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%%%                           SECTION II
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\chapter{\uppercase {Literature Review}}

\label{chap:litreview} 

\section{Introduction}
\noindent The unknown foundations problem in the United States has been studied for decades. Over time, many deterministic methods have been developed for estimating the depth of bridge foundations. In 1996 a National Cooperative Highway Research Program (NCHRP) project was granted to an Engineering firm to investigate the current field testing of bridge foundations \citep{Ols98}.  The testing methods discussed in this report are still being used today, along with a few that have been established after the NCHRP report. Unfortunately, while it has been known for several years that there is an unknown foundation issue and despite the fact that many states are investigating unknown foundations and updating the National Bridge Inventory (NBI), the number of unknown foundations is still growing. The problem still needs to be addressed after bridge construction, when as-built construction plans must be submitted to the NBI and to the State Transportation Department.

\section{Existing Experimental Methods} 
\noindent There are over 9,000 bridges in Texas alone with unknown foundations. The bridges spanning water are of critical importance due to the risk of potential scour. States nationwide are identifying their unknown foundations to update bridge databases; however, current techniques are typically invasive and costly. Many reports with extensive research have been published on current surface and subsurface tests \citep{{Mas98}, {Mercado}, {Ols98}, {Ols05}, {RW08}}. Some of the capabilities and limitations of these methods are presented in this section. 

