Method and Apparatus for Detecting Seismic Events by Identifying Crossings in the NTH Derivative of a Waveform

1. Field of the Invention

This invention relates to marking significant events in a seismic trace. More particularly, it relates to a computer system for automatically detecting significant events in a seismic trace.

2. Description of the Prior Art

The prior art has worked with two different approaches (1) solving for the position of horizons and (2) determining the location of significant changes by amplitude zero-crossings in seismic traces.

Attempts to solve for the position of horizons in the general and practical case have not been successful. In principle, if the seismic source is known exactly and if the earth does not attenuate the seismic signal, the seismic data could be deconvolved to obtain the locations of horizons. However, the source can only be known approximately and, more importantly, the earth attenuates high frequencies in the signal, severely reducing the resolution of seismic data. As a result, any single significant change in a seismic trace is likely to correspond to not just one, but many confused reflections which cannot be distinguished.

Previous approaches that have determined significant changes by amplitude zero-crossings in seismic traces essentially amount to an assertion that the significant changes of interest are primarily determined by points where the trace changes sign. However, this approach is not suitable for mapping seismic events whose amplitudes change sign without a zero-crossing.

The invention of this specification distinctly marks significant changes (turnings) such as local extrema, saddle points, inflections and other high-order changes.

A digital computer system has a memory for storing a seismic trace represented by consecutive points in time, with the corresponding amplitude attached. The system takes the derivative at each point by determining the slope at each point. This process is repeated until the desired Nth derivative has been taken. The system then marks each zero-crossing in the Nth derivative and establishes the amplitude of the trace at the time of the zero-crossing of the Nth derivative by interconnecting the immediately preceding and following points. The amplitude at which the interconnection crosses the time of the zero-crossing determines the amplitude.

The times of the zero-crossings and their associated trace amplitudes are stored for subsequent use. They also may be displayed by way of hard copy, electronic display, or the like. The derivative based on determining the slope at the various points is an approximation. Other techniques, well known, are available. Also, the actual amplitude at the zero-crossing may be computed through the use of a different interpolation. In this preferred embodiment, it has been determined that the linear interpolation approximation described above is adequate for this purpose.

This invention forms a part of the combination invention described and claimed in copending U.S. Pat. No. 4,633,402.

The principle object of this invention is to automatically determine events of interest in a seismic trace.

Another object of this invention is to determine the zero-crossings of the Nth derivative of the trace.

These and other objects will be made evident in the detailed description that follows.

FIG. 1A illustrates a three-dimensional volume of seismic data.

FIG. 1B illustrates three-dimensional horizons produced from the volume of FIG. 1A, using the method and apparatus of this invention.

FIG. 2 is a block diagram of the computer system of this invention.

FIG. 3 illustrates a typical trace is analog form.

FIG. 4 illustrates two analog traces in the application of the mutual nearest neighbor criterion.

FIG. 5 is a reproduction of FIG. 4 except that the effect of also requiring like signs is illustrated.

FIGS. 6A-6H illustrate traces and indicate horizons determined from zeros of successive derivatives taken with N=0 at FIG. 6A and N=7 at FIG. 6H.

FIG. 7 illustrates the low curvature criterion.

FIG. 8A illustrates 3-D continuity of four turnings forming four pairs of turnings.

FIG. 8B, from a top view, illustrates 3-D continuity.

FIG. 9 illustrates vertical uniqueness.

FIG. 10 illustrates transitivity of the pair relation to reformat to the array format.

Three dimensional seismic data is collected typically through the use of vibrators for land data collection and air guns for sea data collection. In either event, the data is collected, processed, and presented as a three-dimensional volume formed of digitized traces. The traces are represented as a series of points, each point being described digitally. By examining the series of points for significant seismic events and by interrelating significant events between traces, three dimensional horizons may be developed.

The processed three-dimensional seismic data sections traditionally have been interpreted by human experts. A tedious and time-consuming task that the interpreter must perform is to pick seismic events through the volume and to map the time and amplitude structure of the events. This invention addresses the problems of automatically picking and mapping three-dimensional seismic event horizons.

FIG. 1A illustrates a typical processed three-dimensional seismic data volume 40. The gradations in shade are generally proportional to changes in amplitude of the seismic signal returned as a result of a seismic source such as a vibrator or air gun having been activated.

