Fuzzy logic and image processing techniques for ... - Semantic Scholar

Home

Search

Collections

Journals

About

Contact us

My IOPscience

Fuzzy logic and image processing techniques for the interpretation of seismic data

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Geophys. Eng. 8 185 (http://iopscience.iop.org/1742-2140/8/2/006) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 192.100.180.19 The article was downloaded on 08/01/2013 at 23:21

Please note that terms and conditions apply.

IOP PUBLISHING

JOURNAL OF GEOPHYSICS AND ENGINEERING

doi:10.1088/1742-2132/8/2/006

J. Geophys. Eng. 8 (2011) 185–194

Fuzzy logic and image processing techniques for the interpretation of seismic data M G Orozco-del-Castillo1 , C Ortiz-Alemán1 , J Urrutia-Fucugauchi2 and A Rodr´ıguez-Castellanos1 1

Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, Mexico, DF 07730, Mexico Programa Universitario de Perforaciones en Océanos y Continentes, Instituto de Geof´ısica, Universidad Nacional Autónoma de México, Mexico, DF 04510, Mexico 2

E-mail: [email protected], [email protected], [email protected] and [email protected]

Received 10 June 2010 Accepted for publication 10 January 2011 Published 10 March 2011 Online at stacks.iop.org/JGE/8/185 Abstract Since interpretation of seismic data is usually a tedious and repetitive task, the ability to do so automatically or semi-automatically has become an important objective of recent research. We believe that the vagueness and uncertainty in the interpretation process makes fuzzy logic an appropriate tool to deal with seismic data. In this work we developed a semi-automated fuzzy inference system to detect the internal architecture of a mass transport complex (MTC) in seismic images. We propose that the observed characteristics of a MTC can be expressed as fuzzy if-then rules consisting of linguistic values associated with fuzzy membership functions. The constructions of the fuzzy inference system and various image processing techniques are presented. We conclude that this is a well-suited problem for fuzzy logic since the application of the proposed methodology yields a semi-automatically interpreted MTC which closely resembles the MTC from expert manual interpretation. Keywords: fuzzy logic, mass transport complexes, seismic interpretation, image processing

1. Introduction The growth in the amount of seismic data requiring interpretation in the past few years has motivated research in the field of automated interpretation. There is also a need to increase geoscientist productivity by freeing him or her from the often tedious and repetitive task of seismic interpretation. The problem relies on the uncertain and imprecise nature of the geophysical data (Nikravesh and Aminzadeh 2001) that complicates efforts to develop fully automated interpretation methods. In recent years, it has been shown that uncertainty may be due to fuzziness (Aminzadeh 1991) rather than chance; this is why fuzzy logic has been considered to be appropriate to deal with the nature of uncertainty in human error (Nikravesh and Aminzadeh 2001). Fuzzy logic, first introduced by Zadeh (1965), understands problems as having a degree of truth, i.e. ‘fuzzy sets of true and false’, unlike classical logic which is based on ‘crisp sets of true and 1742-2132/11/020185+10$33.00

false’ (Aminzadeh and Chatterjee 1984). In contrast to a classical set, a fuzzy set is a set without crisp boundaries, that is the transition from ‘belong to a set’ to ‘not belong to a set’ is gradual, and this smooth transition is characterized by membership functions (MFs) that give fuzzy sets flexibility in modelling commonly used linguistic expressions (Jang et al 1997). Despite its name, fuzzy set theory is not one that permits vagueness (Zadeh 1992). It is a methodology that was developed to obtain an approximate solution where the problems are subject to vague description (Zadeh 1997). Since the human brain interprets imprecise and incomplete sensory information provided by perceptive organs, and since fuzzy set theory provides a systematic calculus to deal with such information linguistically, fuzzy logic has been used as a very useful tool to tackle uncertainty and to handle imprecise information in a complex situation. The numerical computation on which fuzzy set theory relies is performed by using linguistic labels stipulated by their MFs. The definition

