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\chapter{\uppercase{Reliability for Unknown Bridge Foundations}}
\label{chap:Stats}

\section{Introduction}
\noindent It is shown in Chapter~\ref{chap:Experiment} that IP is successful at imaging reinforced concrete bridge foundations and steel H-piles.  ERI is not capable of successfully imaging long, slender piles like those tested herein, but it can image large reinforced concrete structures.  \citet{Briaud} notes that imaging foundations less than 1 m and deeper than 5 m is very difficult with ERI.  Long, slender foundations are tested for this research because they are the common construction practice in the Bryan TxDOT district.  Even though it has been shown that ERI is not capable of imaging these foundations \citep{rung}, the ERI data are included as an indicator of the quality of the IP data because the IP model is dependent on the ERI model.  Because the ERI data is not used to estimate the bridge foundation depth, only the IP data is post processed with MATLAB\textregistered, a numerical computing environment, and used to generate the probability of nonexceedance curve.  The MATLAB\textregistered post processing is performed to automate the foundation depth estimation to provide a quantitative prediction.  The post processed data set and the actual dimensions of foundations are used to calculated the probability of nonexceedance curve.  Each of these processes and results are described in this chapter.

\section{Inverted IP Section Post Processing}
\noindent The AGI EarthImager\texttrademark commercial software is very useful for processing collected data in the field.  The software calculates the apparent resistivity pseudosection from a homogeneous half space model, this forward model is then compared to the measured apparent resistivity pseudosection and the data misfit between the two pseudosection is computed.  Next, the linearized inverse problem is solved, the model is updated, and again the data misfit is calculated.  If the data misfit is within the stop criteria the process is stopped and the inversion is complete.  If the data misfit is above the stop criteria, the process is repeated.  This process is similar for the IP data, but only one iteration is possible and the data misfit is calculated using the ERI model instead of the IP model.  The final results are images of the spatial variations of resistivity and chargeability.  These images are useful for inspecting the subsurface resistivity and chargeability distributions but they must be interpreted with engineering judgment.  This post processing procedure is used to automate a qualitative prediction of the estimated foundation depth so predictions do not vary from one person to the next.  These automated predictions are used in conjunction with the known foundation dimensions in Chapter~\ref{chap:Experiment} to create the probability of nonexceedance curve.

When EarthImager\texttrademark calculates and inverts the apparent resistivity and  apparent chargeability pseudosections the calculated data and inverted data are saved in an output text file that the program uses to visually display the results.  This text file includes the following information in the order listed below: 
\begin{enumerate}
\item \label{itm:one} Headings
\item \label{itm:two} Summary of data and electrodes
\item \label{itm:three} Initial settings
\item \label{itm:four} Surface electrode locations
\item \label{itm:five} Commands, raw V/I, Geometric Factor, Apparent Resistivity, IP (V/V), Correlation
\item \label{itm:six} Mesh size
\item \label{itm:seven} Node location in X (m)
\item \label{itm:eight} Node location in Y (m)
\item \label{itm:nine} Timing
\item \label{itm:ten} Convergence
\item \label{itm:leven} Ip Convergence 
\item \label{itm:twerve} Output of data and model of all iterations
\end{enumerate}
The inputs necessary for the post processing code are Item~\ref{itm:six}, Item~\ref{itm:seven}, Item~\ref{itm:eight}, and Item~\ref{itm:twerve}.  These are copied and pasted into the MATLAB{\textregistered}  m--file in the labeled sections.  The code is attached in Appendix~\ref{app:matlab}.  The following inputs are necessary from the user for the code to run: number of elements, number of elements in the $x$ direction, number of elements in the $y$ direction, and the range of $x$--nodes where the foundation is located.  The last item is found using the known horizontal distance of the foundation at the time of testing.  For example, if the foundation is located 6.5 m from the start of the line and is 1 m wide the nodes corresponding to a horizontal distance from 6.5 - 7.5 m in the $x$--node section are used.  The node number is first, followed by the distance in the $x$ direction in m.  If node 35 is at 6.5 m and node 39 is at 7.5 m 35:39 should be used as the input for the foundation.  No other inputs are necessary for the code to run.

