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\chapter{\uppercase{Forward Models}}
\label{chap:Model}

\section{Introduction}
\noindent Prior to field experimentation forward modeling is done to identify some limitations of the methods relative to field geometry such as electrode spacing and the spacing between multiple foundations.  Forward modeling is defined as ``the process of predicted the data on the basis of the known distribution of model parameter, electrode configuration, and model'' \citep{agi}.  Forward models are useful for planning induced polarization (IP) and electrical resistivity (ERI) surveys but the results should be interpreted with caution.  They are very simplified from field conditions.  Despite this, these models are very useful for directly comparing some field conditions with requiring multiple foundations in situ without different spacing and sized foundations.
\subsection{Model Background}
\noindent Prior to this research, IP was not used for bridge foundations.  Therefore, before a parametric study is conducted, field tests are necessary to obtain the value of chargeability for a foundation.  Reinforced concrete foundations are exclusively modeled.    The test is conducted at one of the National Geotechnical Experimentation Sites (NGES).  This NGES is located on the Texas A\&M Riverside campus and was sponsored by the National Science Foundation and the Federal Highway Administration (FHWA) in 1992.  This site contains a ``sand site'', shown below, which consists of five spread footings and five reaction shafts.  There are two 3 x 3 x 1.2 m$^3$ footings, one 2.5 x 2.5 x 1.2 m$^3$ footing, one 1.5 x 1.5 x 1.2 m$^3$ footing, and one 1 x 1 x 1.2 m$^3$ footing along with five drilled straight shaft foundations varying in length from 10.4 m to 15.8 m.  Each straight shaft foundation is 0.91 m in diameter.  One of the drilled shafts and three of the spread footings are shown in Fig.~\ref{fig:pile} and Fig.~\ref{fig:sand}.

\begin{figure}[!ht]
\centering 
\begin{subfigure}[Deep foundation]
{\includegraphics[width=70mm, height = 60 mm]{figures/pile}\label{fig:pile}}
\end{subfigure}
~~
\begin{subfigure}[NGES sand site]
{\includegraphics[width=70mm, height = 60 mm]{figures/sand}\label{fig:sand}}
\end{subfigure}
\caption{Foundations at Texas A\&M Riverside Campus} \label{fig:site}
\end{figure}
The general soil layering at the sand site consists of a medium dense tan silty fine sand to 3.5 m below the surface, medium dense silty sand with clay and gravel from 3.5 to 7 m, medium dense silty sand to sandy clay with gravel from 7 to 11 m and very hard dark gray clay from 11 to 33 m.  The water table at the sand site is located 4.9 m below the surface \citep{Gib}.  Several trial experiments are conducted at the sand site to ensure that reinforced concrete does, in fact, possess a measurable chargeability.  From these field tests, it is determined that the chargeability of reinforced concrete is 80 -- 100 mV/V when the current injection time is two seconds, measurement delay is 200 ms, and the measurement window width is 200 ms.  Caution should be taken when chargeability values are compared, it is a function of all these parameters as well as the amount of current that one can get into the subsurface.  The assumed model parameters used in this chapter are listed in Table~\ref{tab:model}.  

\begin{table}[!ht]
\centering
\caption{Forward Model Parameters}
\begin{tabular}{|l|c|}
\hline
Parameter & Assumed Value \\
\hline
ERI Soil & 50 $\Omega$m\\
ERI Foundation & 100 $\Omega$m \\
IP Soil & 60 mV/V\\
IP Foundation & 100 mV/V\\
\hline
\end{tabular}
\label{tab:model}
\end{table}

Note that these model parameters are assumed from the control test in Fig.~\ref{fig:control}.  The range of ERI data for different soil types and for reinforced concrete is well documented and the response from the control test fall within the typical ranges.  The inverted resistivity distribution around the foundation is approximately 120 $\Omega$m.  The assumed ERI response of the foundation is taken as 100 $\Omega$m.  IP is not an intrinsic property therefore the IP model parameters are assumed from the control test.  The control test on the 1 x 1 x 1.2 $m^3$ footing is shown in Fig.~\ref{fig:control}.  A detailed explanation of the inversion process that yields the images in Fig.~\ref{fig:control} is in Section~\ref{sec:inversions}.
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/controlb2}}
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{Control test at the NGES:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:control}
\end{figure}

