\includegraphics[width=0.99\textwidth]{figures/climatology-Junin-sd.pdf}\\
{\footnotesize BayGEN as grey boxplots; GLMGEN as white boxplots; observed as lines.}
\end{center} 

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{DSSAT}
The Decision Support System for Agrotechnology Transfer 
%(DSSAT) is a software package made up of many independent computer programs and crop simulation models that operate together to facilitate the efficient analysis of complex alternative decisions in agriculture management and adaptation. 


\begin{itemize}
	\item[] Required inputs:
		\begin{itemize}
			\item Land use and soil type
			\item Minimum, maximum temperatures ({\it daily})
			\item Precipitation ({\it daily})
			\item Solar radiation ({\it daily})
		\end{itemize}
	\item[] Optional inputs:
		\begin{itemize}
			\item Dew point ({\it daily})
			\item Wind speed ({\it daily})
		\end{itemize}
	\item[] Outputs:
		\begin{itemize}
			\item Crop yield ({\it seasonal})
			\item Flowering and harvest date
			\item Biomass ({\it daily})
			\item Soil water content ({\it daily})
			\item Root depth ({\it daily})
		\end{itemize}
\end{itemize}

{\footnotesize \emph{[Jones et al., 1998]} }
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}{Coupling with crop simulation model}
%\begin{center}
%	\includegraphics[width=0.95\textwidth]{figures/DSSAToutput-and-seatot.pdf}\\
%{\small Simulated seasonal precipitation (left) and soybean yield (right).}
%\end{center}
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Coupling with crop simulation model}

{\small 100 simulations from BayGEN, GLMGEN:}\\~\\
\begin{center}
	\includegraphics[width=0.95\textwidth]{figures/DSSAT-cdf-space.pdf}\\~\\
%{\small Simulated seasonal precipitation (left) and soybean yield (right).\\~\\}

{\small ``Break even'' production risk: 30\% (BayGEN), 16\% (GLMGEN).\\~\\
{\it Nonlinear} relationship between weather and crop yield.}

\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Contributions}

\begin{itemize}
	\tick Development of Bayesian weather generator.\\$\rightarrow$ Address non-treatment of uncertainty.\\~\\ \pause
	\tick ``Cascade of uncertainty.''\\~\\ \pause
	\tick Uncertainty propagates to production risk.
\end{itemize}
\vfill
{\tiny Verdin et al., (2016, In Preparation). BayGEN: A Bayesian space-time stochastic weather generator. {\it Water Resources Research}.}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\scshape Chapter 5}
\subsection{A statistical metamodel for monthly groundwater fluctuations}
\begin{frame}{Process based models}
Agricultural planning in the Pampas:
\begin{itemize}
	\item Crop simulation models (DSSAT).
	\item Analyze different management strategies.
	\item Correct crop yields with WTD  (MIKE-SHE).
	\item[] {\small $\rightarrow$ MIKE-SHE has great computational costs. (5km x 5km, daily)}
\end{itemize}
\vfill
\begin{center}
\includegraphics[width=0.95\textwidth]{figures/first-flowchart.png}
\end{center}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{MIKE SHE}
\begin{columns}[t]
\begin{column}[T]{5cm}

{\small
Rainfall-runoff model
\begin{itemize}
	\item Deterministic
	\item Physically based
	\item Distributed
	\item Solves PDE for mass flow \& momentum transfer.
\end{itemize} }

\end{column}
\begin{column}[T]{5cm}
\begin{center}
{\tiny \emph{MIKE by DHI (2012)} }
\includegraphics[width=0.9\textwidth]{figures/MIKE-SHE.png}
\end{center}
\end{column}
\end{columns}

{\small
Inputs:
		\begin{itemize}
			\item Spatially-distributed land use (e.g., leaf-area index, root depth), soil data, precipitation, reference evapotranspiration, river networks
		\end{itemize}
Outputs:
		\begin{itemize}	
			\item Water table depth (WTD), real evapotranspiration (ET), overland flow, unsaturated zone flow, etc.%water content in the unsaturated zone, groundwater flow
		\end{itemize}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{MIKE-SHE calibration}
\begin{center}
\includegraphics[width=0.99\textwidth]{figures/Figure_03a.png}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{MIKE-SHE}
Motivating issues.
	\begin{itemize}
		\item Computationally expensive (5km x 5km, daily).
		\item Only 4 simulations at a time.
		\item Proprietary.
		\item For agricultural planning, interested in WTD.
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Statistical metamodel}
\begin{itemize}
	{\bf \item[4.] { An efficient water table depth model for seasonal agricultural planning.}}\\~\\
\end{itemize}
Model WTD using weather and land use data? \\ $\rightarrow$ Reduce computational costs of distributed hydrologic models.\\~\\
Propose stochastic hierarchical metamodel for {\it monthly} ET and WTD. 
{\small
	\begin{itemize}
		\item[] {\bf Level 1:} ET modeled with land use and weather, PET.
		\item[] {\bf Level 2:} WTD modeled with ET and weather.
	\end{itemize}