While there are many components of a foundation that are important for its stability, the depth of embedment is the most pertinent when evaluating the bridge for scour potential and safety. The most effective method for successfully identifying the type and depth of a foundation is conventional excavation; exposing a portion of the bridge foundation for visual conformation. As can be expected, this is not typically done. First, it is extremely expensive, especially when the bridge is over water, requiring the area to be sheeted and dewatered. If the bridge is large, excavations may also be difficult due to a lack of access, and if the bridge is over water, costly barges may be necessary. Also, excavation can be very dangerous and difficult for workers. Precautions have to be taken so that the foundation is not undermined and the bridge does not collapse.  These reasons as well as others have emphasized the need for nondestructive testing methods.
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\subsection{Surface Methods} 
\noindent In general, surface methods are preferred over subsurface methods if possible. They are more economical, without the upfront cost of a borehole; however, they are often dependent on access to bridge foundations. The simplest nondestructive testing method is a driving rod test.  
\subsubsection{Driving Rod Test}
\noindent The driving rod test is used to determine if the foundation is shallow or deep.  \cite{RW08} used a 1.27 cm diameter steel rod driven with a 6.8 kg hammer
dropped 0.61 m near the bridge foundation. Blow counts per 0.3 m are recorded until the rod cannot be driven further. This test can provide the depth to weathered rock, dense sand layers, or soft layers. Shallow foundations are presumed if the results indicate a strong material or rock near the surface. 
\subsubsection{Sonic Echo Test}
\noindent The sonic echo test was developed to evaluate the integrity and length of newly constructed piles. For new piles, the source and receiver are placed on the top of the pile. The source generates a compression wave of known magnitude.  The velocity of the wave can be estimated with the impulse magnitude. Using the known velocity and the known dimensions of the recently constructed pile, defects in the pile can be detected. Sonic echo can determine the depth of bridge foundations by placing a receiver on the top or side of an exposed pile underneath a bridge. If the top is not exposed, at least two receivers are placed on the side of the structure.  Compression waves are generated with a hand held hammer. The depth of the foundation is determined using the arrival time of the reflected compression wave to the receiver. The method is very effective for columnar substructures. Unfortunately, it is limited because the compression waves generated by the impulse are often diffused into other components, like the pile cap shown in Fig.~\ref{fig:bridge}. Also, has been determined, as a rule of thumb, that when embedded length to diameter ratios are greater than 20:1 or 30:1 in stiffer soils, there will be no identifiable bottom echoes due to excessive damping of the compression wave energy in the sonic echo \citep{Ols98}.
\subsubsection{Bending Wave Test}
\noindent The bending wave test is similar to the sonic echo test; however, it uses flexural (bending) waves instead of compression waves.  Pioneered by Darrin Holt in the early 1990's, bending waves had not yet been investigated for foundation testing prior to this research \citep{holt}.  For a bending wave test, at least two horizontal receivers are placed on the same side of the pile as where it will be impacted, noted by "The Strike" in Fig.~\ref{bend}. 
\begin{figure}[!ht] 
\centering 
{\includegraphics[width=0.5\textwidth]{figures/bend}} 
\caption{Bending wave testing principle \citep{holt} \label{bend}} 
\end{figure} 
Once the bending wave is initiated, the time of arrival of the echoes of the wave is measured. The wave velocity, which is dependent on the material of the foundation and the force of the impact can be calculated.  The depth of the foundation can then be calculated based on the arrival time of the wave. Unlike compressional waves, bending waves are highly dispersive. This dispersive nature sometimes hinders the use of the bending wave method because the impulse
must be a certain amplitude. Another constraint of the bending wave method is the length of the pile that must be exposed above the ground surface for the two transducers. The transducers must be spaced over 1 m apart and there must still be space available over the uppermost transducer for the strike impact. The bending wave test is also only effective for columnar substructures.
\subsubsection{Ultraseismic Test}
\noindent Ultraseismic (US) is another wave propagation test; however, it uses geophysical digital data processing to analyze the propagation of the waves. Unlike the sonic echo test, multiple channels of data are used, so the test requires
access to at least 1.5 m of the surface of the substructure. US was investigated to address the issues that sonic echo and bending wave have on non-columnar foundations. US again uses a hand held hammer to generate the waves, but the arrivals and the reflections of both the compression and bending waves are analyzed using seismic processing. The compression waves are generated by striking the foundation vertically and the bending waves are generated by striking it horizontally. \cite{Ols98} determined that the US method has the broadest applications in determining unknown foundations of all the surface methods investigated the NCHRP study.  
\begin{figure}[!ht] \centering 
{\includegraphics[width=\textwidth]{figures/wave}} 
\caption{Impedance of wave propagation by complex foundations \citep{Ols98} \label{wave}} 
\end{figure}
\noindent While US has proved to be more successful at testing more complex substructures than the previously described methods, the waves are still not able to travel through pile caps or other components on top of piles.
\subsubsection{Dynamic Foundation Response Method}
\noindent Waves impeded by piles caps may lead to false results that indicate shallower foundations than exist, as shown in Fig.~\ref{wave}. Another surface method, the Dynamic Foundation Response Method \citep{Ols98}, uses an impulse hammer with a built in force transducer to excite the bridge in the vertical and horizontal directions along the frame of the substructure as a form of modal analysis. It has been introduced to differentiate between shallow and deep
foundations, as other surface methods are unable to detect piles beneath a pile cap. In the Dynamic Foundation Response Method, the substructure is struck with a known force and the accelerometers measure the resulting vibrations. Using these data, the transfer function is calculated and plots of the transfer function versus the frequency of the structure are created. This is done under the principle that, all other components of the structuring being the same, a
shallow foundation will exhibit a lower resonance frequency than a deep foundation. The researchers have noted, however, that they need more powerful vibrations to excite the bridge to low frequencies comparable to the natural frequencies identified in numerical modeling. They have suggested using other methods to excite the bridge because of this observation. In a later study, \cite{Ols05} again attempted to excite bridges using hand held devices.
Researchers used a 1.36 kg impulse hammer, a 45.4 kg vibrator, and a 5.45 kg impulse sledge hammer; again these did not generate enough energy to excite the bridge.