© 2011 Nanjing Geophysical Research Institute Printed in the UK

185

M G Orozco-del-Castillo et al

of MFs allows the construction of fuzzy if-then rules, which form the key component of fuzzy inference systems (FIS). A FIS is a system which integrates the concepts of fuzzy set theory, fuzzy if-then rules and fuzzy reasoning so that it can effectively model human expertise in a specific application. Fuzzy logic has been successfully applied to solve problems subject to uncertainty and vagueness, playing a key role in several disciplines ranging from medicine (Aldridge et al 2009), chemistry (Jensen et al 1997), meteorology, atmospheric and earth sciences (Baum et al 1997, Gourley et al 2007, Mitra et al 1998, Chang et al 2001) to ebusiness (Luo et al 2003). Chang et al (2001) proposed fuzzy techniques to identify river water quality. They concluded that these techniques might successfully harmonize inherent discrepancies and interpret complex conditions, in contrast to the outputs generated by conventional procedures. Application of fuzzy techniques to robotics constitutes one of the major trends in current research (Li and Huang 2010, Venayagamoorthy et al 2009, Zhou and Meng 2003, Park 2003, Aguirre and Gonzalez 2000). Traditional work has tried to overcome difficulties by focusing on the design of robotic mechanisms, sensors and environment, but fuzzy logic has provided features which allow the modelling of uncertainty and imprecision, two proper characteristics of human environments. A more detailed review can be found in Saffioti (1997). The uncertain, imprecise and linguistic nature of geophysical and geological data makes it a good candidate for interpretation through fuzzy set theory (Nikravesh and Aminzadeh 2001). Fuzzy logic can mimic human performance by combining the quantitative data with qualitative information and subjective observations in order to automate or semiautomate subjective processes. For example, Chappaz (1977) and Bois (1983, 1984) proposed to use fuzzy set theory as a pattern recognition tool for reservoir analysis and interpretation of seismic sections. Bois (1983, 1984) concluded that fuzzy set theory can be used for interpretation of seismic data which are imprecise, uncertain and include human error. This type of error and vagueness cannot be taken into consideration by deterministic techniques, but they are perfectly seized by fuzzy set theory. He also concluded that using fuzzy set theory and seismic data, geological information can be determined; therefore, one can predict the boundary of a reservoir in which hydrocarbon exists. Baygun et al (1996) used fuzzy logic as a classifier for delineation of geological objects in a mature hydrocarbon reservoir with many wells. They have shown that fuzzy logic can be used to extract dimensions and orientation of geological bodies so that the geologist can use such a technique for reservoir characterization in a very quick way, bypassing several tedious steps. Ilkhchi et al (2006) used a fuzzy logic approach to estimate permeability and rock type from conventional well log data. The results of their fuzzy c-means clustering technique and Takagi–Sugeno FIS were successful in the Iran offshore gas fields. Recently, Hashemi et al (2008) have developed a semi-supervised method for detecting random noise using the fuzzy Gustafson–Kessel clustering. Fuzzy clustering can be better described as a non-supervised intelligent tool, rather 186

than a statistical pattern recognition method. This procedure requires seismic specialist supervision interactively to preserve the amplitude and phase information. Their results led them to conclude that there are still many open possibilities for future studies based on application of soft computing, particularly fuzzy logic, in seismic data processing. Throughout the years, several other geophysical and geological applications of fuzzy logic have been proposed (Aminzadeh and Chatterjee 1984, Chen et al 1995, Adams et al 1999a, 1999b, Tamhane et al 2002). Since fuzzy logic can be appropriately used as a tool for the interpretation of seismic data, the aim of this work was to develop a FIS to semi-automatically detect the internal architecture of a mass transport complex (MTC). This FIS was constructed by fuzzy if-then rules consistent with those an actual interpreter would follow to visually detect this feature. The construction of these rules implies the definition of several MFs which relate to some vague and imprecise linguistic expressions commonly used for the detection of a MTC (e.g. strong, chaotic, near).

2. An overview 2.1. Fuzzy logic The nucleus of fuzzy logic is the concept of fuzzy sets which, contrary to classical sets, are not required to have sharp boundaries that distinguish their members from other objects (Nikravesh et al 2003). The construction of a fuzzy set depends on two things: the identification of a suitable universe of discourse and the specification of an appropriate MF. The specification of MFs is subjective, which means that the MFs specified for the same concept by different persons may vary considerably; this subjectivity comes from individual differences in perceiving or expressing abstract concepts and has little to do with randomness (Jang et al 1997) and probability theory, the objective treatment of random phenomena. Fuzzy sets, which usually carry names that conform to adjectives appearing in our daily linguistic usage, are called linguistic values; thus, the universe of discourse is often called the linguistic variable (Jang et al 1997). The membership in a fuzzy set is not a matter of affirmation or denial, but of degree. Degree of membership of objects in fuzzy sets is most commonly expressed by real numbers in the interval [0, 1]. Each fuzzy set, say A, is formally defined by a function μA of the form (Nikravesh et al 2003) μA : X → [0, 1],

(1)

where X denotes the classical set of all objects that are relevant in the context of a particular application, the universe of discourse. The function μA is called a MF of the fuzzy set A. For each x ∈ X, μA (x) specifies the degree of membership of element x in the fuzzy set A. A fuzzy set A in which either μA (x) = 0 or μA (x) = 1 for all x ∈ X is called a crisp set (Nikravesh et al 2003). A distinct notation to denote MFs, where the symbol of the fuzzy set is not distinguished from the symbol of its MF, is also commonly employed in the literature (Nikravesh et al 2003, Jang et al 1997).