The MATLAB{\textregistered} code has four main sections: loading the data, organizing the elements into a numerical pseudosection, isolating the elements around the bridge foundation, and calculating the vertical ratio of these elements as well as the gradient of these elements.  The vertical ratio of the elements and the slope of the elements are used instead of the inverted chargeability values because chargeability is not an intrinsic value.  Therefore a range of typical chargeability for reinforced concrete is be misleading.  As an example, the data from Fig.~\ref{fig:caldwell2b} are used; the second pile tested on the Davidson Creek Bridge.  Recall that the depth of the data gathered for this experiment is 5.2 m below the surface and the foundation is 6.08 m deep using the as-built bridge plans and accommodating for scour.  First, the ratio of the elements are calculated as
\begin{equation}
R_{i} = \epsilon_{i}/\epsilon_{i+1}
\end{equation}
\noindent where $R_{i}$ is the ratio value, $\epsilon_{i}$ is the $i^{th}$ element, and $\epsilon_{i+1}$ is the element directly below it.  Only the elements that coincide with the location of the foundation are plotted.  The chargeability ratio plot from Davidson Creek Bridge 2 is shown in Fig.~\ref{fig:chargedavid2}.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering
\begin{picture}(18.89,12)
\put(1.2,0){\includegraphics[width=0.85\textwidth]{figures/chargedavid2}}
\put(5,3.74){Element 1}
\put(5,3.2){Element 2}
\put(5,2.66){Element 3}
\put(5,2.12){Data Depth}
\put(5,1.58){Foundation Depth}
\put(3.2,0.75){0}
\put(4.2,0.75){0.2}
\put(5.42,0.75){0.4}
\put(6.67,0.75){0.6}
\put(7.93,0.75){0.8}
\put(9.22,0.75){1.0}
\put(10.42,0.75){1.2}
\put(11.71,0.75){1.4}
\put(12.94,0.75){1.6}
\put(14.2,0.75){1.8}
\put(15.59,0.75){2}
\put(2.5,1.2){-10}
\put(2.8,2.14){-9}
\put(2.8,3.125){-8}
\put(2.8,4.10){-7}
\put(2.8,5.075){-6}
\put(2.8,6.05){-5}
\put(2.8,7.06){-4}
\put(2.8,8.03){-3}
\put(2.8,9.01){-2}
\put(2.8,9.98){-1}
\put(2.8, 10.95){ 0}
\put(1.75,4.5){\rotatebox{90}{Depth (m)}}
\put(9.27,0){$R$}
\end{picture}
\caption{Chargeability ratio for Davidson Creek Bridge pile two}
\label{fig:chargedavid2}
\end{figure}
The horizontal black line in Fig.~\ref{fig:chargedavid2} is the depth to which the data is collected and the horizontal blue line is the depth of the foundation.  The first criterion for estimating the foundation depth is the element depth directly beneath $R_{\textrm{max}}$.  For  Fig.~\ref{fig:chargedavid2}, $R_{\textrm{max}}$ occurs at 4.55 m beneath the surface.  The element beneath this is -5.15 m.

