RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 10, ES1001, doi:10.2205/2007ES000278, 2008

5. FCARS: Global Level

[43]  Like DRAS and FLARS, FCARS (Fuzzy Comparison Algorithm for Recognition of Signals) uses at a local level the procedure described in section 3.1 and providing FTS rectification. At a global level, the FCARS search for oscillating rises in a rectification can be described as follows. Significant vertical spikes are first detected in the rectification. Their fairly dense clusters and adjacent features are of interest. Points lying inside such clusters are considered as anomalous without regard for values taken by the rectifying function at these points. Small rectification values at these points can imply only a short-term weakening of a signal due to its inhomogeneity. Such points form central parts of rises in accordance with clusters of the aforementioned vertical spikes.

[44]  Points lying to the left and to the right of the dense clusters can be of two types: these are either quiet, background points near a given cluster or disturbed points of the record that are not necessarily extremal in the rectification and form the initial and final stages of the signal.

[45]  Thus, we can draw the following conclusion: anomalies in a record y correspond to oscillating rises of the rectification Fy. Bases of the anomalies are connected sets in the initial recording interval that consist of points extremally horizontally close to vertically extremal points of the rectification.

[46]  Precisely this definition of a rise serves as a basis for the FCARS global level. FCARS modeling is based on fuzzy comparisons and monolithicity [Bogoutdinov, 2006]. Fuzzy comparisons are instrumental to a correct formulation of the notion of vertically extremal spikes in a rectification. The degree of extremal horizontal proximity to the spikes is described in terms of proximity measures. Further, likewise with the help of fuzzy comparisons, the shell of the rise (anomaly) base is formed by filling gaps in dense clusters of vertically anomalous spikes with horizontally extremal points.

[47]  The properties of proximity measures are used for locating the central part of the rise in this shell. The rise foot is finally extracted after the localization of the side parts of the rise using fuzzy logic and fuzzy comparisons. Below we present an exact description of the FCARS.

5.1. FCARS: Vertical subdivision (the first variant).
[48]  In this case, the vertical measure of anomalousness m v(k)in [-1, 1] at the point k is defined as a fuzzy comparison of Im Fy with the rectification value at this point:

m v(k) = n( ImFy, Fy(k)),

where n(A, a) is defined by formulas (2)-(4) (i.e. n is the binary, gravitational, or t -extension of the fuzzy comparison ng, n(a, b) ).

[49]  Let as aw) be the strong (weak) level of extremality with respect to the modulus of Im Fy, that is as (aw) is the solution of the equation

n( ImFy, as) = 0.5(n( ImFy, aw) = 0).

Definition 5

[50]  (a) The point k is of the vertically background type if m v(k) < 0 Fy(k) < aw.

[51]  (b) The point k is vertically anomalous if m v(k)ge 0.5 Fy(k)ge as.

[52]  (c) The point k is vertically potentially-anomalous if m v(k) in [0, 0.5] Fy(k)in [aw, as].

Let v B, v A, and v P, denote respective sets of vertically-background, vertically anomalous, and vertically potentially-anomalous points. Then the recording period under consideration can be represented as T = v Bcup v Acup v P.

5.2. FCARS: Vertical subdivision (the second variant).
[53]  In this case, the vertical measure of anomalousness m v(k) at the point k is the FLARS measure, defined by (5), and

T = v Bcup v Acup v P,

where

B = {kin T:m v(k)<0},

A = {kin T: m v(k)ge 0.5},

and

v P = {kin T: m v(k)in [0, 0.5]}.

5.3. FCARS: Horizontal subdivision.
[54]  We introduce the left and right measures of proximity to the vertically anomalous subset v A in the model of local survey eq036.gif

eq037.gif(6)

where

eq038.gif

[55]  Note. Measures (6) are connected with the DRAS standard background measures Las and Ras [Gvishiani et al., 2003] via fuzzy negation:

L v AFy = 1 - LasFy, R v AFy = 1 - RasFy.

[56]  The fuzzy disjunction

m v A(k) = max(L v AFy(k), R v AFy(k))

is a measure of anomalousness with respect to v A in T and formally characterizes the bilateral proximity to v A in T. The measure m v A in turn serves as a basis for the construction of the horizontal anomalousness measure mh in FCARS:

mh(k) = n( Im(m v A), m v A (t)).

Definition 6.
[57]  (A) The point k is of the horizontally background type if mh(k) < 0.