The control test is a 13 m long array with electrodes spaced at 1 m.  The foundation is located between the 7th and 8th electrodes as drawn on the image.  The footing is founded 0.76 m in the sand \citep{Gib}.  The SS electrodes and NP electrodes are parallel to each other with 0.3 m between the two lines.  The inverted resistivity section converges within eight iterations with an RMS of 4.38 \%.  An RMS of less than 5\% is considered good quality data.  A detailed description of the inversion process is in Section~\ref{sec:inversions} along with a definition of the RMS shown in the inversions.  The IP data is only inverted once with an RMS of 1.21 mV/V.  No definitive foundation is seen on the inverted resistivity section, most likely because there is not a large contrast between the soil below and the foundation at this site.  The inverted IP section shows a strong IP response within the boundaries of the foundation.  This test is the data from which the model parameters are assumed with the exception of the ERI background.  The assumed soil background for the ERI models is 50 $\Omega$m.  This allows for a greater contrast between the soil and the foundation and is in the range of resistivity for sand (1--10,000 $\Omega$m).

\section{The Inversion Process}
\label{sec:inversions}
\noindent The inversions shown in this chapter use the commercial software AGI EarthImager 2D\texttrademark.  EarthImager 2D\texttrademark is capable of creating synthetic data (the forward model) from a user assumed subsurface resistivity and chargeability distribution and a command file.  Next the inverse simulation is created from the forward model to reconstruct the subsurface resistivity and IP distributions.  Fig.~\ref{fig:model} shows the three step process for an ERI survey: creating the assumed subsurface model, the forward model, and the inverse simulation.  ERI and IP Survey Planner simulations are processed simultaneously.  
\begin{figure}[!ht] 
\setlength{\unitlength}{0.8cm}
\begin{picture}(18.89,15)
\put(0,0){\includegraphics[width=\textwidth]{figures/planner}}
\put(0,13){(a)}
\put(0,8.25){(b)}
\put(0,4){(c)}
\end{picture}
\centering 
\caption{Example of survey planner: (a) 1 x 1 m$^2$ model; (b) calculated apparent resistivity pseudosection; (c) inverted resistivity section \label{fig:model}} \end{figure} 

Fig.~\ref{fig:model} shows the three step process for a 1 x 1 m$^2$ foundation with an assumed soil bulk resistivity of 50 $\Omega$m and a foundation of 100 $\Omega$m.   The synthetic data points are represented by the black dots in the middle image of Fig.~\ref{fig:model}.  The plotting depth and location for each of these data points are approximated and displayed on the resistivity pseudosection.  As shown in Fig.~\ref{fig:pseudo}, the measured (or in this case, calculated) apparent resistivity associated with a pair of current electrodes and a pair of potential electrodes is plotted at the intersection of two lines originating from the middle of the electrode pairs that are rotated 45$^\circ$ from the surface.

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{figures/plottingdepth}
\caption{Dipole-dipole array apparent resistivity pseudosection \citep{Briaud}}
\label{fig:pseudo}
\end{figure}

The point plotted is where the two 45$^\circ$ lines intersect.  the actual depth of penetration also depends on the subsurface resistivity distribution.  For example, a conductive overburden will decrease the penetrating depth dramatically.  This is an approximate representation of the bulk resistivity of the subsurface.  An apparent resistivity measurement should be regarded as a weighted sum of the resistivity distribution of the entire earth.  The pseudosection is displayed by color contouring the apparent resistivity data.  Also, note that the shape of the apparent resistivity shaded contours is triangular.  The triangular shape is related to the shape of Fig.~\ref{fig:pseudo}, showing that the measured data has the greatest sensitivity to the ground bound by the triangular region in Fig.~\ref{fig:pseudo}.  The outer blank region in (b) of Fig.~\ref{fig:model} is the regions where the data is minimally affected.