The most basic operations on classical sets are union, intersection and complement, and similar operations can be implemented for fuzzy sets. As opposed to classical set theory, the operations on fuzzy sets are not unique (Nikravesh et al 2003). Each of them consists of a class of functions that satisfy certain properties. According to Zadeh’s original work, the union of two fuzzy sets A and B is a fuzzy set C, written as C = A ∪ B whose MF is related to those of A and B by μC (x) = max(μA (x), μB (x)) = μA (x) ∨ μB (x).

(2)

A more intuitive definition of union is the ‘smallest’ fuzzy set containing both A and B. The intersection of two fuzzy sets A and B is a fuzzy set C, written as C = A ∩ B whose MF is related to those of A and B by μC (x) = min(μA (x), μB (x)) = μA (x) ∧ μB (x).

(3)

Thus, the intersection of A and B is the ‘largest’ fuzzy set which is contained in both A and B. The complement of a ¯ is defined as fuzzy set A, denoted by A, μA¯ (x) = 1 − μA (x).

(4)

However, it is understood that these functions are not the only possible generalizations of the crisp set operations. For each of the aforementioned three set operations, several different classes of functions with desirable properties have been proposed subsequently in the literature (Jang et al 1997). A commonly used expression for the intersection of two fuzzy sets, as suggested by Zadeh (1965), is given by the algebraic product as μA (x) ∧ μB (x) = μA (x)μB (x)

∀x ∈ X.

(5)

The most convenient and concise way to define a MF is to express it as a mathematical formula, i.e. as a triangular, trapezoidal, Gaussian, sigmoidal, etc function. The choice for the formula is completely dependent on the application and context. Conventional techniques for system analysis are intrinsically unsuited for dealing with humanistic systems, whose behaviour is strongly influenced by human judgment, perceptions and emotions (Zadeh 1965). As an alternative approach to modelling human thinking, Zadeh (1965) proposed the concept of linguistic variables, which can be thought of as variables commonly employed by humans to describe vague concepts. A clear example of a linguistic variable is ‘age’ (Jang et al 1997). The linguistic variable age, depending on its numerical value, can be assigned to linguistic values like young, old, not very young, not very old, etc. Linguistic variables and values are the basis for the creation of fuzzy if-then rules, which are themselves the basis for the construction of FIS that can effectively model human expertise in a specific application. A fuzzy if-then rule assumes the form: if x is A then y is B, where A and B are linguistic values defined by fuzzy sets on universes of discourse X and Y, respectively. Fuzzy reasoning is an inference procedure that derives conclusions from a set of fuzzy if-then rules and known facts. The basic rule of inference in classical logic is modus ponens, according to which we can infer the truth of a proposition B from the truth of A and the implication: if A, then B. However, in much human reasoning modus ponens is

employed in an approximate manner, i.e. if ‘more or less’ A, then ‘more or less’ B, or if A , then B , where A is close to A and B is close to B. When A, B, A and B are fuzzy sets of appropriate universes, the foregoing inference procedure is called approximate or fuzzy reasoning (Jang et al 1997). 2.2. Mass transport complexes One of the successful applications of seismic data is for the modelling of deep-water geomorphology. One important aspect of deep-water geomorphology is the interpretation of MTCs. Weimer (1989) originally defined the term MTC as chaotic appearing sediments that occur at the base of sequences and are overlain and/or onlapped by channel and levee sediments. They commonly overlie an erosional base up fan, become mounded down fan, are externally mounded in shape and pinch out laterally. In its original usage, the term MTC has a sequence stratigraphic connotation that was used to distinguish it from the generic term slide. However, Weimer and Shipp (2004) pointed out that the term MTC has evolved somewhat from its original definition, and is now used routinely in industry to describe a wide spectrum of mass transport-related sediments, including slides, slumps and debris flow. From a review of the literature, they proposed that all mass transport-related deposits be called MTCs. MTCs can reach tenths of kilometres across, hundreds in length, thousands of square kilometres in area and hundreds of metres in thickness. MTCs are believed to comprise more than 70% of the entire stratigraphic column of all continental margins, and they are found in all kinds of tectonic settings. A usual way to look for MTC material is to look for chaotic, discontinuous, low-amplitude reflections underneath the sea floor in contrast to the overlying levee–channel complex, where high-amplitude and continuous reflectors are dominant. Also, the upper and lower boundaries of MTCs tend to be marked by high-amplitude seismic reflectors. Determining origins or the physical characteristics of the MTCs is beyond the focus of this work (e.g., see Moscardelli et al (2006), Moscardelli and Wood (2008)).

