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Abstract

Manual interpretation of large volumes of seismic data is a tedious and timeconsuming process. Seismic interpretation relies on extensive expert knowledge of geological rules, as well as strong geophysical interpretation skills. Moreover, in modern approaches to reservoir characterization, a single, deterministic reservoir model interpreted from seismic information is commonly of less interest than the multiple potential interpretations that can arise from the data.Hence, an understanding of the uncertainty associated with seismic interpretation and its quantification are extremely important to reservoir management and decision analysis. In this chapter, several new tools are presented, most of them based on statistical pattern recognition, that can aid the interpreter in constructing a seismic-based reservoir model and provide some uncertainty quantification. Two groups of methods based on unsupervised and supervised pattern detection are discussed. A new geostatistical approach termed feature-based geostatistics is introduced, the aim of which is to accurately reproduce facies shapes. All methods are validated using an interpreted seismic data set representing channel facies from a turbidite sequence in Gabon, west Africa.

Introduction

Seismic data bring an important constraint to geological variability in geostatistical reservoir characterization. Manipulation of seismic and rock physics data requires skills in both seismic data processing and interpretation. Such interpretation is a tedious, time-consuming task and is based on certain geological and geophysical rules emanating from years of experience. However, the challenge is not to replace traditional seismic interpretation by an automatic tool, but to design automated devices, requiring little expertise in their use, that can aid and complement manual expert interpretations.

A major advantage of automated seismic interpretation is that it allows uncertainty to be quantified, a task that is essentially impossible to accomplish with manual interpretation. Having several experts repeat the same interpretation would allow uncertainty quantification, but this is simply impractical and too costly. In addition, expert interpretations of the same reservoir are commonly inconsistent, which means that a single seismic feature could be manually interpreted as different geological events. Moreover, the seismic data itself leave a large amount of ambiguity about facies types. Local variation in fluid saturation, layer geometries, and large variabilities in petrophysical properties can result in large ambiguities or uncertainties about the presence of geobodies. However, uncertainty quantification with an automated device is possible because learning from expert knowledge is a typical exercise in statistical artificial intelligence.

The first part of this chapter deals mainly with the seismic data itself. A tool is presented that can automatically recognize distinct patterns from seismic amplitudes. A local three-dimensional (3-D) window is defined within which the seismic data are sampled. Each sample represents a local seismic feature. This large data set of seismic features is then clustered into seismic clusters using an unsupervised clustering technique. Next, the clusters are calibrated to facies observations obtained from wells. The methodology is virtually entirely automatic: the clustering technique does not require the number of clusters to be known a priori, yet human experts can intervene and decide on which seismic clusters to retain for further interpretation.

The second part of the chapter concentrates on both seismic and facies geometries. A feature-based probabilistic approach to classifying facies geometries from seismic data is introduced, based on partial expert interpretations. The expert manually interprets a small volume of seismic data by relating facies features to seismic patterns. A neural network structure inspired by the human vision system attempts to mimic this process. Once the neural net is trained, it can be applied to parts of the reservoir that have not yet been interpreted to obtain fast interpretations of geobodies from seismic data without relying on time-consuming expert work. A crucial part of the automated methodology is the reproduction of channel shape morphologies. For this purpose, a new concept in geostatistics termed feature-based geostatistics is introduced.

Both approaches are demonstrated on a seismic amplitude data set from a turbidite sequence in Gabon, west Africa.

Unsupervised Pattern Recognition

Typical to seismic interpretation is the qualitative nature and art of detecting geological patterns in the seismic data. In this section, a quantitative interpretation of seismic data is developed. A statistical approach is proposed that can automatically recognize seismic patterns from amplitude data and then relate them to facies variations. The methodology can be used in a stand-alone form or can guide the geologist and geophysicist in the interpretation.

Methodology

There are two general approaches to this pattern recognition problem: supervised and unsupervised. The supervised approach uses well data and seismic data to directly build a relation between facies and lithology and seismic data. In the traditional colocated approach in geostatistics (e.g., Deutsch and Journel, 1998), this amounts to modeling the conditional probability of a facies indicator,

 

formula

with k = 1,…, K, where sk are the facies categories and k is the number of facies, given the colocated seismic datum s(u). This conditional probability is mathematically denoted as Pr{I(u, sk) = 1|s(u)}. In the unsupervised approach, one attempts to first detect certain patterns or features in the seismic data without relying on any well logs for guidance.