Next, the slope of the elements are calculated such that 
\begin{equation} 
\frac{\textrm{d} \epsilon}{\textrm{d} y} = \frac{\epsilon_{i + 1}-\epsilon_{i}}{\textrm{d} y}
\end{equation}
where $\epsilon$ are the vertical elements and ${\textrm{d} y}$ is vertical change in the $y$ direction.  The criterion for the slope shown in Fig.~\ref{fig:slopedavid2} is the element depth beneath where all of the elements on the foundation are the minimum.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering
\begin{picture}(18.89,12)
\put(1.2,0){\includegraphics[width=0.85\textwidth]{figures/slopedavid2}}
\put(5,3.74){Element 1}
\put(5,3.2){Element 2}
\put(5,2.66){Element 3}
\put(5,2.12){Data Depth}
\put(5,1.58){Foundation Depth}
\put(2.85,0.75){-40}
\put(4.4,0.75){-30}
\put(6,0.75){-20}
\put(7.55,0.75){-10}
\put(9.4,0.75){0}
\put(10.8,0.75){10}
\put(12.425,0.75){20}
\put(14,0.75){30}
\put(15.5,0.75){40}
\put(2.5,1.2){-10}
\put(2.8,2.14){-9}
\put(2.8,3.125){-8}
\put(2.8,4.10){-7}
\put(2.8,5.075){-6}
\put(2.8,6.05){-5}
\put(2.8,7.06){-4}
\put(2.8,8.03){-3}
\put(2.8,9.01){-2}
\put(2.8,9.98){-1}
\put(2.8, 10.95){ 0}
\put(1.75,5.5){\rotatebox{90}{Depth (m)}}
\put(9.27,0){$\frac{\textrm{d} \epsilon}{\textrm{d} y}$}
\end{picture}
\caption{Slope for Davidson Creek Bridge two}
\label{fig:slopedavid2}
\end{figure}
The minimum slope for all elements occurs at 4.55 m.  The element depth directly beneath this is 5.15 m.  The two criteria match at a prediction of 5.15 m, 0.93 m more shallow than the actual foundation depth.  In cases where the two criteria do not match, the average is taken.  

The number of elements that cover the foundation is dependent on size of each element.  In Fig.~\ref{fig:chargedavid2} and Fig.~\ref{fig:slopedavid2} the element mesh division is four, meaning that for a 1 m wide electrode spacing, each element is 0.25 m wide.  The effects of mesh refinement is also investigated.  The maximum refinement is eight.  Increasing the mesh refinement to six and eight increases computational time and yields more conservative predictions using the criteria described.  For this reason, a mesh refinement of four is used for all inversions and foundation predictions.

This criteria is used for each of the bridges in Chapter~\ref{chap:Experiment}, the large unknown foundation from Chapter~\ref{chap:Experiment}, as well as the shallow foundation control test from Chapter~\ref{chap:Model}.  The deep foundations and steel H-piles from the NGES are not used because the RMS values are above to acceptable range to trust the model.  While several of the bridges are also above the acceptable RMS (Figs.~\ref{fig:navasota},	\ref{fig:borrishalfb},~\ref{fig:caldwell1b},~\ref{fig:gumcreekb}) they are included because they are full scale tests.  Also, even though the models are not a good fit for the data, when the stop criteria is used the predictions are less then the foundation.  Therefore, if the data quality is low due to uncontrollable factors, like the soil type, predictions can still be made for the NBI because they are conservative predictions.  Table~\ref{tab:bridge} summarizes the predicted depths using this post processing and the actual depth of the foundations .  All plots for these 13 cases are included in Appendix~\ref{app:postp}.
\begin{table}[!ht]
\centering
\caption{Post Processing Results}
\begin{tabular}{|l|c|c|c|}
\hline
Name & Fig. & Prediction (m) & Actual (m) \\
\hline
Control Test & Fig.~\ref{fig:control} & 1.04 & 0.76\\
3 x 3 m$^2$ Unknown Depth & Fig.~\ref{fig:matfoundation} & 1.98 & 1.0 \\
Beason Creek Bridge & Fig.~\ref{fig:navasota} & 4.52 & 4.88 \\
Clear Creek Bridge 1 m spacing & Fig.~\ref{fig:borrisb} & 1.47  & 4.6\\
Clear Creek Bridge 0.57 m spacing & Fig.~\ref{fig:borrishalfb} & 1.88 & 4.6\\
Davidson Creek Bridge pile one & Fig.~\ref{fig:caldwell1b} & 1.55 & 5.75\\
Davidson Creek Bridge pile two & Fig.~\ref{fig:caldwell2b} & 5.15 & 6.08\\
Davidson Creek Bridge pile three & Fig.~\ref{fig:caldwell3b} & 5.43 & 6.19\\
Davidson Creek Bridge pile four & Fig.~\ref{fig:caldwell4b} & 7.30 & 6.0\\
Davidson Creek Bridge pile five & Fig.~\ref{fig:caldwell5b} & 3.32 & 6.12\\
Davidson Creek Bridge pile six & Fig.~\ref{fig:caldwell6b} & 6.17 & 6.27\\
Gum Creek Bridge pile one & Fig.~\ref{fig:gumcreekb} & 1.94 & 3.03\\
Gum Creek Bridge pile two & Fig.~\ref{fig:gumcreekb} & 1.94 & 3.03\\
\hline
\end{tabular}
\label{tab:bridge}
\end{table}