Once the synthetic data are generated a 2.5D forward model is created.  The model is considered 2.5D because the model is 2D but the electrical field created to gather the data is 3D.   The governing 2D partial differential equation is solved to get the forward solution:
\begin{equation}
\frac{\partial}{\partial x}\left ( \sigma \frac{\partial V}{\partial x} \right) + \frac{\partial}{\partial z}\left ( \sigma \frac{\partial V}{\partial z} \right) - k^2 \sigma V = -I \cdot \partial\left ( x \right ) \cdot \partial(z),
\end{equation}
\noindent where $\sigma$ is electrical conductivity (the inverse of electrical resistivity) as a function of $x$ and $z$, $V$ is scalar electric potential, $k$ is the wave number, and $I$ is electric current.  Once the forward model is created, the elements in the model are compared to the measured data or the synthetic data.  The goodness of fit of the calculated data is characterized by the Root Mean Square (RMS) error,
\begin{equation}
RMS = \sqrt{\frac{\displaystyle \sum_{i=1}^N \left ( \frac{ d_i^{\textrm{Pred}} - d_i^{\textrm{Meas}}}{d_i^{\textrm{Meas}}} \right)^2}{N}} \times 100\% ,
\end{equation} 
\noindent where $d_i^{\textrm{Pred}}$ is the calculated data, $d_i^{\textrm{Meas}}$ is the measured data (or synthetic when Survey Planner is used), and $N$ is the total number of data points.  The RMS is the average misfit over all the data points.  As a rule of thumb, and RMS less than 3\% is considered an excellent fit, 3--5\% is a good fit, 6--10\% is an OK fit but should be interpreted with caution, and greater than 10\% is a low quality model for the gathered data.  After calculating the RMS, the inverted resistivity section is updated with the latest forward model pseudosection.  The inverted section transforms the earths response to the user defined injection of an electric field to the true resistivity subsurface distribution.  The depth maximum depth of the inverted resistivity section is determined by the median depth defined by Edwards \citep{Ed77} and a user defined depth factor.  The median depth is discussed in Section~\ref{sec:depth}.  

The model in Fig.~\ref{fig:model} converges to an RMS of 4.37\% in two iterations.  Thus the calculated data are compared to the synthetic data, the RMS is higher than the stop criteria of and RMS of 5\% set before the process is started, the calculated data are updated for the second inversion after which the RMS is lower than the stop criteria RMS so the inverse simulation is generated.  The IP simulation is very similar except the forward model is not calculated based on the measured or synthetic IP data, it is calculated using the forward model from the resistivity data.  The resistivity model is used to significantly reduce the numerical errors in the IP inversion process.  This is why, upon inspection, the RMS for the inverted IP section has units of mV/V.  Additionally, in Chapter~\ref{chap:Experiment} it is shown that ERI is not successful at imaging slender foundations like those typically used in the Bryan TxDOT District.  ERI is included for these surveys to provide additional information pertaining to the quality of the IP model.  IP data that are correlated with ERI data with a low RMS are better than IP data correlated with ERI data with a high RMS, even if the IP RMS are less than the ideal 3\% previously mentioned.

\section{Numerical Models}
\noindent Using the model parameters in Table~\ref{tab:model}, 48 models are created showing both IP and ERI data.  Each model uses 28 electrodes, the number of electrodes available at TAMU for an ERI/IP survey using NP electrodes.  The dipole-dipole array is used exclusively in this study.  The dipole-dipole array yields the highest quality data in terms of low levels of noise for IP surveys (proven in Chapter~\ref{chap:Experiment}).  The foundation and survey parameters that are varied in this study are the depth of the foundation, the width of the foundation, the number of foundations, and the electrode spacing totaling 24 models.  Each foundation is fictitious and intended for a parametric study only.