To capture such patterns from seismic data, a window approach (originally described by Ayvazyan, 1998; see also Caers and Ma, 2002) is proposed. The window W(u) is formed by a series of neighbor locations that include a central location u. It captures information about the local spatial pattern of seismic amplitudes and is termed a seismic feature. In the window approach, one aims at modeling the conditional probability of all facies categories given a window of seismic data s(u).Figure 1 provides an example of such a window that is strictly vertical; yet any 3-D window geometry can be used.

The approach taken here is an unsupervised one consisting of two steps: an unsupervised clustering of seismic features in distinct clusters (this step does not involve any well data) and calibration of the clusters against well data to determine the conditional probabilities. These steps are described more rigorously as follows.

Step 1

Using a predefined template or window, the seismic-amplitude data s(u), uA, where A is the study area, are scanned to obtain realizations of the random vector S(u) = {S(u), S(u + h1),…, S(u + hT)}, where {hβ, (β = 1, ),…, nt] is a vector defining the template geometry. Using an unsupervised clustering method, the samples s(u) = (s(u), s(u + h1),…,s(u + hnt)}, uA, obtained by scanning the seismic data are clustered into an unknown number of clusters tm, m = 1,…, M. M itself is an unknown quantity. Once each seismic feature is clustered into a specific cluster, the cluster label m is assigned to the original location u of the seismic feature s(u). The result of such clustering is a 3-D cube of cluster labels, mathematically denoted as

Figure 1.

A simple vertical template.

Figure 1.

A simple vertical template.

 

formula

Step 2

The interpreted sequence of facies from well logs is tied to the 3-D cube of cluster labels; hence, a frequency table of facies occurrences per cluster label can be generated. The entries in this frequency table are the desired facies probabilities given seismic data.

There are three main advantages to this two-step approach:

  1. Seismic features are clustered without being influenced by well data.

  2. Some clusters of seismic features have a direct geological significance, whereas others may be attributed to recording artifacts or noise. Therefore, the clustering may serve as a filter of the seismic information. Experts can use these classes as a tool to guide further manual interpretation.

  3. The clustering method proposed does not rely on a priori knowledge of M, the number of classes. M is determined from the data itself.

Unsupervised Clustering

Clustering aims at grouping a set of observed seismic features into separate clusters. Each feature is described by a set of nt + 1 amplitudes. In the case of supervised clustering, each seismic feature would have a cluster label already attached; hence, the aim of supervised clustering is to find a classification system or model that is able to classify future seismic features that do not have class labels. In such a case, the set of seismic features’ measurements and labels serves as a training set of information from which a classification device can learn on. In statistical terms, a classification system is a regression model because it builds a relationship (most likely nonlinear) between amplitudes in a window and a cluster label. In the application phase, amplitudes of new features are provided, and the regression model is used to determine the missing cluster labels. Supervised classification is well known, and many techniques have been developed (see Ripley, 1996, for a general overview).

The problem becomes much more complex in unsupervised classification, where cluster labels are not available. First, the feature database now has to be mined to find clusters in the data. Second, the number of classes is not known a priori and needs to be determined from the database itself. This mining process is not likely to be trivial and will actually depend on certain criteria or strategies by which the search process is conducted. Several existing unsupervised clustering methods (such as the widely popular k means and mixture distribution models; see Ripley, 1996) rely on the knowledge of the existing number of clusters. However, in many real cases (such as the geobody detection problem), the number of existing clusters is unknown a priori, and any wrong assumption might lead to incorrect geobody determination.

An unsupervised clustering routine is used here that does not rely on a priori knowledge of the number of classes or clusters. As mentioned above, some measure of similarity between measurements needs to be defined to produce clusters that contain similar seismic features. The so-called minimum message length (MML) clustering method (Wallace and Boulton, 1986; Wallace and Dowe, 1994) relies on principles of information theory to measure a goodness of a clustering, which is found to be well suited for the geobody detection problem.

Clustering is essentially a dimensionality reduction problem, reducing the number of data (nt + 1) x N, with N the number of features, into a set of discrete classes M, M ≪ (nt + 1) x N. In the geobody detection problem, one essentially reduces the dimension of order 10 × 106 ((nt + 1) x N) into one of dimension 100 (M). This allows the seismic interpreter to make more efficient use of large data sets.