\section{Probability of Non-Exceedance}
The results of the post processing predictions are used for creating the probability of non-exceedance curve.  The probability of non-exceedance curve is created by fitting a lognormal cumulative distribution function (CDF) to the defined variable $\xi$:
\begin{equation}
\xi = \frac{D_{\textrm{actual}}}{D_{\textrm{predicted}}}
\end{equation}
\noindent where $D_{\textrm{actual}}$ is the actual depth of the foundation and $D_{\textrm{predicted}}$ is the depth predicted by IP using the previously described decision process.  To verify that the lognormal distribution is a good fit for these data over the normal distribution, the normal distribution CDF of $\xi$ and the  $\log (\xi)$ are plotted as described by \citet{BC}.  A linear fit indicates that the distribution is a good fit for the data.  
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering
\begin{picture}(18.89,11.75)
\put(1.2,0){\includegraphics[width=0.85\textwidth]{figures/normcdf}}
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\put(6.55,0.75){1.5}
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\put(15.5,0.75){4}
\put(2.9,6.25){0.5}
\put(4.95,6.25){1}
\put(6.55,6.25){1.5}
\put(8.3,6.25){2}
\put(10.1,6.25){1.5}
\put(11.85,6.25){3}
\put(13.75,6.25){3.5}
\put(15.5,6.25){4}
\put(2.35,1.15){-0.2}
\put(2.35,2.25){-0.1}
\put(2.95,3.25){0}
\put(2.6,4.35){0.1}
\put(2.6,5.5){0.2}
\put(2.95,6.75){0}
\put(2.6,8.15){0.5}
\put(2.95,9.55){1}
\put(2.6, 10.95){1.5}
\put(1.5,9.2){$p$}
\put(1.5,3.25){$d$}
\put(9.45,0){$\xi$}
\put(0,11.5){(a)}
\put(0,6){(b)}
\end{picture}
\caption{Normal distribution CDF of $\xi$: (a) probability of $\xi$ from a normal distribution; (b) residuals of the linear fit}
\label{fig:normcdf}
\end{figure}
In Fig~\ref{fig:normcdf}, $p$ is the probability that a single observation from a normal distribution with a mean, $\mu$, and standard deviation, $\sigma$, will fall in the interval $(-\infty \ \textrm{x}]$ or
\begin{equation}
p = F(x|\mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}}\int_{-\infty}^x \textrm{e}^{\frac{-(t-\mu)^2}{2\sigma^2}}\mathrm d t
\end{equation}
\noindent where $\mu$ and $\sigma$ have been defined, $t$ is each value of $\xi$, and $x$ is the interval of the data.  The norm of the residuals is used as an estimator of the goodness of fit of the linear fit to the data.  The norm is defined as
\begin{equation}
norm = \sqrt{\displaystyle \sum_{i=1}^N d_i^2}
\end{equation}
where $d_i$ is the difference between the $i_{th}$ predicted data and empirical data.  For the normal CDF, the linear norm of residual is 0.19.  A perfectly linear set of data has a linear norm of residual of 0.
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering
\begin{picture}(18.89,11.75)
\put(1.2,0){\includegraphics[width=0.85\textwidth]{figures/lognormcdf}}
\put(2.9,0.75){-0.3}
\put(4.15,0.75){-0.2}
\put(5.6,0.75){-0.1}
\put(7.05,0.75){0.0}
\put(8.45,0.75){0.1}
\put(9.85,0.75){0.2}
\put(11.25,0.75){0.3}
\put(12.65,0.75){0.4}
\put(14,0.75){0.5}
\put(15.5,0.75){0.6}
\put(2.9,6.25){-0.3}
\put(4.15,6.25){-0.2}
\put(5.6,6.25){-0.1}
\put(7.05,6.25){0.0}
\put(8.45,6.25){0.1}
\put(9.85,6.25){0.2}
\put(11.25,6.25){0.3}
\put(12.65,6.25){0.4}
\put(14,6.25){0.5}
\put(15.5,6.25){0.6}
\put(2.35,1.15){-0.2}
\put(2.35,2.25){-0.1}
\put(2.95,3.25){0}
\put(2.6,4.35){0.1}
\put(2.6,5.5){0.2}
\put(2.95,6.75){0}
\put(2.6,7.5){0.2}
\put(2.6,8.4){0.4}
\put(2.6,9.3){0.6}
\put(2.6,10.1){0.8}
\put(2.95, 10.95){1}
\put(1.5,9.2){$p$}
\put(1.5,3.25){$d$}
\put(9.45,0){$\log \xi$}
\put(0,11.5){(a)}
\put(0,6){(b)}
\end{picture}
\caption{Normal distribution CDF of $\log \xi$: (a) probability of $\log \xi$ from a normal distribution; (b) residuals of the linear fit}
\label{fig:lognormcdf}
\end{figure}
Comparing Figs.~\ref{fig:normcdf} and~\ref{fig:lognormcdf}, the normal distribution of $\log (\xi)$ is more linear with a norm of 0.03.  The lognormal distribution is used because it is more linear than the normal distribution, each experiment is independent, and the result cannot be a negative number as in the normal distribution.  