FIG. 1B indicates typical three-dimensional horizons 50a-50e that lie within three-dimensional seismic volume 40 where time is the vertical dimension. These horizons are the end result of this invention and may be displayed as shown on paper, or may be displayed in the face of a cathode ray tube, or any other display apparatus. The horizons may be viewed from any direction, of course, and other information about the horizon, such as the seismic amplitudes on the surface, may also be displayed. Horizon, in the strict geophysical sense, means "geological interface". As used in this invention, horizon means "interpreter's sketch of a surface through seismic data." The wavelengths of the source typically imparted into the earth in seismic exploration are of such a magnitude as to preclude the detection of all "geological interfaces". Therefore, the horizons that are referred to in this specification are surfaces in the three-dimensional seismic volume 40 that are estimated to be everywhere locally parallel to the "geologic interfaces".

The automatic production of the horizons 50a-50e is accomplished in four steps:

Step 1: Determine and describe significant points of change along each seismic trace.

Step 2: Determine pairs of turnings such that the turnings within pairs lie on the same three-dimensional seismic event.

Step 3: Determine seismic event surfaces (horizons).

Step 4: Produce physical maps of the seismic event surfaces (horizons).

FIG. 2 illustrates the LISP machine of this invention, a LAMBDA machine made by LISP Machines, Inc. The LAMBDA system and the use of the LISP programs in the Appendix, are set out in detail in the following manuals of LISP Machine, Inc.:

"Introduction to the LISP System"--1983

"ZMACS-Introductory Manual"--1983

"ZMAIL Manual"--April 1983;

"Window System Manual"--August 1983;

"Lisp Machine Manual"--January 1983;

"System 94 Release Notes"--Undated

In block form, a LISP processor 11 is connected via bus 17 (a Texas Instruments NuBus.TM. type bus) to a Motorola 68000 processor 12, main memory 13, display controller 14 and system diagnostic unit 15. Display controller 14 controls video display 20 which is connected to RS232 interface 18. The video display 20 may be used for displaying the representation of the three-dimensional section 40 and for the display of the horizons 50a-50e. Keyboard 22 and mouse 21 are connected to video display 20 for interaction therewith. SMD disk controller 23 controls disk drive 25 which is used for storage as is main memory 13. Printer 30 is connected to RS232 port 29 which in turn is connected through the system diagnostic unit 15 to LISP processor 11, main memory 13 and SMD disk controller 23. The printer 30 is therefore accessible to a storage unit for receiving a representation of the horizons 50a-50d for printing hard copies of the desired images.

Turning now to FIG. 3, Step 1 as carried out by system 10 will be described. For the purpose of description, each significant point of change along a seismic trace is a turning. The description of a turning includes its position in the volume, the amplitude of the seismic trace at that position, and its sign (negative or positive). Trace 51 is represented in an analog manner. Points 52-58 represent turning points in the case where turnings are defined by zero-crossings in the first derivative. In fact, trace 51 is represented by a series of digital numbers. Each number signifies an amplitude. Turning points 52, 54, 57 and 58 have positive amplitudes and points 53, 55, and 56 have negative amplitudes. However, the signs of the turning are determined by the (N+1)th derivative of the trace and thereof the sign of turnings 54, 55 and 56 are positive while the sign of turning points 52, 53, 57 and 58 are negative. The turning points are determined using zero-crossings in the Nth derivative. The turnings have the advantage of distinctly marking significant changes such as local extrema, saddle points, inflections and other high-order changes. This is in contrast to prior art methods of looking at the zero crossings of the original trace as significant. Exactly which changes are marked can be controlled by varying the value of N, a parameter of step 1. The derivative at each sample point, t0, is approximated as follows:

1. Let t-, t0, and t+ be the times of three consecutive sample points.

2. Let A (t) be the amplitude of the trace at time t.

3. Then the derivative, D, of the trace at t0 is computed by:

This approximation for computing the derivative is derived from the parabola P, passing through (t-, A(t-)), (t0, A(t0)), and (t+, A(t+)). D (t0) is the slope of P at t0. The solution is simple because the sample points are regularly spaced at unit intervals, allowing the following property of parabolas to be exploited:

Theorem: Let P(x)=AX.sup.2 +BX+C. Then the derivative of P at a value half way between two values, v and w, is the slope of the line through (v, P(v)) and (w, P(w)). That is:

Higher order-derivatives are computed by iterative applications of the scheme for calculating the derivative as shown above.