Methodology of the MML Clustering

The MML methodology relies on the well-known Shannon principle for message coding. To optimally code (in terms of smallest number of bytes) a simple message, such as a sentence, one should attach the shortest code to the most frequent words in a sentence. A similar principle can be envisioned in clustering seismic features: if the entire database of seismic features sampled from a seismic cube is considered as a message, then to present that message in its shortest form, one should attach the shortest (cluster) name to the cluster of the most commonly appearing features. At the same time, the variability or spread of features in a single cluster should not be too large because it would require a cost to code (explain) that variability. Minimum message length therefore attempts to transfer the entire database of seismic features into a new database containing the following elements:

  1. the cluster label to which each seismic feature belongs

  2. some summary of the average properties of each cluster (mean, standard deviation)

  3. the deviation of each measurement from the average properties

Each of these elements is bit-coded and can be attributed to a cost calculated in bytes. Using an optimization method, MML searches for the optimal clustering of features such that the number of bytes used to represent this new database is as small as possible. To understand this process better, consider two extremes: (1) all seismic features are clustered into one single cluster, and (2) there are as many clusters as seismic features. In the first case, the cost of labeling clusters is small, but the spread in the single cluster is extremely large, hence requiring a large coding cost; in the second case, the variability in each cluster is zero, but the cost, in terms of bytes, to represent each cluster label is large. Minimum message length searches for a balance between these two extremes.

The application of information-based clustering to the problem of geobody detection is motivated as follows: if a clear geological body exists, then the seismic response is likely to be similar over that entire geobody. Hence, local seismic-amplitude measurements will tend to group into a single cluster. This means that the entire cluster of seismic-amplitude information can be largely compressed into information provided by the cluster label and the average properties of amplitude features in a cluster. The aim of MML clustering is to represent the amount of information into the briefest possible representation of all amplitude measurements.

Application

Seismic data obtained from the lower Miocene age Baliste-Crecerelle canyon fill deposits off the African coast in Gabon are used to illustrate the proposed interpretation tool. The complete canyon fill deposits extend for 120 km (75 mi) basinward from the paleoshelf edge with a width of as much as 10 km (6 mi) and a thickness of 500 m (1600 ft). Wonham et al. (2000) investigated the sedimentary architecture of the canyon fill using high-resolution seismic stratigraphy, seismic attribute analysis, and borehole data calibrated to seismic data. The gOcad software (Mallet, 2002) has been used to integrate and interpret these data to develop a model of stratigraphic architecture from the scale of the canyon (4 km; 2.5 mi) to the scale of individual channels (200 m; 660 ft).

Individual channels are located through bodycheck picks of high-amplitude reflectors. More than 100 individual channel morphologies have been manually created and discretized on a regular stratigraphic grid. A large seismic cube (more than 40 million cells) and corresponding interpreted channel facies interpretations are available.

The clustering approach is applied to the entire seismic data cube to test its capabilities against the expert interpretation. Only results of selected but representative channel bodies are shown. Figure 2 shows the expert interpretation: a single channel is apparent, running from east to west. The seismic data are not shown.

The five-pixel vertical template in Figure 1 is used to scan the seismic-amplitude data set. Using this window, a set of seismic features s(u) is compiled by scanning the entire 3-D cube. The MML algorithm clusters these features into 27 clusters. Recall that the number 27 is unknown a priori and obtained as a result of the MML clustering. The cluster label of each feature s(u) is mapped at the location u, generating the cluster label cube shown in Figure 3. A horizontal body can be distinguished in Figure 3b and f corresponding mainly to cluster label 26. This feature is present in most horizontal slices although represented by different cluster labels. In fact, in Figure 3a, c, and e, cluster labels 3–5 correspond to the same channel feature. Comparing the results to the interpreted reservoir horizontal slices in Figure 2, the geobody obtained from clustering corresponds well to the interpretation model. Next, the cluster labels can be calibrated with well data (step 2 in the algorithm) to obtain a probability of channel facies for each grid cell in the reservoir (results not shown).

Figure 2.

Horizontal slices of a selected region of the expert seismic interpretation. (a) is the top slice, (f) is the bottom slice.

Figure 2.

Horizontal slices of a selected region of the expert seismic interpretation. (a) is the top slice, (f) is the bottom slice.