Assumptions:
\begin{itemize}
		\item[] Land use constant for calendar year (from database).
		\item[] $\rightarrow$ Pasture, soybean, maize, wheat, soybean/wheat, sunflower.
		\item[] Homogeneous soil type (i.e., not parameterized.)
	\end{itemize}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Statistical metamodel}
Equally spaced ``knot'' locations $\bs$. (20km x 20km)\\$\rightarrow$MIKE-SHE grid cells. (5km x 5km)\\$\rightarrow$Analog to stations.
\begin{flalign}
&Y_{ET}(\bs,t) = \bX_{ET}(\bs,t)'\bbeta_{ET}(\bs)& \\ ~\nonumber \\
&Y_{WT}(\bs,t) = \bX_{WT}(\bs,t)'\bbeta_{WT}(\bs) + \epsilon& \\
&\epsilon \sim GP(0,C(t))& \\ ~\nonumber \\
&\bX_{ET}(\bs,t) = (X_P(\bs,t),X_X(\bs,t),X_N(\bs,t),X_E(\bs,t),X_{L1}(\bs,t),&\nonumber \\ 
& ~~~~~~~~~~~~~~~~~~~~~~~~X_{L2}(\bs,t),X_{L3}(\bs,t),X_{L4}(\bs,t),X_{L5}(\bs,t),X_{L6}(\bs,t))& \\ ~\nonumber \\
&\bX_{WT}(\bs,t) = (Y_{WT}(\bs,t-1),X_P(\bs,t),X_X(\bs,t),X_N(\bs,t),Y_{ET}(\bs,t))&
\end{flalign}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Study region \& data}
	\begin{center}
		\includegraphics[width=0.95\textwidth]{figures/studyregion-with-wells-and-grid.pdf}\\
		1961--2013 daily precipitation, maximum and minimum temperature,
		interpolated to MIKE-SHE grid; MIKE-SHE estimates of ET and WTD.
	\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Basin average behavior}
Jan 1961 -- Dec 2013
	\begin{center}
		\includegraphics[width=0.995\textwidth]{figures/basin-average.pdf}\\
		Ensemble of 100 simulations vs. MIKE-SHE for historic period.
	\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Spatial summary statistics}
Jan 1961 -- Dec 2013
	\begin{center}
		\includegraphics[width=0.895\textwidth]{figures/metamodel-sumstats.pdf}\\
		Ensemble mean vs. MIKE-SHE.
	\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Wet \& dry periods}
\begin{center}
MIKE-SHE ~~~~~~~~~~~~~~~ Ensemble mean ~~~~~~~~~~~~~~~ Difference
		\includegraphics[width=0.95\textwidth]{figures/metamodel-wet-and-dry-months-and-diff-raster.pdf}
	\end{center}
Top row: Dec 2000 (SE floods); Bottom row: Feb 2012 (NW depletion).
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Seasonal forecast}
	\begin{center}
		\includegraphics[width=0.725\textwidth]{figures/IRI-OND2013.pdf}\\
	\emph{Probabilistic forecasts:}\\