3. Development The seismic profile in figure 1(a) contains a MTC, which is highlighted as manually interpreted by an expert in figure 1(b). The objective of this work was to develop a FIS capable of semi-automatically detecting the internal architecture of the MTC by establishing a set of fuzzy if-then rules consistent with those an actual interpreter would follow to visually detect this feature. These fuzzy rules are dependent on several fuzzy MFs related to different linguistic values. A summary of these functions listing their linguistic value, respective equation and figure where its graph is plotted is shown in table 1. The type of MF each equation represents, along with the heuristically defined parameters for each one of them, is listed in table 2. The process to detect the MTC implied the detection of another geological feature, the sea floor, and therefore, this methodology is also explained. 187

M G Orozco-del-Castillo et al (a)

(b)

Figure 1. (a) The original seismic profile used to recognize the sea floor and the MTC. Note the MTC that lies underneath the sea floor and between two mud volcanoes. (b) The original seismic profile with the MTC highlighted as manually detected by a human interpreter. Table 1. Linguistic values represented by each membership function used in the FIS, its equation number and figure where the respective graph is plotted. Linguistic value

Equation

Strong

SMF (x) =

Around 130 pixels Very rugose Near 130 pixels

DMF (x) = RMF (x) = NMF (x) =

Underneath the sea floor

UMF (x) =

Not in the mud volcanoes

CMF (x) =

1 1+ea11 (x−c11 )

⎧ ⎪ ⎨

e

⎪ ⎩

e

−(x−c21 )2 2 2σ21 −(x−c21 )2 2 2σ22

x c21

a31 (x−c31 ) −(x−c41 )2 2 2σ41 −(x−c41 )2 2 2σ42

x c41

e ⎪ ⎩ e ⎧ 0 ⎪ ⎪ ⎨ x−a51

(6)

5(a)

(7)

5(b)

(10)

5(c)

(11)

5(d)

(12)

5(e)

(13)

5(f )

x > c41 x a51 a51 x b51 b51 x c51 c51 < x

b51 −a51

⎪ 1 ⎪ ⎩ 0 ⎧ 1 ⎪ ⎨ 1 − 1+ x−c61 2b61 ⎪ ⎩ 1−

Figure

x > c21

1 1+e ⎧ ⎪ ⎨

Equation number

x CIM /2

a61

1 x−c 2b62 1+ a 62

x > CIM /2

62

3.1. Detection of the sea floor A usual task for a seismic interpreter is the detection of the sea floor. Normally this process can be done without further complications manually, but to do so along the whole seismic cube could become a very tedious task. More so, as stated 188

in section 2.2, the detection of the sea floor represents an important component for the detection of the MTC, located underneath it. There are basically two cues to identify the sea floor: a strong reflector located at about 1400 m deep (in this particular image, around 130 pixels. As the actual depth is not relevant, throughout the rest of this work we refer


Table 2. Classification of each membership function used in the FIS and the numerical values for their parameters. Equation SMF (x) = DMF (x) = RMF (x) = NMF (x) =

UMF (x) =

CMF (x) =

Membership function 1 1+ea11 (x−c11 )

⎧ ⎪ ⎨ ⎪ ⎩

e e

−(x−c21 )2 2 2σ21 −(x−c21 )2 2 2σ22

x c21

e

−(x−c41 )2 2 2σ41 −(x−c41 )2 2 2σ42

x c41

⎪ ⎩ e ⎧ 0 ⎪ ⎪ ⎨ x−a51 ⎪ ⎪ ⎩

x a51 a51 x b51 b51 x c51 c51 < x

1 0

⎪ ⎩ 1−

a11 = 2 c11 = 4

Combination of two Gaussian functions

c21 = 130 σ21 = 10 σ22 = 30

Sigmoidal

a31 = 2 c31 = 4

Combination of two Gaussian functions

c41 = 130 σ41 = 10 σ42 = 80

Trapezoidal

a51 = DR − 3 b51 = DR + 3 c51 = RIM

Complement of two generalized bells

a61 = 100 b61 = 8 c61 = 0 a62 = 200 b62 = 8 c62 = CIM

x > c41

b51 −a51

⎧ ⎪ ⎨ 1−

Sigmoidal

x > c21

1 1+ea31 (x−c31 )

