\def\figdir{Fig}
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\title{Attribute combinations for image segmentation} 

\author{Adam Halpert}
\footer{SEP-138}

\lefthead{Halpert}
\righthead{Attribute combinations}

\begin{abstract}
Seismic image segmentation relies upon attributes calculated from seismic data, but a single attribute (usually amplitude) in not always sufficient to produce an accurate result. Therefore, a combination of information from different attributes should lead to an improved segmentation outcome. This paper explores opportunities for combining attribute information at three different stages: before segmentation (by multiplying attribute volumes), after the eigenvector calculation (via a linear combination of individual eigenvectors), and after individual boundaries have been drawn (by using uncertainty calculations to extract the best elements of individual boundaries). Overall, a method that uses uncertainty calculations to determine weights for the eigenvector linear combination produces satisfactory results, while avoiding potential drawbacks of other methods.  This method produces promising results when tested on field data in both two and three dimensions. 

\end{abstract}
\section{Introduction}
%
Image segmentation - an automated process of dividing an image into regions - offers a number of promising applications for seismic data.  Among the most straightforward of these applications is to the task of picking salt bodies on seismic images, a process that can be an ambiguous and time-consuming when undertaken manually, especially for large three-dimensional datasets with complex salt body geometries.  The development of an algorithm for automatically tracking salt boundaries \cite[]{Lomask.sepphd.126,Lomask-1} in many cases allows for the quick, efficient and globally-optimized calculation of a salt interface location.  Such information may then be used, for example, to quickly update a velocity model as part of an iterative migration system \cite[]{Halpert.sep.134.adam1}.

The seismic image segmentation scheme is based on the Normalized Cut Image Segmentation (NCIS) algorithm \cite[]{Shi-1}, which calculates an eigenvector based on specific attributes gleaned from the image; the eigenvector is then used to trace a boundary across the image. Although the most straightforward attribute for delineating salt boundaries on seismic images is amplitude of the envelope, this attribute alone is not always sufficient to produce an accurate calculation of the boundary. In such cases, other attributes may be used for segmentation.  For example, an estimate of dips in a seismic image is often used for interpretation purposes \cite[]{Bednar.sep.95.bednarb}, and strong variations in dominant dips within an image can be indicative of a salt interface.  Additionally, an instantaneous frequency calculation \cite[]{Taner-1} may be applied in order to track seismic waves' behavior as they pass from low-velocity sediments into a high-velocity salt body. \cite{Halpert.sep.136.adam1} provide details on using these attributes for segmentation using a single attribute. Ideally, however, a segmentation algorithm will combine information from these multiple attributes into a single result. In this paper, I discuss three strategies for combining attributes: a multiplication of attribute volumes, a linear combination of individual eigenvectors, and a combination of individually calculated boundaries using an uncertainty measure derived from the eigenvectors. We will see that while the the last method produces satisfactory results, an equally good result may be obtained by using the same uncertainty measurement to instead create weights for the linear combination method. This approach helps to avoid potential implementation problems, especially in three dimensions.
 

\section{Attribute combinations}
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In the segmentation algorithm, the determination of a salt interface takes place in three distinct stages. The first stage is the calculation of attributes that may be useful in indicating a boundary between sediments and a salt body. The second stage involves transforming the attribute volumes into eigenvectors of the image via the construction of a weight matrix based on the attribute values.  Finally, the third stage ``draws" the salt boundary using the eigenvector values. Each of these three stages represents an opportunity for combining information from different attributes. The following sections will explore these three options, and illustrate their advantages and disadvantages with example calculations on a 2D seismic section taken from a 3D Gulf of Mexico field dataset, seen in Figure \ref{fig:dat}.