	{\footnotesize ~~~OND 2013 Precipitation: (25:40:35) (A:N:B) \\
			~~~OND 2013 Temperature: (40:35:25) (A:N:B)}	\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Seasonal forecast}
	\begin{center}

		{\small December 2013 WTD } \\
		\includegraphics[width=0.93\textwidth]{figures/validation-OND2013.pdf}\\
	\end{center}
{\footnotesize (a) MIKE-SHE, (b) ensemble mean, (c) difference, (d) ensemble 97.5$^{\text{th}}$ percentile, (e) 2.5$^{\text{th}}$ percentile, (f) 95\% ensemble spread.}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Contributions}
\begin{itemize}
	\tick Development of statistical metamodel for monthly WTD.\\$\rightarrow$ Use in agricultural planning.\\~\\ \pause
	\tick Address computational costs of distributed hydrologic models.\\~\\ \pause
	\tick Capture dominant mechanisms of WTD variability.\\~\\ \pause
	\tick Downscaling seasonal forecasts (i.e., Chapter 3).
\end{itemize}
\vfill
{\tiny Verdin et al. (2016, In Preparation). A statistical metamodel for monthly groundwater fluctuations. {\it International Journal of Agricultural Management.}}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{\scshape Closing}
%\begin{frame}{Limitations}
%	\begin{itemize}
%		\tick Simulating at unobserved locations.\\~\\ \pause
%		\tick Seasonal forecasts uninformative to decision makers.\\~\\ \pause
%		\tick GLM weather generators neglect uncertainty.\\$\rightarrow$ Not exploring full range of future conditions.\\~\\ \pause
%		\tick Distributed hydrologic models not conducive for near real-time use.
%	\end{itemize}
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\scshape Closing}
\begin{frame}{Summary}
\begin{itemize}
	\tick Developed GLM weather generator.
	\item[] $\rightarrow$ Spatial models on GLM coefficients.
	\item[] $\rightarrow$ Address problem of simulating at unobserved locations.\\~\\ \pause
	\tick Illustrated GLM weather generator for planning purposes.
	\item[] $\rightarrow$ Translate seasonal forecasts; downscale projections.
	\item[] $\rightarrow$ Address disconnect between data \& decision.\\~\\ \pause
	\tick Developed Bayesian weather generator.
	\item[] $\rightarrow$ Quantify uncertainty neglected with traditional methods. 
	\item[] $\rightarrow$ ``Cascade of uncertainty'' to decision space.\\~\\ \pause
	\tick Developed statistical metamodel for hydrologic model.
	\item[] $\rightarrow$ Use in agricultural planning, downscaling seasonal forecasts. 
	\item[] $\rightarrow$ Address computational costs of distributed hydrologic models.
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Contributions}

{\small \begin{itemize}
	\item GLM space-time weather generator (GLMGEN).$^1$
	\item[] {\footnotesize $\rightarrow$ Simulate at unobserved locations.}
	\item Conditional GLMGEN.$^2$
	\item[] {\footnotesize $\rightarrow$ Downscaling tool; useful for climate change studies.}
%	\item[] {\small $\rightarrow$ Address disconnect between data \& decision.}
	\item Bayesian space-time weather generator (BayGEN).$^3$
	\item[] {\footnotesize $\rightarrow$ New, novel, and a significant contribution to the literature.}
	\item Statistical emulator for water table depth.$^4$
	\item[] {\footnotesize $\rightarrow$ Conditioned on climate, land use; downscale seasonal forecasts, make cropping decisions.} 
%	\item[] {\small $\rightarrow$ Downscale seasonal forecasts, make cropping decisions.}
\end{itemize}
}
\vfill
{\tiny $^1$Verdin et al., (2015). Coupled stochastic weather generation using spatial and generalized linear models. {\it Stochastic Environmental Research and Risk Assessment}, 29(2), 347--356.}\\
{\tiny $^2$Verdin et al., (2016). A conditional stochastic weather generator for seasonal to multi-decadal simulations. {\it Journal of Hydrology}.}\\
{\tiny $^3$Verdin et al., (2016, In Preparation). BayGEN: A Bayesian space-time stochastic weather generator. {\it Water Resources Research}.}\\
{\tiny $^4$Verdin et al. (2016, In Preparation). A statistical metamodel for monthly groundwater fluctuations. {\it International Journal of Agricultural Management.}}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Future work}
\begin{itemize}
%	\item Obtain Ph.D. (hopefully)\ldots ~~~ ?:-)
%	\item[] 
	\item Development of CRAN package for GLMGEN.
	\item[] $\rightarrow$ Conditional \& unconditional simulation.
	\item[] 
	\item Refinement of metamodel. 
	\item[] $\rightarrow$ Consider  LAI \& root depth.
	\item[] $\rightarrow$ Finer scales if needed.
	\item[] 
	\item Graduate??
	\item[]
	\item Get married!!