⎧ ⎪ ⎨

Parameters

1 x−c 2b61 1+ a 61 61 1 x−c 2b62 1+ a 62

x CIM /2 x > CIM /2

62

to the location of the sea floor as where it is located in the image). These cues can be expressed as an if-then rule as ‘if the reflector is strong, and if it is located at about 130 pixels, then the reflector corresponds to the sea floor’. By this approach, this is a well-suited problem for fuzzy logic, since the terms ‘strong’ and ‘at about’ describe vague concepts. Nevertheless, it is necessary to quantify these concepts in order to apply the fuzzy inference process. The concept ‘strong’ related to a ‘strong reflector’ corresponds to how well connected a series of points is from one side of the profile to the other. Since the reflectors that could correspond to the sea floor are horizontal by nature, enhancing this feature improves the process. This is achieved by convoluting the original image with a 2 × 4 kernel which is shown, along with the resulting image, in figure 2(a). The resulting image F consists of 8-bit values representing the ‘horizontal-ness’ of each pixel; for visual purposes, dark pixels and low values represent high horizontal components. Working directly with this image, instead of the original one, makes it more appropriate to deal with the horizontal reflectors that could represent the sea floor. Since there are several possible reflectors representing the sea floor, several ‘seed’ pixels were pseudo-randomly (around 130 pixels deep) located to track the reflectors across the profile. For each starting location or seed pixel (rp , cp ), six of the eight neighbours (those on either side of the seed) of each pixel are taken into account to create a path m across all the columns CIM of the seismic profile. The next pixel in this path corresponds to the neighbour with the lowest value, i.e. with the largest horizontal component. This process yields several candidates for the sea floor, as shown in figure 2(b). As a path is created across a seismic profile representing a reflector, we quantify

how ‘strong’ it is by measuring how ‘tough’ it was to create it. Since the values on the image correspond to the horizontal component of a given pixel, the ‘less tough’ it is to create a path, the stronger the reflector. This approach allows us to define the linguistic variable ‘toughness’, calculated for every reflector candidate j as Tj , the mean of the collection of numbers in the path of the reflector, as stated by the process: mcp = rp mi+1 = min(F (ri − 1, ci + 1), F (ri + 1, ci + 1))

∀cp i CIM

mi−1 = min(F (ri − 1, ci − 1), F (ri + 1, ci − 1)) Tj =

1 C

C

F (ri , ci + 1), F (ri , ci − 1),

∀cp > i 1

mi .

i=1

This variable Tj is evaluated into a sigmoidally shaped MF SMF (x) associated with the linguistic value ‘strong’ (as in strong reflector), defined as SMF (x) =

1 . 1 + ea11 (x − c11 )

(6)

Once the strength of a reflector is quantified, the decision whether a reflector corresponds to the sea floor, according to the if-then rule stated earlier, depends on the depth of a reflector. In other words, we are trying to locate the strongest reflector located at about 130 pixels. Since a reflector spans several rows of pixels, we consider its depth as the mean of the rows of all the pixels the path encompasses. It can now be defined as a Gaussian combination MF corresponding to the 189

M G Orozco-del-Castillo et al (a)

(b)

(c)

Figure 2. (a) The top segment of the resulting seismic profile from convoluting the original profile with the filtering mask shown in the image, and (b) several candidates for the sea floor generated from tracking along the profile with different seed pixels. (c) The selected candidate for the sea floor overlapped in the original seismic profile with lower brightness for its better visualization.

linguistic value ‘around 130 pixels’, whose linguistic variable is defined as ‘average depth’. The MF is defined as ⎧ −(x−c21 )2 ⎪ ⎨e 2σ212 x c21 DMF (x) = . (7) −(x−c21 )2 ⎪ ⎩e 2σ222 x > c21 Now that the ‘depth’ and ‘strength’ of all of the possible reflectors defining the sea floor are quantified, the fuzzy inference process is applied to each of them according to the previously stated fuzzy rule: ‘if the reflector is strong, and if it is located at about 130 pixels, then the reflector corresponds to the sea floor’. The conjunction ‘and’ can be thought of as the intersection of both fuzzy sets ‘strong’ and ‘around 130 pixels’, and was implemented as the algebraic product of the membership of every possible reflector to each fuzzy set, as described in equation (5). The reflector with the highest associated value is shown overlaid the original seismic profile in figure 2(c). 3.2. Detection of the MTC internal architecture As stated in section 2.2, a usual way to look for MTC material is to look for chaotic, discontinuous, low-amplitude reflections underneath the sea floor in contrast to the overlying levee–channel complex, where high-amplitude and continuous reflectors are dominant. The keywords for the detection of 190