\activeplot{dat}{width=4in,height=3.0in}{CR}{A migrated seismic section that will be used for examples throughout this paper. Note the discontinous nature of the strong reflector (salt boundary), which will present challenges for the segmentation algorithm. [{\bf ER}]}

The following examples will seek to combine useful information from two attributes - amplitude and dip variability. Figure \ref{fig:eigs} shows eigenvectors derived from these two individual attributes. The eigenvector values range from -1 to +1; in the figures here, negative values are dark and positive values are light. The salt boundary is typically drawn along the zero-contour of the eigenvector, where values pass from negative to positive. Thus, a sharp transition from dark to light colors in the eigenvector indicates a boundary location with relative certainty, while a grey area indicates a slower transition from negative to positive values, and relative uncertainty of the boundary location. Clearly, the amplitude eigenvector provides better information throughout most of the image, although the transition near $x=18000$ suggests significant uncertainty. This is logical, as the original section (Figure \ref{fig:dat}) shows a great deal on discontinuity at this location. Overall, the dip eigenvector shows much less certainty than the one derived from the amplitude attribute; however, the previously mentioned location appears more certain on the dip eigenvector. The boundary calculations corresponding to these two eigenvectors (Figure \ref{fig:bnds}) confirm these observations. Therefore, an obvious goal for combining information from these two attributes is to produce a boundary that uses information from the amplitude attribute in most locations, but incorporates the dip information at this location.

\activeplot{eigs}{width=\textwidth,height=3in}{CR}{Eigenvectors derived from amplitude of the envelope (a) and dip variability (b) attributes.}
\activeplot{bnds}{width=\textwidth,height=3in}{CR}{Salt boundaries corresponding to the eigenvectors in Figure \ref{fig:eigs}}

\subsection{Attribute multiplication}

One approach, suggested by \cite{Lomask.sepphd.126}, is to combine multiple attribute volumes into a single volume via multiplication:
\begin{equation}
A = \prod_{i=\mathrm{all \ attributes}} a_i \ ,
\end{equation}
where $a$ is an individual attribute volume, and then proceed with segmentation normally. Multiplication of the attribute data has the effect of reinforcing information in areas where the attributes ``agree," which can be beneficial. However, it also can have the effect of destroying potentially valuable information if the two attributes are not in agreement.  Figure \ref{fig:attrmult} shows the eigenvector and boundary calculation resulting from this process.

\activeplot{attrmult}{width=\textwidth,height=3in}{CR}{Eigenvector (a) and boundary calculation (b) resulting from segmentation using attribute volumes multiplied together.}

Clearly, in this case the disadvantages of multiplying attribute volumes together outweigh the possible advantages - the process appears to have incorporated the worst information from each of the attributes, resulting in a final boundary that does not improve on either of the individual results (Figure \ref{fig:bnds}) in any location.

\subsection{Eigenvector combinations}
%
A second approach is to use the individual attribute volumes to calculate an eigenvector, and then combine the eigenvectors before determining a boundary.  Following the recommendation of \cite{Shi-1}, a simple way to combine the eigenvectors is via linear combination:

\begin{equation}
E = \sum_{i=\mathrm{all \ attributes}} \lambda_i e_i \ ,
\end{equation}
where $e$ is an individual eigenvector and $\lambda$ is a specific weight value assigned to the attribute in question.  Of course, taking this approach introduces the problem of determining weight values for each attribute. Figure \ref{fig:eqwt} shows the results of this approach if equal weights are given to the amplitude and dip attributes. While the boundary is satisfactory in many locations, the dip attribute clearly has too much influence in some areas where the amplitude attribute provides much better information. While this method shows promise, a mechanism for assigning better weights is necessary; such a scheme will be discussed shortly.