Ultimately, \cite{Ols05} used the Vibroseis truck to excite the substructure from the bridge deck level because of its ability to generate a low frequency output. While the bridge was excited to
necessary frequencies, modeling the data was not yet successful. Olson attempted to use parameter estimation for finding a solution to different pile cases; however, results were consistently incorrect. Also, parameter estimation was used to solve for the first vertical mode of vibration of the structure to measure damage.  It was concluded that this method is not feasible for estimating foundation damage because the effects of the damage are too small to model.
\subsubsection{Structural Response Method}
\noindent \cite{Mas98} believed that the approaches such as dynamic and seismic testing were too complex, requiring specialized knowledge and equipment that are not readily available making some projects not feasible. It was desired to implement a simple and low-cost approach using a stiffness matrix created from measurements taken during loading from dump trucks.  Axle loads were continuously measured while driving slowly across a bridge.  Because the excitation must go through the deck and all the way down the foundation, much more energy must be generated to be effective. The team generated six models to find stiffness matrices for different foundation types and conducted field tests on two bridges with different known foundations. Unfortunately, the measured stiffness matrices did not show similarities to the models. The team did, however, find that for
both the measured and field tests, the stiffness matrices had distinctly different patterns of coefficients for spread footings versus pile foundations.

\cite{Mas98} hypothesized that each foundation type has a unique pattern of stiffness coefficients. Because the bridge was loaded with a truck that has known properties, they assumed the loading and certain known displacements to generate a stiffness matrix. While they had not determined the coefficients at the time of publishing, they showed that the pattern of the coefficients is different for shallow versus deep foundations, and these differences were similar to those predicted by theory. Similarly, \cite{Suz} used the ratios of a theoretical stiffness to indicate scour on bridge piers. Accelerometers were attached to the top of bridge piers to measure the vertical and horizontal accelerations of a pier noting that a change in the horizontal stiffness occurs when scour removes the soil from one side of a foundation. This unbalance of stiffness is being used as a
monitoring system for bridges over water.
  

\subsection{Subsurface Methods}
\noindent In some bridge foundation predictions it is possible to drill a borehole for different testing.  A brief review of three subsurface methods are now described: parallel seismic, borehole sonic, borehole radar, and the induction method.
\subsubsection{Parallel Seismic Test}
\noindent Another way to determine the depth of an unknown foundation of a bridge is by investigating it using a subsurface method after drilling a borehole. Parallel seismic is one of the subsurface methods that is commonly used. Once the hole is drilled, a hydrophone or geophone is lowered at set intervals as the exposed foundation substructure is impacted with an impulse hammer. 
\begin{figure}[h] 
\centering 
\scalebox{0.6}{\includegraphics{figures/parallel}} \caption{Parallel seismic \citep{Ols98}  \label{parallel}} 
\end{figure} The compression waves that travel down the substructure are refracted into the soil and the arrival of the waves is tracked through the receiver. When the receiver notes an apparent change in the waves, the bottom of the foundation is noted. \cite{Ols98} noted that the parallel seismic method has the broadest application for determining unknown foundations of the ten that were investigated. Parallel seismic can not only be used for identifying the depth of bridge foundations, but it can also determine the depth of the scour zone \citep{Mercado}. 
\subsubsection{Borehole Sonic, Borehole Radar, and the Induction Method Tests}
The borehole sonic test is similar to parallel seismic, measuring the reflection of waves from alongside the structure.  Borehole radar is similar to borehole sonic, but it relies on the dielectric constants that cause reflections instead of differences in stiffness. The induction method, which is used in North Carolina
\citep{RW08}, is performed by lowering an induction sensor in a borehole which creates a magnetic field. The field is interrupted by metal objects, such as rebar in a foundation. This disruption allows for the rebar in the foundation to generate its own magnetic field, which is detected by the induction sensor in the borehole. The received voltages read by the induction sensor are measured at a minimum of every 0.31 m. The depth at which the receiver no longer indicates a voltage reading is the estimated foundation length. This method is highly dependent on metal within the substructure.

\subsection{Existing Methods Summary}
\noindent Of all the methods reviewed, vibration based methods are most common due to the ease of instrumenting the bridge and interpreting the results. Unfortunately, many researchers have noted that it is difficult to excite a bridge to the desired frequency for determining unknown foundations \citep{{Mas98}, {Ols98}, {Ols05}}. The Dynamic Foundation Response Method \citep{Ols98} uses a 5.44 kg hammer with a built in force transducer to excite the bridge in the vertical and horizontal direction along the frame of the substructure. It has been noted, however, that more powerful vibrations are needed to excite the bridge to frequencies comparable to the natural frequencies identified in numerical modeling. The researchers suggested using other methods to excite the bridge.  Other researchers have unsuccessfully tried to excite a bridge using heavy traffic, such as loaded dump trucks \citep{Mas98}.  