MTC material are therefore chaotic, discontinuous or irregular, and underneath the sea floor. Note how the region containing the MTC in the original seismic profile can be interpreted as chaotic material near and underneath the sea floor, located between the mud volcanoes on opposite sides of the profile. The aim of this work was to obtain these same results by automatically processing the image and constructing a FIS, consisting of the linguistic rule ‘if an element is chaotic, and if it is located near and underneath the sea floor and between the two mud volcanoes, then the element corresponds to the MTC internal architecture’. For the construction of the FIS it is needed to quantify the fuzzy terms ‘chaotic’, ‘near and underneath’ and ‘between’, so MFs can be applied to each one of them. Chaotic or irregular reflections can be thought of as highfrequency zones, which can be enhanced by applying a highpass filter to the original image. This was implemented by subtracting the results of a usual low-pass filter (Ks ) convoluted to the original image, as done in equation (8): H = |I − (I ∗ Ks )| .

(8)

This new image H, shown in figure 3(a), clearly shows irregular areas as brighter spots in the image than those homogeneous ones, point by point, but fails to address the matter about whole irregular areas on the image, i.e. a pixel is shown dark if its immediate neighbours share its value, even if it is contained in a bigger area which is clearly irregular in a

Fuzzy logic and image processing techniques for the interpretation of seismic data (a)

(b)

Figure 3. (a) The resulting image H after enhancing the high-frequency features as done in equation (8). (b) The rugosity image R obtained by equation (9). Note how the image in (b) represents the image in (a) in a more general (not pixel-by-pixel) manner.

general sense. It was proposed that successive applications (n) of the convolution operation of the same low-pass filter Ks to image H (as shown in equation (9)) can help achieve this more general perception of irregular areas as opposed to pixel-by-pixel analysis obtained by the mere application of equation (8). Following this, a more adequate image is shown in figure 3(b), which we refer to as a ‘rugosity’ image R. It can be seen that larger groups of pixels can be associated, thereby accounting for homogeneous and heterogeneous areas more precisely: R = Ks∗n ∗ H.

(9)

Once ‘rugosity’ has been defined as a fuzzy or linguistic variable and as an image, a ‘very rugose’ (irregular, chaotic) sigmoidally shaped MF was defined as equation (10). The x-axis corresponds to the intensity level of each of the pixels in the image R: 1 . (10) RMF (x) = a 31 1 + e (x−c31 ) The concept of ‘near’ the sea floor is analogous to the already described concept of ‘around 130 pixels’ used to describe the depth of the reflectors (possibly representing the sea floor) as a Gaussian combination MF. The difference resides that in this case the concept of ‘near’ is not as rigorous as in the previously used context. In other words, a reflector at 150 pixels is definitely not ‘near’ the 130 pixel mark, but a point at 150 pixels deep could very well correspond to the

MTC internal architecture, in other words, ‘near’ is used here with distinct comprehensions. Therefore, it is necessary to modify the MF to allow a broader definition of ‘near’, as shown in equation (11): ⎧ −(x−c41 )2 ⎪ ⎨e 2σ412 x c41 . (11) NMF (x) = −(x−c41 )2 ⎪ ⎩e 2σ422 x > c41 To account for the ‘underneath the sea floor’ concept, a trapezoidal-shaped MF was defined. This function behaves practically as a step function, where the value for pixels in rows greater than the average depth of the sea floor, DR , but lower than the rows in the image, RIM, is 1, and 0 otherwise. The complete definition is given by equation (12): ⎧ 0 x a51 ⎪ ⎪ ⎪ ⎪ x−a 51 ⎨ a51 x b51 b51 −a51 UMF (x) = . (12) 1 b51 x c51 ⎪ ⎪ ⎪ ⎪ ⎩ 0 c51 < x Finally, to exclude those elements contained in the mud volcanoes and be able to include only those elements between them, we proposed to use the complement of a combination of generalized bell-shaped functions as MFs. Two generalized bell-shaped MFs, each one accounting for half of the total columns of the image, CIM , were defined. These MFs centred near values corresponding to the first and the last 191


Figure 4. The final results of the methodology, the MTC as automatically detected by the FIS. (a)

(b)

(c)

(d )

(e)

(f)

Figure 5. Plots of the different membership functions used in the fuzzy inference system: (a) ‘strong’, equation (6); (b) ‘around 130 pixels’, equation (7); (c) ‘very rugose’, equation (10); (d) ‘near 130 pixels’, equation (11); (e) ‘under the sea floor’, equation (12); (f ) ‘between the mud volcanoes’, equation (13).