A second example is shown in Figure 4, showing two slices from a 3-D manually interpreted seismic cube next to the results of the MML clustering. In this case, cluster numbers 3–5 appear to be indicating a channel.

Feature-Based Interpretation Of Seismic Data

Motivation

In the previous section, a pixel-based method for detecting patterns from seismic data is presented. Although the method provides satisfactory results locating channel objects, the shape of these channels is poorly reproduced. For example, compare the object-based manual interpretation with the pixelbased classification in Figure 4.

In this section, a novel neural network approach is combined with a feature-based geostatistical-simulation method to overcome the poor-shape reproduction. The proposed method attempts to establish an explicit relationship between seismic patterns and facies geometries instead of pixels. The automated interpretation methodology consists of two phases: a learning phase and a reproduction and generalization phase. In the learning phase, the aim is to understand the relationship between facies geometry and local seismic patterns. In the reproduction phase, the trained neural network is applied to uninterpreted areas of the seismic cube to determine channel probability patterns from seismic data. Finally, a feature-based geostatistical-simulation technique conditioned to the channel probability patterns is used to reproduce channel shapes accurately.

Figure 3.

Cluster labels from the MML clustering (label values have no physical meaning).

Figure 3.

Cluster labels from the MML clustering (label values have no physical meaning).

Figure 4.

(a, b) Expert interpretation; (c, d) cluster labels from the MML clustering.

Figure 4.

(a, b) Expert interpretation; (c, d) cluster labels from the MML clustering.

To model the seismic-to-facies relationship during the learning phase, a training data set is required that consists of a 3-D cube of seismic data and the corresponding expert interpretation. Such data may be obtained from partial seismic interpretations or from partial interpretations near the wells based on well logs.

In geostatistics, the traditional approach is to build the reservoir one pixel at a time (pixel-based geostatistics) or one object at a time (object-based modeling). However, the human expert is not likely to build a channel or a shale body one pixel at a time or one body at a time. Experts look for patterns in the seismic data and relate them to geobody features.

In this terminology, a piece of a large facies body is termed a facies feature. For example, an oxbow can be a feature of a larger channel complex. One can think of features as the pieces of a jigsaw puzzle and feature-based simulation as an attempt to solve this puzzle consistent with the given channel probabilities.

Implementation Of The Neural Network

The neural network accepts a seismic window as its input. The information in seismic data is mainly coded in the vertical direction. Thus, an input window with a much higher aspect ratio in the vertical direction is chosen. The output of the neural network is a pattern of channel probabilities. Typical dimensions for these windows are 5 × 5 × 9 for the input (seismic data) and 3 × 3 × 1 for the output (facies), as shown in Figure 5. The seismic window is 3-D and has a large vertical component, whereas the channel is flat, although in general, any type of window can be selected.

The connections of the neural network loosely follow the human eye neural wiring architecture (Ganong, 1997). Traditionally, back-propagation networks are designed to be fully connected with one or two hidden layers regardless of the task at hand (Haykin, 1999). By borrowing concepts from the human eye, architecture specific for this application is favored over a more traditional, fully connected architecture. The two main concepts adopted from vision science are overlapping receptive fields and hierarchical processing.

In neural network jargon, the term “receptive field” refers to the area of an image or scene that is processed by a single neuron (Palmer, 1999). For the seismic interpretation, the receptive field coincides with a seismic window. The hidden neuron receives input only from its corresponding neighborhood and not from all available input nodes. In other words, the neuron is partially connected to the input layer (see Figure 6a). The human brain introduces redundancy to this structure through neurons with partially overlapping receptive fields. In case a neuron fails, combining information from other neurons can still retrieve any lost information.

Figure 5.

(Left) The template used to scan the seismic cube, and (right) the template used to scan the expert interpretation (channels). The information obtained from seismic data is mainly coded in the vertical direction. Thus, an input window with a much higher aspect ratio in the vertical direction is chosen.

Figure 5.

(Left) The template used to scan the seismic cube, and (right) the template used to scan the expert interpretation (channels). The information obtained from seismic data is mainly coded in the vertical direction. Thus, an input window with a much higher aspect ratio in the vertical direction is chosen.