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
	\begin{center}
		\huge{Thank you!}
	\end{center}
	
\begin{columns}[t]
\begin{column}[T]{5cm}
Committee:
\begin{itemize}
	\item[] {\small Balaji Rajagopalan}
	\item[] {\small Guillermo Podest\'a}
	\item[] {\small William Kleiber}
	\item[] {\small Ben Livneh}
	\item[] {\small Joseph Kasprzyk}
\end{itemize}
NCAR folks:
\begin{itemize}
	\item[] {\small Rick Katz}
	\item[] {\small Linda Mearns}
	\item[] {\small Seth McGinnis}\\~\\~\\
\end{itemize}
\end{column}
\begin{column}[T]{5cm}
Buenos Aires team:
\begin{itemize}
	\item[] {\small Federico Bert}
	\item[] {\small Pablo Garc\'ia}
	\item[] {\small Angel Men\'endez}
	\item[] {\small Santiago Rovere}
\end{itemize}
Funding:
\begin{itemize}
	\item[]{\small NSF: The Dynamics of Coupled Natural and Human Systems (CNH) Program} % promotes interdisciplinary analyses of relevant human and natural system processes and complex interactions among human and natural systems at diverse scales.
\end{itemize} 

\end{column}
\end{columns}
\centering
Carsen, my family, my friends, \ldots\\ and all I forgot to mention.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}{Funding}

%National Science Foundation (NSF) \\~\\ The Dynamics of Coupled Natural and Human Systems (CNH) Program promotes interdisciplinary analyses of relevant human and natural system processes and complex interactions among human and natural systems at diverse scales.

%	\begin{center}
%		\includegraphics[width=0.3\textwidth]{figures/NSF.jpg}
%	\end{center}

%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\centering\Huge Additional slides
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 2}
\begin{center}
Domain extremal precipitation: 1908--2010\\
\includegraphics[width=0.999\textwidth]{figures/extremes-no-outline2.jpg}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 3}
Spatial validation: OND (1961--2013)
	\begin{center}
	Occurrence ~~~~~~~~~ Amounts ~~~~~~~~~ Max Temp ~~~~~~~~~ Min Temp\\
		\includegraphics[width=0.95\textwidth]{figures/Figure3.pdf}
	\end{center}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 4}
Define $\btheta$ as a vector of all model parameters:
\begin{flalign}
&\btheta = (\bbeta_O, \bbeta_A, \bbeta_N, \bbeta_X, C_X(t), C_N(t), C_{\bbeta_O}, C_{\bbeta_A}, C_{\bbeta_X}, C_{\bbeta_N}, \balpha_A, C_{\balpha_A}),&
\end{flalign}
and $\bY$ as a vector of the data:
\begin{flalign}
&\bY = (O(\bs,t), A(\bs,t), Z_N(\bs,t), Z_X(\bs,t)).&
\end{flalign}
Seek the {\it posterior distribution}:
\begin{flalign} \label{eqn:posterior}
&p(\btheta | \bY) = \frac{p(\bY|\btheta)p(\btheta)}{p(\bY)} \propto p(\bY|\btheta)p(\btheta)&\\
\end{flalign}
$p(\bY|\btheta) =$ likelihood function. \\
$p(\btheta) =$ prior joint distribution.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 4}
\begin{center}
BayGEN ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ GLMGEN\\
	\includegraphics[width=0.4999\textwidth]{figures/domain-extrema-min-max-temp.jpg}
	\includegraphics[width=0.4999\textwidth]{figures/WGEN-qqplots.jpg}\\
{\small Domain daily extreme temperature.}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 4}
\begin{center}
BayGEN\\
	\includegraphics[width=0.995\textwidth]{figures/ONDJFM-maxdaily-precip.pdf}\\
GLMGEN\\
	\includegraphics[width=0.995\textwidth]{figures/extremes-WGEN2.pdf}%\\
%{\small Seasonal maximum daily precipitation.}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 4}
BayGEN:
\begin{center}
{\small Precip} ~~~~~~~~~~~~~~~~~~~~~ {\small Max Temp} ~~~~~~~~~~~~~~~~~~~~~ {\small Min Temp}\\
	\includegraphics[width=0.89\textwidth]{figures/Figure-OND-BayGEN.pdf}\\
{\small OND: Ensemble mean (top) and 95\% ensemble spread (bottom).}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 4}
GLMGEN:
\begin{center}
{\small Precip} ~~~~~~~~~~~~~~~~~~~~~ {\small Max Temp} ~~~~~~~~~~~~~~~~~~~~~ {\small Min Temp}\\
	\includegraphics[width=0.89\textwidth]{figures/Figure-OND-WGEN.pdf}\\
{\small OND: Ensemble mean (top) and 95\% ensemble spread (bottom).}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 5}
\begin{center}
Leaf Area Index  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Root Depth
\includegraphics[width=0.49\textwidth]{figures/Figure_02a.png}
\includegraphics[width=0.49\textwidth]{figures/Figure_02b.png}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Chapter 5}
\begin{center}
MIKE-SHE calibration
\includegraphics[width=0.99\textwidth]{figures/Figure_03b.png}
\end{center}
\end{frame}


\end{document}