\subsection{Single Foundations}
\noindent The models discussed initially are single foundation models.  The effects of multiple foundations within one foundation width of each other are addressed later.

\subsubsection{Shallow Foundations}
\noindent The shallow foundations are each constructed with the model parameters from Table~\ref{tab:model} and using the dipole-dipole array.  The term shallow indicates that each foundation is embedded 1 m.  The electrode spacing is 0.5 m, 1 m, or 2 m and the width of the foundation is 1m or 5m.  Every combination is presented and the dimensions of the model foundation are drawn over each inverse simulation.  ERI and IP are both presented.  The apparent resistivity color scale on the right side of each image is broken into 20 segments from 30 $\Omega$m to 150 $\Omega$m spaced logarithmically.  The chargeability color scale is also broken into 20 segments from 35 mV/V to 120 mV/V spaced logarithmically.  A linear color scale can also be used, but plotting the scale logarithmically allows the images to be more clear when there are large differences between the maximum and minimum values. These scales are kept consistent so the inverted sections can be compared to each other as the foundation dimensions and electrode spacings are changed.
\clearpage
Fig.~\ref{fig:S15s} are inverted forward models of a 1 x 1 m$^2$ foundation with 28 electrodes spaced every 0.5 m yielding a 13.5 m long array.  The depth of penetration in the ideal numerical model for an array of this length is 3.27 m.  The inverted resistivity section converges in eight iterations with an RMS of 4.66 \%.  The bulk resistivity response of the foundation is approximately contained within the boundaries of the shallow foundation.  The IP response converges with an RMS of 1.30 mV/V and a high chargeability is seen covering almost the entire foundation dimensions.
\begin{figure}[ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/S15s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 1 m$^2$ foundation with 0.5 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:S15s}
\end{figure}

\clearpage
Fig.~\ref{fig:S11s} is the numerical model of the field test from Fig.~\ref{fig:control} except the background bulk electrical resistivity of the soil is assumed to be much lower than the field test and 28 electrodes are used instead of 14.  The lower resistivity is used to increase the contrast between the soil and foundation so more can be learned from the inversions.  This assumption is reasonable because the lower resistivity of sand is still within the range of sands which can be from 1 - 1,000 $\Omega$m \citep{Mil}.  With 28 electrodes spaced every meter the length of the array is 27 m and the depth of penetration is 6.5 m.  The inverted resistivity section converges to an RMS or 4.88 \% in two iterations.  
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/S11s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 1 m$^2$ foundation with 1 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:S11s}
\end{figure}
A small zone of resistivity of around 100 $\Omega$m is seen at the top of the foundation in the center of the block surrounded by a zone of resistivity higher than the soil approximately within the dimensions of the model.  This stronger resistivity response shows the importance of a strong contrast between the soil and the foundation when compared to Fig.~\ref{fig:control}.  The inverted IP section looks similar to Fig.~\ref{fig:control} with a strong chargeability response around 120 mV/V within the dimensions of the foundation.

Fig.~\ref{fig:S12s} is at 54 m long array with a depth of penetration of 13.1 m and electrodes spaced every 2 m.  Upon inspection, the inverted resistivity background has more anomalies than the prior inversions.  This may be because the size of the foundation related to the model is quite small causing more mathematical anomalies.  Related to this, to response is much lower for both the ERI and IP sections.  
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/S12s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 1 m$^2$ foundation with 2 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:S12s}
\end{figure}\\
The inverted resistivity section converges in eight iterations to an RMS of 4.49\% and the inverted IP section converges to an RMS of 1.44 mV/V.  Both sections show a response within the dimensions of the foundation higher than the background foundation.  The ERI response is over 70 $\Omega$m and the IP response is over 80 mV/V.  This model reflects the rule of thumb that the electrode spacing should be equal to or less than the width of the anomaly to be imaged.  The next three models are still embedded 1 m but the width of each foundation is now 5 m instead of 1 m.