Methods are often hindered by the assumptions researchers make based on typical foundation construction practices. Unfortunately, some of the bridges with unknown foundations may not have been built with standard construction methods. For example, a few years ago, a group of engineers were able to go into a partially collapsed salt mine discovered beneath a bridge to investigate the foundation. While beneath the surface, it was discovered that the concrete substructure contained no reinforcing steel.  These methods that depend on certain components of a substructure that are assumed to always be present could hinder the results for cases when they are not included.  Electrical resistivity imaging (ERI) and induced polarization (IP) do not depend on the structure in any way so they are not hindered like most surface methods.  Also, they do no rely on common construction practices for estimating the depth of foundations so adding them to the testing database will complement the issues of many of the current methods.

\section{Electrical Methods}
\noindent Induced polarization (IP) and electrical resistivity imaging (ERI) are reviewed in this section.  The use of IP and ERI for Geotechnical Engineering is not new.  ERI is used for characterizing shallow subsurface ground conditions to ``see'' between bore holes \citep{{Hilt}, {Wise}, {Groves}}.  ERI testing is also used for landfill mapping \citep{Bern}.  Moreover, the electrical resistivity of reinforced concrete is used to evaluate corrosion of the rebar \citep{Su}.  The electrical resistivity of reinforced concrete varies depending on the degree of corrosion in the rebar, the moisture content, and the material composition \citep{Tuuti}.  In conjunction with parallel seismic and sonic echo testing, ERI has been successfully employed for determining the depth of bridge foundations \citep{Hoss}.

IP has also been used to measure the corrosion of reinforcing bars in concrete.  \citet{Hub} placed electrodes on a concrete surface and successfully measured a change in the IP response as corrosion occurred in the reinforcement.  IP is used for archaeological surveys when searching for buried artifacts \citep{Mey} and for distinguishing between coarse and fine-grained materials in landfill explorations \citep{Ler07}.  IP is also used for differentiating between waste and reinforced concrete at the bottom of waste dumps \citep{Bav}.  It is used for aquifer explorations, locating unexploded ordnances and land mines \citep{Pel}, and is increasingly used for contaminant-plume mapping \citep{Sog06}.

\subsection{Electrical Resistivity Imaging}
\noindent ERI and IP field data collection are very similar, requiring an extra electrode line to collect IP data to ensure minimal noise.  In this dissertation, it is shown that IP is better at imaging long slender piles.  IP data is dependent on ERI data so ERI is included to show the quality of the IP surveys.  In order to fully understand IP principles, a basic understanding of ERI is necessary.  ERI is derived from Ohm's law in an isotropic homogeneous medium, 

\begin{equation}
\vec{J}=\sigma\vec{E}
\label{ohm}
\end{equation}
  
\noindent where $\vec{J}$ is current density vector in A/m$^2$, $\sigma$ is the conductivity in S/m (where S is Siemens, the reciprocal of one ohm), and $\vec{E}$ is the electric field vector in V/m.  Equation~\ref{ohm} is the general equation for the more well known circuit law \begin{equation}\label{vir} V=IR \end{equation} where $V$ is voltage, $I$ is current, and $R$ is resistance.  Resistance should not be confused with resistivity.  If current is injected into a homogeneous half-space, as in Fig.~\ref{half}, the electric field is parallel to the current flow and perpendicular to the equipotential lines (shown as a hemispherical surface of radius $r$ in Fig.~\ref{half}).  
\begin{figure}[!ht]
\centering \includegraphics[width=0.5\textwidth]{figures/hemisphere}
\caption{Current injected into a homogeneous half-space \citep{low}} \label{half} 
\end{figure}  \noindent Using this model, the current density, $\vec{J}$, is the current, $I$, divided by the assumed surface area, $2\pi r^2$.   Note that conductivity, $\sigma$ is the reciprocal of resistivity, $\rho$.  From Equation~\ref{ohm}, the electric field, $\vec{E}$, at a distance $r$ is 
\begin{equation} \vec{E} = \rho \vec{J} = \rho \frac{I}{2 \pi r^2} \end{equation} where all variables have previously been introduced.