columns of the image respectively can describe the concept of whether an element is contained ‘in the mud volcanoes’, so the complement is here used as the negation of it (‘not in the mud volcanoes’). The MF is described by equation (13): ⎧ 1 − x−c1 2b61 x CIM /2 ⎪ ⎪ ⎨ 1+ a 61 61 CMF (x) = . (13) x > CIM /2 1 − x−c1 2b62 ⎪ ⎪ ⎩ 1+ a 62 62

Once every linguistic value has been associated with a fuzzy MF, the inference process to detect the MTC is done analogously as the process of detecting the sea floor in section 3.1. The only conjunction operator in the if-then rule for the detection of the MTC is ‘and’, so only an intersection operation, as opposed to union or complement, is needed. 192

The best-suited operation for the intersection was heuristically determined as the algebraic product of the evaluated MFs. The inference process is done pixel by pixel, so its result is another image where high-valued pixels represent locations of the MTC internal architecture. After binarizing these values with an experimentally defined threshold, the resulting pixels are shown overlaid the original seismic profile in figure 4. Note how the automatically interpreted MTC clearly resembles the manually interpreted one (figure 1(b)).

4. Conclusions Due to the measurement uncertainties in seismic data, the vagueness in the interpretation process, the impreciseness and


the abstract nature of the concepts and thoughts involved in the manual interpretation of MTCs, we consider this kind of seismic interpretation problem well suited for the methodology proposed in this paper. It is shown that the characteristics to detect the internal architecture of a MTC can be expressed as a set of fuzzy if-then rules, where the linguistic nature in the process of manual interpretation can be approximated by a fuzzy MF representing vague concepts. As stated in the previous section, the methodology is rule dependent, i.e. if the rules do not properly represent what is looked for, the FIS will not function adequately. This is not necessarily prohibitive, since inadequate results suggest that the rules are not well established, and their construction can be corrected improving the performance of both manual and automatic interpretations. In this particular application, it can be seen in figure 4 that certain areas inside the MTC are not detected as part of its internal architecture, which implies they do not comply with the established rules; so probably a rule stating that MTCs should be closed areas is missing as part of the definition of a MTC. The promising results of the semi-automated MTC interpretation in comparison to the manual interpretation lead us to believe that several other seismic interpretation tasks can be appropriately handled by means of fuzzy logic.

Acknowledgments This work is part of the collaboration program between the Mexican Institute of Petroleum Research and the National University. This contribution was supported by project IMP/D.00475, Y.00107, SENER-Conacyt 128376. The authors are grateful to the anonymous reviewers for their valuable suggestions which greatly improved the quality of this paper. The authors also thank Denise Trejo for her detailed revision of the grammar on this manuscript.

References Adams R D, Collister J W, Ekart D D, Levey R A and Nikravesh M 1999a Evaluation of gas reservoirs in collapsed paleocave systems Ellenburger Group, Permian Basin, TX, AAPG Annual Meeting (San Antonio TX, 11–14 April) Adams R D, Nikravesh M, Ekart D D, Collister J W, Levey R A and Siegfried R W 1999b Optimization of well locations in complex carbonate reservoirs GasTIPS vol 5, GRI 1999 Aguirre E and Gonzalez A 2000 Fuzzy behaviors for mobile robot navigation: design, coordination and fusion Int. J. Approx. Reason. 25 255–89 Aldridge B B, Saez-Rodriguez J, Muhlich J L, Sorger P K and Lauffenburger D A 2009 Fuzzy logic analysis of kinase pathway crosstalk in TNF/EGF/insulin-induced signaling PLoS Comput. Biol. 5 Aminzadeh F 1991 Where are we now and where are we going? Expert Systems in Exploration ed F Aminzadeh and M Simaan (Tulsa, OK: SEG Publications) pp 3–32 Aminzadeh F and Chatterjee S 1984 Applications of clustering in exploration seismology Geoexploration 23 147–59 Baum B A, Tovinkere V, Titlow J and Welch R M 1997 Automated cloud classification of global AVHRR data using a fuzzy logic approach J. Appl. Meteorol. 36 1519–40