Fig.~\ref{fig:S55s} is a 5 m wide foundation embedded 1 m with 28 electrodes spaced every 0.5 m.  The resistivity section converges to an RMS of 4.98 \% in eight iterations.  A response similar to that in Fig.~\ref{fig:S15s} is seen with a zone of high resistivity (over 100 $\Omega $m )  within the dimensions of the foundation, but this strong response covers much more of the foundation than before.  
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/S55s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 1 m$^2$ foundation with 0.5 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:S55s}
\end{figure}
The well imaged foundation using ERI in Fig.~\ref{fig:S55s} agrees with the finding that ERI is excellent at imaging large foundations, not long slender piles that are 1 m or less \citep{rung}.  The inverted IP section converges with an RMS of 1.61 mV/V.  Almost the entire foundation is imaged using IP with a localized band of chargeability over 100 mV/V.

Fig.~\ref{fig:S51s} is a 28 electrode array spaced every 1 m totaling 27 m with a depth of penetration of 6.5 m.  The inverted sections look very similar to Fig.~\ref{fig:S55s}, with the same strong response for both images.  The inverted resistivity section converges in eight iterations to an RMS of 4.79 \% and the IP converges with an RMS of 1.37 mV/V.  

\begin{figure}[ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/s51s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 1 m$^2$ foundation with 1 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:S51s}
\end{figure}

This model shows that for a large, shallow foundation there is no advantage of longer electrode spacing because the depth of penetration is not important for data quality.  This is an important finding if the area surrounding the foundation is limited and a longer array can not be used.

The final shallow foundation model, Fig.~\ref{fig:S52s} is similar to the array with 2 m spacing in Fig.~\ref{fig:S12s} in that the foundation is imaged by both methods, but the response is much lower than in the previous arrays. 
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/S52s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 1 m$^2$ foundation with 2 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:S52s}
\end{figure}
Also, again more mathematical anomalies are seen in the model back  ground than from more shallow models.  For both of these cases longer arrays are not necessary or appropriate when a larger depth of penetration is not necessary.

\subsubsection{Deep Foundations}
\noindent The same models presented for shallow foundations are regenerated but now they are deeper.  Each of the models in this section are embedded 5 m instead of 1 m.  

Fig.~\ref{fig:D15s} is a 28 electrode array spaced every 0.5 m.  The depth of penetration for a 0.5 m array is only 3.27 m so the model is stretched in order to use a 1 x 5 m$^2$ foundation.  Looking at the inversions, the backgrounds are full of mathematical anomalies that are found when the model is forced to be stretched deeper than the depth of penetration.  
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/D15s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 5 m$^2$ foundation with 0.5 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:D15s}
\end{figure}
For this model, a depth factor of two is used instead of 1.1 as in the other models.  Stretching the model does not help to increase the depth to which the ERI and IP sections are capable of imaging.    Both show a higher response in the expected ranges to 1.5 m beneath the surface.  The resistivity zone of over 100 $\Omega$m extends to 0.75 m and the zone of over 70 $\Omega$m extends to 1.5 m.  Similarly, the zones of chargeability around 120 mV/V only extend to 1 m beneath the surface.  A zone of over 80 mV/V extends down to 1.5 m after which the signal dissipates.  This highlights the importance of gathering data deeper than the foundation depth.

Fig.~\ref{fig:D11s} is a 27 m long array that extends 6.5 m in to the subsurface.  
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/D11s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 5 m$^2$ foundation with 1 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:D11s}
\end{figure}
The ERI section has a zone of resistivity above 100 $\Omega$m that extends 0.8 m down the pile and a zone above 70 $\Omega$m that extends 3.3 m into the subsurface.  The inverted resistivity section converges to an RMS of 4.57\% in eight iterations.  The inverted IP section converges to an RMS of 1.29 mV/V.  The inverted IP section also has a region of chargeability of over 100 mV/V at the top of the pile and chargeability over 80 mV/V down the length of the pile to 3.3 m.  Both of the inverted sections are not capable of imaging the full length of the pile and both contain anomalies in the background.