The lognormal CDF is created by first calculating the mean, $\mu$, and standard deviation, $\sigma$, of the $\log (\xi)$.  The lognormal distribution with a mean, $m$, and variance, $v$, has parameters
\begin{equation}
\mu = \log({m^2} / {\sqrt{v + m^2}})
\end{equation}
\begin{equation}
\sigma = \sqrt{\log(v/m^2 +1)}
\end{equation}
where $\mu$ and $\sigma$ are the mean and standard deviation.  The lognormal CDF is calculated by 
\begin{equation}
p = F(x|\mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}}\int_{0}^x \frac{\textrm{e}^{\frac{-(\ln(t) - \mu)^2}{2\sigma^2}}}{t}\mathrm d t
\end{equation}
in which all variables have been defined.  Using the parameters that define the data $\xi$, $t$ is a vector from 0 to 4 because those are the boundaries of the current data set of $\xi$.  The curve fit to the data and the data are plotted in Fig.~\ref{fig:prob} which shows the probability that the foundation is more shallow than the depth predicted by IP as a function of the ratio $\xi$.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering
\begin{picture}(18.89,11.75)
\put(1.2,0){\includegraphics[width=0.85\textwidth]{figures/poe}}
\put(3.2,0.75){0}
\put(4.5,0.75){0.5}
\put(6.25,0.75){1}
\put(7.6,0.75){1.5}
\put(9.4,0.75){2}
\put(10.75,0.75){2.5}
\put(12.5,0.75){3}
\put(13.85,0.75){3.5}
\put(15.6,0.75){4}
\put(2.95,1.15){0}
\put(2.6,2.15){0.1}
\put(2.6,3.125){0.2}
\put(2.6,4.1){0.3}
\put(2.6,5.075){0.4}
\put(2.6,6.05){0.5}
\put(2.6,7.025){0.6}
\put(2.6,8){0.7}
\put(2.6,8.975){0.8}
\put(2.6,9.95){0.9}
\put(2.95,10.925){1}
\put(1.25,1.9){\rotatebox{90}{Probability that the foundation is more}}
\put(1.75,1.75){\rotatebox{90}{ shallow than the depth predicted by IP}}
\put(7.925,0){$\xi = \frac{D_{\textrm{actual}}}{D{\textrm{predicted}}}$}
\end{picture}
\caption{Full probability of non-exceedance curve}
\label{fig:prob}
\end{figure}