Another concept borrowed from hierarchical processing is a cascading set of layers of hidden neurons with an overlapping receptive field principle applied to each layer (Palmer, 1999). Consider the example in Figure 6b. Here, the hidden neuron connects to a total of four neurons in the previous layer (its immediate receptive field), and each of those neurons connect to their respective overlapping 3 × 3 receptive fields in the input layer. The result of this structure is an effective receptive field of 5 × 5 for the hidden neuron of interest. This 5 × 5 area is said to be processed hierarchically by the four neurons in the first hidden layer and the hidden neuron in the second hidden layer.

The combination of overlapping receptive fields and hierarchical processing poses two main advantages over traditional fully connected networks: (1) by introducing structure to the neural network architecture, the neural network is allowed to detect patterns by looking at small-scale local changes (receptive fields) and by considering the global relations of these local changes to each other (hierarchical processing); and (2) although the structure uses a relatively large number of neurons, the computational performance is not degraded because the connections between these neurons are selective. Typically, the processing power lost because of the missing connections is compensated for by introducing an extra, fully connected layer right before the output (Reed and Marks, 1999).

Figure 6.

(a) The receptive field of a neuron. Instead of fully connecting the neurons to all inputs, a local area is chosen for the neuron to process. (b) A network with two hidden layers, each layer employing the receptive field concept. The highlighted 5 x 5 area corresponds to the effective receptive field of a second hidden layer neuron.

Figure 6.

(a) The receptive field of a neuron. Instead of fully connecting the neurons to all inputs, a local area is chosen for the neuron to process. (b) A network with two hidden layers, each layer employing the receptive field concept. The highlighted 5 x 5 area corresponds to the effective receptive field of a second hidden layer neuron.

The application-specific architecture of the neural network is critical to the success of the proposed seismic interpretation methodology. The relation between seismic-amplitude data and the facies occurrence is not a direct relation. A particular seismic-amplitude datum does not contribute much to its colocated facies interpretation. Instead, facies are determined on the basis of change in neighboring amplitude reflections. The receptive field concept allows the network to capture the small-scale local changes in the seismic data, and the hierarchical processing relates these small-scale changes to the overall large-scale change in the entire seismic window. Traditional, fully connected networks commonly lack the capability to capture the nature of seismic-amplitude patterns.

Because of the application-specific nature of the neural network, the final architecture depends on the size and shape of both the seismic template and the channel template. In this chapter, the templates shown in Figure 5 are used. Use of such templates provides a network architecture shown in Figure 7.

This partially connected neural network structure is trained using the classical online back-propagation learning algorithm. A presentation of the backpropagation algorithm is beyond the scope of this chapter. The reader is referred to Haykin (1999).

During the training phase, the content of the seismic window is given to the neural network as the input and its corresponding expert interpretation as the desired output. These expert interpretations are binary in nature for this application: channel vs. no channel. Because interpretations are never certain, a probabilistic neural network modeling the probability of observing certain facies from seismic data is required. This network should be capable of producing channel probabilities even when trained only by binary channel data. Caers and Ma (2002) suggest one such neural network structure for pixel-based estimations. Their methodology is adapted to the problem at hand and is reviewed briefly in the next section.

Figure 7.

The neural network structure used in this chapter. The vertical information content of seismic data is high, and therefore, it is emphasized in the structure by introducing more hierarchical levels along the Z-axis.

Figure 7.

The neural network structure used in this chapter. The vertical information content of seismic data is high, and therefore, it is emphasized in the structure by introducing more hierarchical levels along the Z-axis.

A Probabilistic Neural Network Methodology

Traditionally, neural networks are used for modeling a complex, nonlinear interaction between a set of variables. Commonly, neural network aims at predicting a set of target variables Y (e.g., petrophysical properties) from a set of input variables X (e.g., well-log measurements). The neural network models a specific nonlinear function g(X) between these two types of variables. The neural network above is an example of a regression-type neural network because it models E[Y|X] (conditional expectations). The traditional framework, however, lacks the commonly important capability of modeling the uncertainty about these predictions (i.e., P(Y |X), the conditional probabilities).

Caers and Ma (2002) recently published a general methodology for turning deterministic neural networks into probabilistic neural networks, the output of such being the conditional probability P(Y|X). Seismic data are known to potentially leave a large degree of uncertainty about the presence of reservoir facies (smoothing and tuning effects). Uncertainty quantification is needed, because such ambiguities commonly cannot be resolved. In this chapter, the approach of Caers and Ma (2002) is applied to the neural network architecture presented in the previous section.