Fig.~\ref{fig:D12s} is a 54 m long array with a depth of penetration of 13.1 m.  A zone of high resistivity (80--90 $\Omega$m) is at the top of th pile extending to only 0.25 m.  The rest of the pile shows a bulk resistivity of around 60 $\Omega$m, but pockets of around 60 $\Omega$m are also seen beneath the pile and around the background.  The inverted IP section shows a response that penetrates down 75\% of the pile.  The zone of a high IP response of around 99 mV/V is at the top of the pile that dissipates down the length of the pile to 70 mV/V.  This inversion again shows the resolution issues when the electrode spacing is increased, these inversions are not as clear as the shallow foundations but they are also not as clear as the models when the electrode spacing is equal to the pile diameter.
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/D12s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 5 m$^2$ foundation with 2 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:D12s}
\end{figure}

\newpage Fig.~\ref{fig:D55s} is again the 0.5 m spaced 28 electrode array with a 5 x 5 m$^2$ foundation that has a depth factor of two to stretch the mesh in order to insert the model.  Again, this is not recommended for data quality.  Like the 5 m long shallow foundations, both the ERI and IP inversions image more of the foundation. 
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/D55s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 5 m$^2$ foundation with 0.5 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:D55s}
\end{figure}
 The depth of penetration of the strong signal from the shallow foundations is much deeper for the large foundations as well.  The inverted resistivity section has an RMS of 4.85\% that converges in seven iterations.  The ERI inversion shows a strong apparent resistivity that extends to 3.75 m of over 100 $\Omega$m.  Similarly, the IP inversion penetrates to 4 m with a zone of over 90 mV/V with an RMS of 1.68 mV/V.  Again, the IP is capable of imaging deeper than the ERI inversion with a stronger response. 
 
\newpage Fig.~\ref{fig:D51s} is the 5 x 5 m$^2$ foundation with electrodes spaced 1 m.  The inverted resistivity section converges in eight iterations to an RMS of 4.49\% and the inverted IP section converges with an RMS of 1.45 mV/V.  Again, the quality of the ERI and IP response of the lager foundation is much higher than the narrow foundation that is the same depth with the same electrode spacing in Fig.~\ref{fig:D11s}, particularly when comparing the inverted resistivity sections.  Both responses are not as high as Fig.~\ref{fig:D55s}, but for foundation imaging purposes the difference is negligible.  
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/D51s2b}} 
 \put(0,10){(a)}
 \put(0,4.6){(b)}
\end{picture}
\caption{5 x 5 m$^2$ foundation with 1 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:D51s}
\end{figure}

\newpage The last single foundation model shows the advantage of using a larger electrode spacing when a large foundation is imaged.  Fig.~\ref{fig:D52s} is a 5 x 5 m$^2$ with electrodes spaced every 2 m.   
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/D52s2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 5 m$^2$ foundation with 2 m electrode spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:D52s}
\end{figure}
The inverted resistivity section has a strong apparent resistivity (approximately 150 $\Omega$m) in the center of the foundation with a surrounding zone of over 70 $\Omega$m nearly filling the dimensions of the foundation.   Similarly, the IP section foundation is nearly filled with a chargeability of over 100 mV/V in the boundaries of the foundation with the remaining space filled with a chargeability of over 80 mV/V.  For large, deep foundations, an electrode spacing that gathers deeper data while still being half the width of the foundation is advantageous. 