Baygun B, Luthi S M and Bryant I D 1996 Applications of neural networks and fuzzy logic in geological modeling of a mature hydrocarbon reservoir Artificial Intelligence in the Petroleum Industry vol 2 (Paris, France: Pennwell) chapter 5 pp 125–38 Bois P 1983 Some applications of pattern recognition to oil and gas exploration IEEE Trans. Geosci. Remote Sens. 21 416–26 Bois P 1984 Fuzzy seismic interpretation IEEE Trans. Geosci. Remote Sens. 22 692–7 Chang N B, Chen H W and Ning S K 2001 Identification of river water quality using the fuzzy synthetic evaluation approach J. Environ. Manag. 63 293–305 Chappaz R J 1977 Application of the fuzzy sets theory to the interpretation of seismic (abstract) 47th SEG Meeting (Calgary, Canada) Paper R-9 Chen H C, Fang J H, Kortright M E and Chen D C 1995 Novel approaches to the determination of Archie parameters: II. Fuzzy regression analysis SPE 26288-P Gourley J J, Tabary P and Chatelet J P 2007 A fuzzy logic algorithm for the separation of precipitating from nonprecipitating echoes using polarimetric radar observations J. Atmos. Ocean. Technol. 24 1439–51 Hashemi H, Javaherian A and Babuska P 2008 A semi-supervised method to detect seismic random noise with fuzzy GK clustering J. Geophys. Eng. 5 457 Ilkhchi A K, Rezaee M and Moallemi S A 2006 A fuzzy logic approach for estimation of permeability and rock type from conventional well log data: an example from the Kangan reservoir in the Iran Offshore Gas Field J. Geophys. Eng. 3 356 Jang J, Sun C T and Mizutani E 1997 Neuro-Fuzzy and Soft Computing—A Computational Approach to Learning and Machine Intelligence (Englewood Cliffs, NJ: Prentice Hall) Jensen O N, Mortensen P, Vorm O and Mann M 1997 Automation of matrix-assisted laser desorption/ionization mass spectrometry using fuzzy logic feedback control Anal. Chem. 69 1706–14 Li T H S and Huang Y C 2010 MIMO adaptive fuzzy terminal sliding-mode controller for robotic manipulators Inf. Sci. 180 4641–60 Luo X, Jennings N R, Shadbolt N, Leung H F and Lee J H M 2003 A fuzzy constraint based model for bilateral, multi-issue negotiations in semi-competitive environments Artif. Intell. 148 53–102 Mitra B, Scott H D, Dixon J C and McKimmey J M 1998 Application of fuzzy logic to the prediction of soil erosion in a large watershed Geoderma 86 183–209 Moscardelli L and Wood L 2008 New classification system for mass transport complexes in offshore Trinidad Basin Res 20 73–98 Moscardelli L, Wood L and Mann P 2006 Mass-transport complexes and associated processes in the offshore area of Trinidad and Venezuela AAPG Bull. 90 1059–88 Nikravesh M and Aminzadeh F 2001 Past, present and future intelligent reservoir characterization trends J. Pet. Sci. Eng. 31 67–79 Nikravesh M, Aminzadeh F and Zadeh L A 2003 Soft Computing and Intelligent Data Analysis in Oil Exploration (Amsterdam: Elsevier) Park J H 2003 Fuzzy-logic zero-moment-point trajectory generation for reduced trunk motions of biped robots Fuzzy Sets Syst. 134 189–203 Saffioti A 1997 Fuzzy logic in autonomous robotics: behavior coordination Proc. 6th Int. Conf. on Fuzzy Systems pp 573–8 Tamhane D, Wong P M and Aminzadeh F 2002 Integrating linguistic descriptions and digital signals in petroleum reservoirs Int. J. Fuzzy Syst. 4 586–91 Venayagamoorthy G K, Grant L L and Doctor S 2009 Collective robotic search using hybrid techniques: fuzzy logic and swarm intelligence inspired by nature Eng. Appl. Artif. Intell. 22 431–41

193


Weimer P 1989 Sequence stratigraphy of the Mississippi Fan (Plio-Pleistocene), Gulf of Mexico Geo-Mar. Lett. 9 185–272 Weimer P and Shipp C 2004 Mass transport Complex: Musing on past uses and suggestions for future directions Offshore Technology Conference (Houston, TX) Paper presented at OTC paper 16752 Zadeh L A 1965 Fuzzy sets Inf. Control 8 33353 Zadeh L A 1992 The calculus of fuzzy if-then-rules AI Expert 7 23–7

194

Zadeh L A 1997 The roles of fuzzy logic and soft computing in the concept, design and deployment of intelligent systems Software Agents and Soft Computing: Towards Enhancing Machine Intelligence, Concepts and Applications ed H S Nwana and N Azarmi (London: Springer) pp 183–90 Zhou C and Meng Q 2003 Dynamic balance of a biped robot using fuzzy reinforcement learning agents Fuzzy Sets Syst. 134 169–87