Results Of The Neural Network Application

The neural network structure above is applied to the reservoir studied in the previous part of this chapter (Figure 4). Figure 8 shows the training results; that is, the neural network is trained using the expert interpretation in Figure 8a and the seismic data (not shown). The templates of Figure 5 are used. To check the performance, the trained neural network is applied to the same training set. Figure 8b shows the network prediction, and Figure 8c is the overlaid comparison of the network prediction with an actual expert interpretation. Figure 8c confirms that the neural network manages to predict the location of the channels well, at least in conformance with the expert interpretation. However, the results lack the crisp shape reproduction desired.

Figures 9 and 10 contain the test results for two test areas. Test results are obtained by applying the trained neural network to other parts of the seismic cube where no expert interpretation is available. An independent expert interpretation is also available in the test area that can be compared with the neural network result.

Figure 10 is a typical test output that one would expect from a neural network predictor. The result is not as good as the training area result, but the channel locations are still easily identifiable. The more interesting result is that of Figure 9. The neural network seems to predict certain channel-like structures near the top-left comer of the grid, although the expert interpretation does not contain any channels in that area. In this region, a salt dome is known to exist, yet the neural network cannot distinguish between seismic reflections belonging to the salt dome and those belonging to the channel body. It is hard to avoid such misinterpretations because the original neural network is never trained on a salt dome seismic response. Next, a method is developed to overcome such problems.

Figure 8.

Neural network results for the training area.

Figure 8.

Neural network results for the training area.

A Feature-Based Geostatistical-Simulation Algorithm

The neural network estimator is essentially a standalone module that can be used to estimate facies channel probabilities. As shown in Figures 8–10, it is not sufficient to accurately identify facies features and geometries. Although the neural network explicitly considers the channel shapes using a facies template (right side of Figure 5), the results lack the crispness of channel objects. The neural network is applied independently on each window of seismic amplitudes, not considering the spatial continuity of the channel objects.

In this section, a novel geostatistical approach that restores the channel object geometry is presented. This step uses the channel probabilities of Figures 810 as conditioning soft data. A feature-based conditional simulation method is used for this purpose.

Feature-based geostatistics deals with patterns instead of single pixels (pixel-based geostatistics) or entire objects (object-based modeling). Versatile feature-based simulation algorithms (Arpat et al.,2002), as well as various approaches to the use of features that are not simulation based (Arpat et al., 2001), have been published by the authors before. In this chapter, a new, relatively simple feature-based simulation method is developed to process the channel probabilities estimated by the neural network estimated. The methodology is illustrated on a simple example and then applied to the case study.

A Simple Example

Consider the simple 10 × 10 expert interpretation in Figure 11a depicting a small channel. This image contains 35% channel facies. Assume that a channel probability map derived from seismic data using a neural network as above is also given (Figure 11b). Note that this map lacks the crispness of an actual channel body.

Figure 9.

Neural network results for a test area.

Figure 9.

Neural network results for a test area.

Figure 10.

Neural network results for another test area.

Figure 10.

Neural network results for another test area.

Step 1: Extracting Features

The algorithm starts by extracting features from the expert interpretation. The feature template is placed on this image and scanned in a sliding fashion. Figure 11c illustrates the scanning process. All features extracted from expert interpretation are given in Figure l1d.

Step 2: Calculating Skins

Once the features are extracted, the training image is rescanned to calculate the so-called skins belonging to each feature. Feature skins constitute pixels in the immediate neighborhood of a feature. These skins serve to quantify the direction of continuity and connectivity associated with each feature. For example, in Figure 12a, the 3 × 3 feature has a skin of two pixels in each direction. The pixel values of the skin express the probability of channel occurrence for any feature in the feature set. The skin of a feature is calculated by averaging the neighboring pixels of all occurrences of that feature in the expert interpretation. Figure 12b illustrates this process. If one considers features as puzzle pieces, then the skin corresponds to the way the puzzle pieces interconnect with each other.

Step 3: Simulation

A random path is determined. The leftmost map of Figure 13a shows the first node visited in the grid. At this node, feature assignment is constrained by the channel probabilities obtained from seismic data (Figure 11b). The assigned feature needs to be selected from the existing feature set. This selection will retain the feature that is most similar to the colocated channel probability pattern of Figure 11b. The Tanimoto distance function is used as the similarity metric:

 

formula

where n is the total number of pixels in the feature body and skin (7 × 7 in this case), a, are the pixel values of the probability pattern, and bi, are the pixel values of the candidate feature. A sample calculation for the Tanimoto metric is illustrated in Figure 14. Other distance measures could be used, such as Euclidean or Manhattan distance (Duda et al.,2001). For the particular application to channel reservoirs, a sensitivity study has found that the Tanimoto distance function promotes the channel continuity best.