\newpage
\subsection{Multiple Foundations}
\noindent The single foundation inversions are used to narrow down the necessary inversions for multiple foundations.  For both shallow and deep foundations, the 1 m electrode spacing yields the highest quality inversions.  For the multiple foundation study the same four foundations are used (1 x 1 m$^2$, 5 x 1 m$^2$, 1 x 5 m$^2$, and 5 x 5 m$^2$ )  but the spacing between the multiple foundations is changed.  Each foundation dimension is used in conjunction with different spacing between the foundations: 1 foundation width and 2 foundation widths apart.
\subsubsection{Shallow Foundation}
\noindent Fig.~\ref{fig:s11mb} represents five foundations with a 1 m gap between each foundation.  Comparing this model to Fig.~\ref{fig:S11s}, the area of the response for each foundation is the same as it is for the single foundation model, i.e. a response higher than the model background within the boundaries of the foundation model.  The influence of multiple foundations is apparent when comparing the value of the resistivity and IP response.  In the single foundation model, there is a zone of approximately 100 $\Omega$m in the center of the foundation and a zone of around 110 mV/V in the same location.  The overall response is lower when multiple foundations are in the model, but for shallow foundations this reduction in terms of identifying the foundation is negligible.  

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/s11m2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 1 m$^2$ foundation with one foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:s11mb}
\end{figure}
\newpage
Fig.~\ref{fig:s12mb} is the same model as Fig.~\ref{fig:s11mb} but the spacing between the foundations is 2 m instead of 1 m.  The affect of moving the foundations farther apart is seen when comparing the inverted resistivity sections, but it is not as noticeable in the IP sections.  The inverted resistivity section in Fig.~\ref{fig:s12mb} has zones of higher resistivity on the foundations than Fig.~\ref{fig:s11mb}, particularly on the exterior foundations.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/s12m2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 1 m$^2$ foundation with two foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:s12mb}
\end{figure}


Fig.~\ref{fig:s51mb} is three 5 m long, 1 m deep foundations spaced 5 m apart.  The inverted resistivity section converges in four iterations to an RMS of 4.7 \%.  The foundations in the resistivity section are identified by zones of resistivity of over 70 $\Omega$m with thin layers of resistivity of over 100 $\Omega$m at the bottom of the foundations.  Comparing this to Fig.~\ref{fig:S51s}, the resistivity response within the foundation is lower.  In the single foundation model, the resistivity of over 100 $\Omega$m covers most of the foundation.  Also, the multiple foundation model has additional zones of resistivity over 70 $\Omega$m beneath the exterior foundations.  The IP models, on the other hand, have negligible differences between the single foundation and multiple foundations.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/s51m2b}} 
 \put(0,10){(a)}
 \put(0,4.6){(b)}
\end{picture}
\caption{5 x 1 m$^2$ foundation with one foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:s51mb}
\end{figure}

In Fig.~\ref{fig:s52mb} the distance between the foundations is increased to 10 m which means that with the foundation centered the exterior foundation models are only 1 m in the model due to limited space.  Even so, there is little difference between the centered foundation in Fig.~\ref{fig:d51mb} and Fig.~\ref{fig:d52mb}.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/s52m2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 1 m$^2$ foundation with two foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:s52mb}
\end{figure}
\clearpage
\subsubsection{Deep Foundations}
\noindent Fig.~\ref{fig:d11mb} is the same foundation as Fig.~\ref{fig:D11s} but with 5 foundations spaced every 1 m.  The effects of multiple foundations are noticeable in both the ERI and IP sections.  The highest resistivity on the multiple foundation model is around 100 $\Omega$m compared to around 120 $\Omega$m on the single foundation.  Also, the single foundation has a resistivity of around 80 $\Omega$m to 3.3 m down the pile.  This depth of higher resistivity is still seen on Fig.~\ref{fig:d11mb} but not on all the piles and it is also spread across and in between the piles, no longer contained within the foundation model dimensions.  The IP response stays on the foundations instead of spreading across and in between them but again the value of the response is much lower than the single foundation.  Also, the depth of penetration for the higher IP response is only to 1.5 m with multiple foundations versus 3.3 m for a single foundation.
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/d11mb}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 5 m$^2$ foundation with one foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:d11mb}
\end{figure}

When the foundations from Fig.~\ref{fig:d11mb} are spaced twice as far apart (every 2 m) as in Fig.~\ref{fig:d12mb} the spread out effect of the resistivity response is no longer seen but the depth of penetration of the 80 $\Omega$m bulk resistivity of the piles is varied.  The far left pile and the next pile have a resistivity of 80 $\Omega$m down to around 2.5 m but the three remaining piles only have this resistivity to less than 1.6 m.  There are also additional background anomalies in this model.  Again, the IP foundation responses are confined to the dimensions of the foundations but the response is still not as high as it is in Fig.~\ref{fig:D11s}.  The high response also appears to dissipate from the foundations when comparing them left to right.