The voltage measured at a point, P, on surface within the radius $r$, is equal to the work done by a vector field to move a test charge from infinity to P.  Work is fundamentally the product of force and distance, where the force experienced by a test charge is proportional to the electric field, $\vec{E}$.  Using these fundamental definitions, the voltage at P is defined as the integral
\begin{equation}\label{volt} 
V_{\textrm{P}} = \int_{r}^{\infty}\vec{E}\cdot \mathrm{d}r = \int_{r}^{\infty}\frac{I\rho}{2\pi r^2}\mathrm{d}r = \frac{I\rho}{2\pi r}
\end{equation}
where all variables have been defined.  This is the basic equation of electrical resistivity.  

Fig.~\ref{voltdiff} is the same as Fig.~\ref{half} but two additional electrodes are added to the surface for measuring the voltage.  
\begin{figure}[!ht]
\centering 
\includegraphics[width=\textwidth]{figures/voltdiff}
\caption{Voltage measured between points P and Q for a point source of electric current injected into a half-space of uniform resistivity, $\rho$ \citep{Briaud}}\label{voltdiff} 
\end{figure}
\noindent The voltage measured across the terminals P and Q is the difference between the two voltages.  Using Equation~\ref{volt}, the voltage potential measured across terminals P and Q is
\begin{equation}\label{potential} V_{\textrm{PQ}} = V_{\textrm{P}} - V_{\textrm{Q}} = \frac{I\rho}{2\pi}\left\{\frac{1}{r_\textrm{P}} - \frac{1}{r_\textrm{Q}}\right\} \end{equation}
where $r_\textrm{P}$ is the distance from the current source to the potential electrode P and $r_\textrm{Q}$ is the distance to potential electrode Q.

All derivations so far have been made assuming that the subsurface is homogeneous, but in reality the spatial resistivity of the Earth is heterogeneous.  Apparent resistivity, $\rho_\textrm{a}$, is the resistivity that would have been measured if the Earth was, in fact, homogeneous.  Apparent resistivity is found by rearranging Equation~\ref{potential} so that
\begin{equation} \rho_\textrm{a} = \frac{2\pi V}{I}\left\{\frac{1}{r_\textrm{P}} - \frac{1}{r_\textrm{Q}}\right\}^{-1}\end{equation}
which is how the measured apparent resistivity data for this research is estimated should the electrode configuration in Fig.~\ref{voltdiff} be used.  For this research, instead a four electrode configuration is used as in Fig.~\ref{4array} with a current source and current sink the apparent resistivity is
\begin{equation}\label{rhoa} \rho_\textrm{a} = \frac{2\pi V}{I}\left\{\frac{1}{r_{\textrm{AP}}} - \frac{1}{r_\textrm{AQ}} - \frac{1}{r_\textrm{BP}} + \frac{1}{r_\textrm{BQ}}\right\}^{-1}\end{equation}
which is the general equation for all arrays that use four electrodes.  Equation~\ref{rhoa}  can be simplified for established array types where the distances are specified between current and potential electrodes.
\begin{figure}[!ht]
\centering\includegraphics[width=\textwidth]{figures/4array}
\caption{Voltage measured between points P and Q for a point source A and point sink B of electric current \citep{Briaud}}\label{4array}
\end{figure}