\activeplot{eqwt}{width=\textwidth,height=3in}{}{Eigenvector (a) and boundary (b) resulting from taking an equally-weighted linear combination of individual eigenvectors.}

\subsection{Boundary combinations}
%
Finally, a third ``domain" in which information from different attributes may be combined is after individual boundary calculations have already taken place. This method requires a measure of uncertainty along each individual boundary, so that a new boundary can be created by incorporating the ``most certain" boundary at each location in the image. As discussed previously, such an uncertainty measure may be gleaned from the zero-crossing of the eigenvector. A sharp transition - quantitatively, a large difference between the last negative and first positive value of the eigenvector at that location - signifies relative certainty, while a slow transition or small difference signals uncertainty. In this case, the measurement is taken perpendicular to the calculated boundary, so as to avoid the assumption that the boundary is in all locations locally horizontal. After such calculations are made at all locations for each boundary, a combined boundary is formed by taking the most certain boundary location (depth value) at each horizontal location. Figure \ref{fig:bnd-hrzcombo} shows the result of this process.

\activeplot{bnd-hrzcombo}{width=4in,height=3in}{}{Boundary obtained by combining the ``most certain" elements of the individually calculated boundaries shown previously.}

This approach performs very well in this example. Information from the amplitude attribute is honored nearly everywhere, and the dip information is incorporated only where it is superior to the amplitude information.  However, the manner in which this approach is implemented could lead to problems in some circumstances. Taking the best elements of different boundaries could easily lead to erratic, ``either/or" behavior in the combined boundary; indeed, some indications of this behavior may be seen in the jaggedness of the boundary where the dip information plays a significant role. It is likely that this behavior would be even more troublesome in three dimensions.

\subsection{Weighted eigenvector combinations}
%
An approach that has the potential to preserve the advantages of the previous approach, while mitigating the disadvantages discussed above, is to take the uncertainty measurement scheme utilized previously for the boundary combination approach and use it to assign attribute weight values for a linear combination of eigenvectors. This has the effect of following the recommendation of \cite{Shi-1} for combining information from different sources, while at the same time taking advantage of a ``built-in" method for estimating uncertainties. 

Since the eigenvectors range in value from -1 to +1, the eigenvector difference across one of  the boundaries can never be greater than two. Thus, the value
\begin{equation}
\omega = \frac{1}{2} (d_{amp} - d_{dip}) \ ,
\end{equation}
where $d$ is the difference across a calculated boundary at a particular $x$ location, will range from 0 to 1. If we want to heavily penalize uncertainty in one of the eigenvectors, we set the weight values as follows:
\begin{eqnarray}
W_{amp} & = & \left\{ \begin{array}{rl}
\omega^2 &\mbox{ if $\omega < 0.5$} \\
\sqrt{\omega} &\mbox{ if $\omega > 0.5$}
\end{array} \right. \\
W_{dip} & = & 1 - w_{amp} \ .
\end{eqnarray}

The results of assigning weight values in this manner to create an eigenvector are shown in Figure \ref{fig:eig-wt}. Although the weighting scheme causes significant variation in the outlying eigenvector values (apparent from the visible vertical striping), the zero-contour boundary remains relatively smooth. The boundary successfully follows the salt interface everywhere the amplitude-only boundary does, and incorporates the dip information only where the amplitude boundary fails. Furthermore, we do not see the erratic behavior in areas where the dip information is most significant, as we did for the boundary combination method.

\activeplot{eig-wt}{width=\textwidth,height=3in}{}{Eigenvector (a) and boundary (b) calculated using a linear combination of the dip and amplitude eigenvectors, weighted by uncertainties across their respective zero-contour boundaries.}

\section{Conclusions}
%
Seismic image segmentation using a single attribute is not always sufficient to produce an accurate salt boundary calculation. Several possibilities exist for combining information from different attributes. Multiplying attribute volumes together has the effect of enhancing information common to all attributes, but overall gives an inferior result due to potentially useful information being destroyed. Combining eigenvectors from individual attributes is a second option, but a linear combination of eigenvectors requires the assignment of weight values to information from individual attributes. Such weights may be derived using an uncertainty measure calculated from the eigenvectors and the individual zero-contour boundaries. An eigenvector produced using weights calculated in this manner leads to a boundary that incorporates the best elements of individual attribute boundaries, without displaying erratic behavior that could arise from simply taking elements from individual boundaries on a point-by-point basis.


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