\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/d12m2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{1 x 5 m$^2$ foundation with two foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:d12mb}
\end{figure}

The foundations in Fig.~\ref{fig:d51mb} are 5 x 5 m$^2$ spaced 5 m apart.  Comparing this to Fig.~\ref{fig:D51s}, the center foundations do not have as much of a high response in terms of the area of the foundation covered as the single foundations.  The exterior foundations show significantly less of a high response and the green zones (around 80 $\Omega$m) that have previously been constrained to the area inside the foundation extend well beyond the foundations of the exterior blocks causing the bottom of the foundation to be unidentifiable.  Additionally there is a zone of high resistivity on the left corner of the far left foundation that makes this foundation even more difficult to identify.
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/d51m2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 5 m$^2$ foundation with 1 foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:d51mb}
\end{figure}

\newpage The final model is again the 5 x 5 m$^2$ foundation but now with another 5 m deep foundation 10 m to the left and right of the center model.  Increases the distance between the large foundations greatly reduces the background mathematical anomaly and enhances the quality of both the ERI and IP sections.  This may be attributed to the fact that there is only enough space in Fig.~\ref{fig:d52mb} for 1 m width for the extra two foundations.  Despite this, the model shows that the researcher can successfully image a large foundation if there is at least twice the width of that foundation before the next one.  Note that this model is a very large foundation, specifically for bridges, so it is unlikely that multiple of these be spaced close together.
\begin{figure}[!ht]
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,11)
\put(0,0){\includegraphics[width=\textwidth]{figures/d52m2b}} 
\put(0,10){(a)}
\put(0,4.6){(b)}
\end{picture}
\caption{5 x 5 m$^2$ foundation with two foundation width spacing:  (a) inverted resistivity section; (b) inverted IP section}
\label{fig:d52mb}
\end{figure}

\clearpage
\section{Forward Model Summary}
\noindent The conclusions from these models are used for planning field surveys using ERI and IP for bridge foundations.  In these models the effects of electrode spacing relative to the foundation dimensions and of multiple foundations on the inversion quality for identifying the depth of bridge foundations are investigated.  Several key field setup procedures when testing a bridge foundation have been identified.  

\begin{enumerate}
\item Only test one foundation in an array
\item Electrode spacing should be equal to or less than the width of the foundation
\item Depth of penetration of a survey should be deeper than the embedment depth of the foundation
\end{enumerate}

Note that the depth of penetration is related to the length of the array.  The farther the pair of current electrodes are from the voltage potential electrodes, the deeper data may be gathered.  The longest the array can be using the Texas A\&M equipment is 54 m, the same as the longest forward model array.  The maximum length of the array is limited because only 28 NP electrodes are available at this time and the spacing between the cable take-outs on the equipment cables is two meters, yielding a maximum of 54 m long.  If it is determined that the depth of penetration is not as deep as a truly unknown foundation, more cables and electrodes should be acquired so deeper data can be gathered.

Additionally, while these forward models are useful tools for planning a field investigation they should be interpreted with caution.  Each model is made with an ideal soil background that is homogeneous.  If the soil surrounding a foundation in the field has a highly conductive layer within the stratification, for example, the depth of penetration may be affected as well as the results.  From the control test it is observed that the contrast between the soil and the foundation is not great enough for the foundation to be imaged; this is another uncontrollable factor in the field that will affect the inversion.  Most importantly, IP is not an intrinsic property.  The IP value used for these models is assumed from the control test which is run using a 2 second delay time with 200 ms time windows.  If the delay time or the window time are changed the IP values should not be compared to these models. 






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