Figure 13a shows the selected feature for the first visited node. Once this most similar node is determined, corresponding skin values are pasted into the simulation grid, replacing the original channel probabilities from seismic data as shown in the last column of Figure 13a.

Figure 11.

(a) A 10 × 10 expert interpretation used for the simple example. (b) The channel probability from seismic data used for the simple example. For demonstration purposes, it is assumed that this map is obtained from a neural network structure as explained in the text. (c) Various steps of feature extraction from expert interpretation. (d) All features extracted from the expert interpretation.

Figure 11.

(a) A 10 × 10 expert interpretation used for the simple example. (b) The channel probability from seismic data used for the simple example. For demonstration purposes, it is assumed that this map is obtained from a neural network structure as explained in the text. (c) Various steps of feature extraction from expert interpretation. (d) All features extracted from the expert interpretation.

Next, another node along the random path is selected, for example, the one right next to the first visited node. Figure 13b shows the location of this node. Again, the most similar feature is selected, considering the channel probabilities from seismic data and any previously determined features (sequential simulation). Figure 13b shows the selected feature and the updated probability map. Figure 13c shows some intermediate steps during the simulation. The final result, after all nodes have been simulated, is shown in the last column of Figure 13c. If another realization is required, then the random path needs to be changed.

Figure 12.

(a) A 3 × 3 feature with two units of skin, (b) Two samples showing how the skin of a feature is calculated. Simply, all occurrences (twice only in the above examples) are added up and then divided by the number of occurrences.

Figure 12.

(a) A 3 × 3 feature with two units of skin, (b) Two samples showing how the skin of a feature is calculated. Simply, all occurrences (twice only in the above examples) are added up and then divided by the number of occurrences.

Results

The proposed feature-based simulation method is applied to the channel probabilities obtained using the neural network (Figure 8). Figure 15 shows the results for the training area. The feature-based simulation manages to convert the channel probability patterns into actual channel shapes successfully. Notice that wherever there is a discontinuity in the probabilities, the simulation algorithm fills in the channel and promotes continuity.

Figure 13.

The steps of the feature-based simulation. In (a) and (b), the location of the node dictated by the random path is shown, followed by the most similar feature selected, followed by the final pasting of the skin values. (c) An illustration of the intermediate steps and the final realization.

Figure 13.

The steps of the feature-based simulation. In (a) and (b), the location of the node dictated by the random path is shown, followed by the most similar feature selected, followed by the final pasting of the skin values. (c) An illustration of the intermediate steps and the final realization.

Figure 14.

Sample Tanimoto distance calculation.

Figure 14.

Sample Tanimoto distance calculation.

Figure 15.

Final results for the training area.

Figure 15.

Final results for the training area.

Figure 16 presents the results for the test area containing the salt dome. This problematic test area produces results without the salt dome probabilities converted into actual channels. This is possible because the salt dome probabilities do not correspond to a channel pattern found in the expert interpretation. The feature-based algorithm naturally recognizes the difference between the channel shapes and the salt dome response.

Conclusion

Building a single or multiple facies model from seismic data should rely on the existing multiple-point relation between seismic attribute data and the continuity of the underlying facies. Expert interpretations are explicit representations of the multiple-point relationship. Such interpretations are mappings of seismic features (consisting of multiple pixels or voxels) into channel shapes or object pieces.

In this chapter, an automated interpretation tool is presented that capitalizes on this notion. Building on ideas from human vision science, a neural network approach is developed that maps seismic features into facies features. However, because neural network approaches commonly neglect spatial continuity, an additional geostatistical step is warranted.

The geostatistical-simulation algorithm developed is also somewhat nontraditional in the sense that features (parts of objects) are simulated instead of pixels and objects. The feature-based algorithm attempts to build larger objects by connecting smaller object pieces constrained to the presence of facies detected in the seismic data by the neural network.

Figure 16.

Final results for a test area.

Figure 16.

Final results for a test area.

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197
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Acknowledgments

The authors appreciate the opportunity provided by TotalFinaElf to work on the submarine-channel data set. Comments on previous parts of this work by Andre Haas are also appreciated.