[58]  (B) The point k is horizontally anomalous if mh(k)ge 0.5.

[59]  (C) The point k is horizontally potentially-anomalous if mh(k)in[0, 0.5].

[60]  Let hB, hA, and hP, denote respective sets of horizontally-background, horizontally anomalous, and horizontally potentially-anomalous points. This triad provides a horizontal subdivision of the recording interval

eq039.gif

[61]  The set hB being considered as background, its complement hAcup hP in T is the disjunctive union of nonbackground intervals eq040.gif      The FCARS processes on such of them that intersect with hA (Pmcup hA oslash): it is the points of hA that are considered as actually anomalous because they lie in a neighborhood where the concentration of points with vertically extremal values of the rectification Fy is horizontally extremely high.

[62]  Thus, we have Pmcap hA oslash. We will determine the boundaries of an anomaly s in Pm, i.e. the rise platform in the rectification Fy corresponding to this anomaly. For this, we need the following statement according to which an anomalous interval in Pm necessarily contains points that are anomalous both vertically and horizontally.

[63]  Statement 7. If Pm = [b, e] in hAcup hP, Pmcap hAoslash then Pmcap v Acap hA oslash.

[64]  Proof. Let k*in [b, e]cap hA and k* v A. If k* is anomalous on the left, the interval [b, k*] necessarily contains vertically anomalous points because otherwise all points to the left of k* up to the point b -1 inclusively would be horizontally anomalous, which contradicts its vertical background property. If k** is a point vertically anomalous in [ b, k* ] and closest to k*, then L v AFy(k**) > L v AFy(k*). Consequently, k** is a horizontally and vertically anomalous point: k**in Pmcap v Acap hA. The right-hand case is treated analogously. Thereby, the statement is proven.

[65]  Let bA (eA) is the first (last) point in the intersection Pncap hAcap v A. The segment [ bA, eA ] in FCARS is considered as the central part of the signal s, which means that its boundaries bs and es lie in the intervals [b, bA] and [eA, e], respectively.

5.4. FCARS: anomaly boundaries.
[66]  As regards verticality, two types of points are present in the interval [b, bA]: background points with m v(k) < 0 and nonbackground points with m v(k)ge 0 (see Definition 5). The "logic'' of the beginning of an anomaly s is formulated as follows: the point bs in the interval [b, bA] should lie as far as possible to the right of background points and to the left of nonbackground points.

[67]  Now we formalize this logic on the basis of fuzzy comparisons. Let

C = {k in [b, bA]: m v(k) < 0}

and

D = {kin [b, bA]: m v(k)ge 0}.

The function n(C, k) (n(k, D)) is the measure of the fact that k lies the to right of C (to the left D ). Their fuzzy conjunction min(n(C, k), n(k, D)) is the measure of the fact that k lies to the right of C and to the left of D. Therefore, the anomaly beginning bs can be naturally set equal to the absolute maximum of this conjunction in the interval [ b, bA ]:

eq041.gif

2007ES000278-fig03
Figure 3
[68]  The anomaly end es in the interval [ eA, e ] is defined quite analogously:

eq042.gif

Figure 3 presents results obtained by FCARS processing of a seismic record.

[69]  Free parameters of FCARS are the rectifying functional F, local survey window 0 < Dll |T|, and the fuzzy comparison ( n ) of positive functions. Accordingly, the algorithm can also be written as FCARS( F, D, n ).

5.5. Comparative analysis of FCARS with DRAS and FLARS.

[70]  1. Constructively, proximity measures in FCARS coincide with background measures in DRAS.

[71]  2. The set of anomalies hAsubset T in FCARS is formed "almost in the same way'' as in FLARS.

[72]  3. As distinct for DRAS and FLARS, the choice of the free parameters a and b in FCARS is fully automated.

[73]  4. FCARS differs basically from DRAS and FLARS by the block determining anomaly boundaries (subsection 5.4.): it uses the vertical subdivision T = v B + v A + v P (Definition 5).


RJES

Citation: Gvishiani, A. D., S. M. Agayan, Sh. R. Bogoutdinov, E. M. Graeva, J. Zlotnicki, and J.  Bonnin (2008), Recognition of anomalies from time series by fuzzy logic methods, Russ. J. Earth Sci., 10, ES1001, doi:10.2205/2007ES000278.

Copyright 2008 by the Russian Journal of Earth Sciences

Powered by TeXWeb (Win32, v.2.0).