The current and potential electrodes in Fig.~\ref{4array} can be arranged in many different combinations, but there are traditional four-electrode configurations.  The array in Fig.~\ref{fig:DD} is the dipole-dipole array.  Arrays vary by the spacing between the two sets of electrodes and the spacing in between the sets.  
\begin{figure}[!ht] 
\setlength{\unitlength}{0.8cm}
\centering 
\begin{picture}(18.89,3.5)
\put(2.5797,0){\includegraphics[width=0.75\textwidth]{figures/DDfigure}} 
\put(4.4,0.3){$a$}
\put(9.4,0.3){$na$}
\put(14.65,0.3){$a$}
\put(4.35,2.9){V}
\put(14.65,2.9){I}
\end{picture}
\caption{Dipole-dipole array \label{DD}} 
\label{fig:DD}
\end{figure}   
\noindent Each array has advantages; some are capable of gathering deeper data, some are better at lateral changes, etc.  Similarly, every array has disadvantages.  The dipole-dipole array is used exclusively in this research as it is shown to yield the highest quality results for foundations using IP (this is shown in Chapter~\ref{chap:Experiment}).  In the dipole-dipole array the spacing between the current electrodes and the potential electrodes remains constant; the two sets of electrodes are separated by a factor of this spacing, $n$.

\subsection{Induced Polarization Imaging} 
\noindent Induced polarization (IP) is a near surface geophysical imaging technique that is similar to electrical resistivity testing.  It was first observed by Conrad Schlumberger in the early 1900s while he was using resistivity equipment in the field \citep{Schlum}.  The use of IP for infrastructure has not been widely documented other than considering the infrastructure a nuisance.  \cite{Seig} stated that "strong (nonsulphide) IP responses were almost invariably found in the vicinity of man-made metal structures, such as grounded metal fences, pipelines, and railway tracks" which was seen as a burden that lead to erroneous conclusions.  

Similar to ERI, IP also utilizes four electrodes consisting of two current and two voltage potential reading electrodes.  For IP measurements, the current is introduced into the subsurface for a predetermine time window (two seconds in this research) and then turned off.  After this sudden stop, the voltage measured across the potential electrodes does not immediately drop to zero.  Instead, the primary voltage, $V_\textrm{p}$, drops significantly to the secondary voltage, $V_\textrm{s}$, followed by an approximately logarithmic voltage decay (or transient voltage), $V(t)$, as shown in Fig.~\ref{fig:curve}. This behavior can be loosely considered analogous to the charging and discharging of a capacitor.  

 \begin{figure}[!ht]
 \setlength{\unitlength}{0.8cm}
      \centering
      \begin{picture}(18.89,10)
      \put(2.5797,0.5){\includegraphics[width=0.75\textwidth]{figures/Curve}}
      \put(14,0){Time (s)}
      \put(2,5){\rotatebox{90}{Voltage (V)}}
      \put(8.6,3.4){$V_\textrm{p}$}
      \put(9.8,2){$V_\textrm{s}$}
      \put(14.3,2.75){$V(t)$}
\end{picture}
      \caption{IP decay curve}
      \label{fig:curve}
 \end{figure}

Likewise, when current is turned on the voltage jumps to an initial increase, followed by a similar build-up of current until the original amplitude is reached.  According to \citet{Sum} the IP effect is due to electrochemical effects.  When current is introduced into the ground, the current flow can cause chemical reactions with certain minerals.  The chemical reaction in turn causes energy to be stored as chemical energy.  When the current flow is disrupted, the chemical reactions reverse while returning some of the stored energy as current, which decays in time.  These chemical reactions are composed of two effects: electrode and membrane polarization \citep{dece}.  

Electrode polarization occurs when the soil matrix has metallic components in its pores. When current is applied, electrolyte ions concentrate at the interface between the metallic grains and electrolytes. When the current is turned off, these ions relax and return to their undisturbed distribution. The response shown on the inversions for this research is considered to be an effect of strong electrode polarization due to the steel in reinforced concrete acting as a metallic component in the concrete matrix of the foundation. Membrane polarization occurs when there is no metal, but instead ions in the electrolyte move through electrically open and restricted pores. Membrane polarization is found in clay rich environments. In order to determine which mechanism controls, an unreinforced foundation is tested in a sand rich environment and it showed little IP response indicating that electrode polarization is the controlling measurement for reinforced concrete.

The IP effect lies in measuring the decay and growth curves of voltages measured at the surface. These decay and growth curves are approximately logarithmic.  This was determined by \cite{ScottWest} using synthetic low porosity samples and an oscilloscope to visualize the curves.  Prior to their work, it was thought that decay curves were exponential.  Even today, visualizing these decay curves can be a useful tool for quality control of data.   