Figures & Tables

Figure 1.

A simple vertical template.

Figure 1.

A simple vertical template.

Figure 2.

Horizontal slices of a selected region of the expert seismic interpretation. (a) is the top slice, (f) is the bottom slice.

Figure 2.

Horizontal slices of a selected region of the expert seismic interpretation. (a) is the top slice, (f) is the bottom slice.

Figure 3.

Cluster labels from the MML clustering (label values have no physical meaning).

Figure 3.

Cluster labels from the MML clustering (label values have no physical meaning).

Figure 4.

(a, b) Expert interpretation; (c, d) cluster labels from the MML clustering.

Figure 4.

(a, b) Expert interpretation; (c, d) cluster labels from the MML clustering.

Figure 5.

(Left) The template used to scan the seismic cube, and (right) the template used to scan the expert interpretation (channels). The information obtained from seismic data is mainly coded in the vertical direction. Thus, an input window with a much higher aspect ratio in the vertical direction is chosen.

Figure 5.

(Left) The template used to scan the seismic cube, and (right) the template used to scan the expert interpretation (channels). The information obtained from seismic data is mainly coded in the vertical direction. Thus, an input window with a much higher aspect ratio in the vertical direction is chosen.

Figure 6.

(a) The receptive field of a neuron. Instead of fully connecting the neurons to all inputs, a local area is chosen for the neuron to process. (b) A network with two hidden layers, each layer employing the receptive field concept. The highlighted 5 x 5 area corresponds to the effective receptive field of a second hidden layer neuron.

Figure 6.

(a) The receptive field of a neuron. Instead of fully connecting the neurons to all inputs, a local area is chosen for the neuron to process. (b) A network with two hidden layers, each layer employing the receptive field concept. The highlighted 5 x 5 area corresponds to the effective receptive field of a second hidden layer neuron.

Figure 7.

The neural network structure used in this chapter. The vertical information content of seismic data is high, and therefore, it is emphasized in the structure by introducing more hierarchical levels along the Z-axis.

Figure 7.

The neural network structure used in this chapter. The vertical information content of seismic data is high, and therefore, it is emphasized in the structure by introducing more hierarchical levels along the Z-axis.

Figure 8.

Neural network results for the training area.

Figure 8.

Neural network results for the training area.

Figure 9.

Neural network results for a test area.

Figure 9.

Neural network results for a test area.

Figure 10.

Neural network results for another test area.

Figure 10.

Neural network results for another test area.

Figure 11.

(a) A 10 × 10 expert interpretation used for the simple example. (b) The channel probability from seismic data used for the simple example. For demonstration purposes, it is assumed that this map is obtained from a neural network structure as explained in the text. (c) Various steps of feature extraction from expert interpretation. (d) All features extracted from the expert interpretation.

Figure 11.

(a) A 10 × 10 expert interpretation used for the simple example. (b) The channel probability from seismic data used for the simple example. For demonstration purposes, it is assumed that this map is obtained from a neural network structure as explained in the text. (c) Various steps of feature extraction from expert interpretation. (d) All features extracted from the expert interpretation.

Figure 12.

(a) A 3 × 3 feature with two units of skin, (b) Two samples showing how the skin of a feature is calculated. Simply, all occurrences (twice only in the above examples) are added up and then divided by the number of occurrences.

Figure 12.

(a) A 3 × 3 feature with two units of skin, (b) Two samples showing how the skin of a feature is calculated. Simply, all occurrences (twice only in the above examples) are added up and then divided by the number of occurrences.

Figure 13.

The steps of the feature-based simulation. In (a) and (b), the location of the node dictated by the random path is shown, followed by the most similar feature selected, followed by the final pasting of the skin values. (c) An illustration of the intermediate steps and the final realization.

Figure 13.

The steps of the feature-based simulation. In (a) and (b), the location of the node dictated by the random path is shown, followed by the most similar feature selected, followed by the final pasting of the skin values. (c) An illustration of the intermediate steps and the final realization.

Figure 14.

Sample Tanimoto distance calculation.

Figure 14.

Sample Tanimoto distance calculation.

Figure 15.

Final results for the training area.

Figure 15.

Final results for the training area.

Figure 16.

Final results for a test area.

Figure 16.

Final results for a test area.

Contents

GeoRef

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