Fig.~\ref{fig:prob} is used to provide a level of confidence with the inversion output after post processing.  For example, if the predicted depth is 8 m from the post processing of the IP data previously described and if 8 m is taken as the actual depth, $\xi$ is 1.  Fig.~\ref{fig:prob} indicates that there is 39.01 \% probability that the actual foundation is embedded less than 8 m.  This curve can be used, therefore, to adjust the predictions made by IP for an acceptable probability of failure, where failure is over predicting the foundation depth because this could be hazardous when accounting for scour.  Fig.~\ref{fig:problog} is plotted on a semi-log scale so probabilities of failure that geotechnical engineers are more comfortable with can be seen.  A probability of $10^{-6}$, for example, is a one in a million chance that the depth predicted by IP is more shallow than the actual depth of the foundation.  Reducing an IP prediction by this much may be necessary on a major interstate with a high daily traffic.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering
\begin{picture}(18.89,11.75)
\put(1.2,0){\includegraphics[width=0.85\textwidth]{figures/poelog}}
\put(3.2,0.75){0}
\put(5.425,0.75){0.5}
\put(8.15,0.75){1}
\put(10.425,0.75){1.5}
\put(13.15,0.75){2}
\put(15.4,0.75){2.5}
\put(2.3,1.2){$10^{-6}$}
\put(2.3,2.9){10$^{-5}$}
\put(2.3,4.5){10$^{-4}$}
\put(2.3,6.1){10$^{-3}$}
\put(2.3,7.8){10$^{-2}$}
\put(2.6,9.4){0.1}
\put(2.95,11){1}
\put(1.05,1.9){\rotatebox{90}{Probability that the foundation is more}}
\put(1.55,1.75){\rotatebox{90}{ shallow than the depth predicted by IP}}
\put(7.925,0){$\xi = \frac{D_{\textrm{actual}}}{D{\textrm{predicted}}}$}
\end{picture}
\caption{Probability of non-exceedance curve in log scale}
\label{fig:problog}
\end{figure}

\newpage
\section{Reliability Summary}
The post processing analysis that has been created for IP surveys on bridge foundations is explained in this chapter and shown to be a reliable decision process for estimating the depth of the foundations.  This approach can be used for data with RMS above the acceptable range for ERI and IP surveys, but high RMS data should still be interpreted with caution as it is a measure of the goodness of fit of the numerical model to the measured data.  

This post processing criteria are then used for creating a probability of non-exceedence curve showing the probability that the actual depth of the foundation is less than the depth predicted by IP.  Using probability of non-exceedance curve the depth predicted by IP can be reduced or increased depending on the acceptable level of confidence for the bridge that is being tested.  A major interstate, for example, may require a very low probability that the actual depth is less than that predicted by IP, so the predicted depth is reduced by a determined factor.  The predicted depth may also need to be reduced for a lower probability that the actual depth is less than that predicted by IP on scour critical bridges.

The recommendations for analyzing an unknown foundation are as follows:
\begin{enumerate}
\item Conduct an ERI/IP survey and process the data
\item Estimate D$_{\textrm{pred}}$ using the post processing algorithm
\item Decide on the target probability of non-exceedance for the infrastructure considered
\item Read $\xi (p) $ from the probability of non-exeeedance curve
\item Write $D_{\textrm{actual}} = \xi (p) D_{\textrm{pred}}$
\item State the results as, ``there is a (target probability value) probability that the foundation is at least $\xi (p) D_{\textrm{pred}}$''
\end{enumerate}