The fundamental law of IP relates the polarization, $\vec{P}$,  with current density, $\vec{J}$,

\begin{equation}\label{fund}
\vec{P}=-M\vec{J}
\end{equation}

\noindent where $\vec{J}$ is current density in A/m$^2$.  The chargeability, $M$, is dimensionless.  From Equation~\ref{fund}, $\vec{P}$ is proportional to the negative of the secondary voltage, $V_\textrm{s}$, and $\vec{J}$ is proportional to $V_\textrm{p}$, the primary voltage, so that

\begin{equation}\label{Charge}
 M =  \frac{V_\textrm{s}}{V_\textrm{p}}
\end{equation}

\noindent which is the fundamental, and theoretically, observable measurement of the IP effect.  Even though chargeability is a dimensionless quantity, it is often defined in terms of millivolts per volt (mV/V).  According to \cite{Sum}, while the fundamental law holds true for ideal and theoretical measurements, in practice chargeability is measured as an area under the decay curve rather than by Equation~\ref{fund}.  The equipment used for this research, the Advanced Geosciences, Inc (AGI) SuperSting\texttrademark R8/IP multi-channel imaging system (see Section~\ref{Equipment}), estimates the chargeability by measuring the transient curve and projecting the curve to zero time so $V_\textrm{s}$ can be estimated and Equation~\ref{Charge} is used.  

Chargeability is also presented as \cite{Sum} suggested; the integrated area under the transient decay curve in Fig.~\ref{fig:curve} normalized by $V_\textrm{p}$.  Equation~\ref{Chargeint} more easily shows the relationship of chargeability with a capacitor as mentioned earlier,    

\begin{equation}\label{Chargeint} M =  \frac{1}{V_\textrm{p}} \int_{t_1}^{t_2} V(t)\,\mathrm{d}t \end{equation}
\noindent where $V_\textrm{p}$ is the measured voltage when the current is turned on and $V(t)$ is the transient voltage integrated over a time window between \emph{t$_1$} and \emph{t$_2$}.   The integrated chargeability is in ms.   Equations~\ref{Charge} and~\ref{Chargeint} are considered equal due to the Newmont M331 standard. The standard is derived from a standard pulsed square wave and the integration time of the secondary voltage \citep{dolan}. Using the Newmont Standard the two units of measurement are interchangeable, however mV/V
are used exclusively for this study.  Note that IP can be measured in both the time domain and the frequency domain; time domain is used exclusively in this research. 


\subsection{Depth of Investigation}
\label{sec:depth}
\noindent A common source of confusion for geophysical methods is the depth of investigation. As a rule of thumb, the penetrating depth for ERI and IP surveys is 15\% - 25\% the largest array length for a four electrode array, however the actual depth of penetration also depends on the subsurface resistivity distribution. The maximum depth of the inverted section using AGI's EarthImager 2D\texttrademark is determined by the median depth defined by \cite{Ed77},

\begin{equation}\label{depth} n(n + 1)(n + 2) \{ [ n^2 + u]^{-1/2} - 2[(n + 1)^2 + u]^{-1/2} + [(n + 2)^2 + u]^{-1/2} \} = 1 \end{equation}
\noindent where \begin{equation} \label{variable} u = 4(z_{\textrm{med}}/a)^2 \end{equation}
in which $n$ and $a$ are the electrode spacing variables from Fig.~\ref{DD}.  This depth of investigation equation was found by empirically deriving depth coefficients and applying them to both theoretical and actual field cases.  The depth of investigation was applied to field cases where there were multiple data sets with different dipole lengths.  This formula was used so the data sets could be combined to make one complete set.  Initially, the full depth where the elementary volume of earth contributed the max at the surface was used but discrepancies from the ideal dipole occurred at low values of \emph{n}.   This issue was resolved by instead using $z_{\textrm{med}}$, the depth where exactly half one half of the total signal originates from above and below the point.