In principle, the first derivative defines turnings at local extrema and saddles of the seismic trace. The first derivative therefore captures attributes of an important turning point quite well. The second derivative defines turnings at inflections and straight segments of the trace. In that case, the term "turning" is somewhat of a misnomer, but the inflections may serve just as well or better than the local extrema and saddles for a given trace. As N becomes longer, smoother and more complete, three-dimensional horizons are produced. In practice, N=7 has the advantage of marking changes or mapping that are ordinarily not noticed by human inspection. This may be seen by an inspection of FIGS. 6A-6H where FIG. 6A illustrates traces with N=0. FIG. 6B illustrates N=1, FIG. 6C illustrates N=2, and so on with FIG. 6H illustrating N=7. The implementation of this step 1 may be seen in the appendix beginning on Page 2 thereof under the title "STEP 1: DETERMINE TURNING POINTS".

By specifying only those points that are turning points in the Nth derivative, data compaction is accomplished. Of course, a variety of other techniques, including frequency doman filters, may be employed to approximate the derivatives.

FIG. 4 illustrates traces 60 and 61 having the mutual nearest neighbor criterion applied. The mutual nearest neighbor criterion was originally devised and used for solving problems in computer vision as set out in the Ph.D dissertation "A Computational Theory of Spatio-Temporal Aggregation for Visual Analysis of Objects in Dynamic Environments" by Bruce E. Flinchbaugh. The Ohio State University, June 1980.

This mutual nearest neighbor criterion has been applied in the past to seismic interpretation relation to two dimensional seismic sections. This prior art technique involved the two-dimensional lineation of wavelets. Wavelets are defined as that portion of a wiggle trace between amplitude zero-crossings when N=0. A lineation refers to a two-dimensional horizon in a 2-D seismic section. However, a technique is applied herein for the first time to a three-dimensional volume 40. That is, the mutual nearest neighbor criterion is applied in the X direction and in the Y direction to each and every turning using Nth derivative of every trace making up the three-dimensional volume 40.

In FIG. 4, turning point A is shown connecting to turning point B. That is, the nearest neighbor to point A of trace 60 in trace 61 is point B. Likewise, the nearest neighbor to point D of trace 60 is point E of trace 61. The nearest neighbor to point F of trace 60 is point G of trace 61. In a similar manner, the nearest neighbor to point B of trace 61 is point A of trace 60. The nearest neighbor of point E of trace 61 is point D of trace 60. The nearest neighbor to point G of trace 61 is point F of trace 60. The nearest neighbor of point C of trace 60 is point B of trace 61, but because point B's nearest neighbor is not point C, point C does not have a mutual nearest neighbor, thus point C is not paired with any turning. It should be noted that while these traces are shown in a 2-D analog representation, the mutual nearest neighbor is based only on the time dimension which is the vertical dimension as shown in FIG. 4.

Turning point D has been paired with turning point E. Turning point E has a radically different character from turning point D because its (N+1)th derivative is positive while turning point D's is negative. Likewise, turning point G's (N+1)th derivative is negative while turning point F's is positive.

FIG. 5 illustrates that turning points D and E may not be linked nor may turning points F and G. This prohibition against such links is made based on the assumption that the character of a three-dimensional seismic horizon, as represented by the sign of the turning cannot change radically between adjacent traces. Therefore, the criterion is that the signs must be homogeneous. The implementation of the mutual nearest neighbor and homogeneous signs criteria may be seen in the appendix on page 3 under the heading "STEP 2: DETERMINE PAIRS OF TURNINGS ON THE SAME HORIZON", extending to page 6 to the heading "APPLY-LOW-CURVATURE-CRITERION".

Each turning point, through this implementation, has now been identified as being paired with up to four other turnings, one in each of the four directions from the turning (the positive and negative X and Y directions).

As part of Step 2, the curvature of the path formed by the paired turnings in successive digital traces is assessed in both the X and Y directions. In FIG. 7 line segments are drawn between paired turnings to represent path 70 through turnings H, J, K, L, and M. The curvature of the path at K is assessed by considering the slope, m1, of the segment between J and K, and the slope, m2, of the segment between K and L. If m1 and m2 differ by more than a predetermined value, the path 70 is "cut" at K by eliminating the pairs (J, K) and (K, L). The predetermined value is a parameter of the method. In other words, a pair can only occur when the curvature of the path through the turnings of the pair, together with the turnings of an incident pair in the same direction, is less than a maximum curvature. Restricting pairs of turnings in this manner helps to rule out implausible three-dimensional seismic horizons. The implementation of the low curvature criterion may be studied in the Appendix beginning at page 6 under the heading "APPLY-LOW-CURVATURE-CRITERION".

Step 2 also includes a requirement of three-dimensional continuity. This criterion requires that a pair can only occur wherever the pair is a link in a closed loop of turnings. Restricting pairs of turnings in this manner helps to assure that pairs are on seismic events that have three-dimensional extent as a horizon. This criterion is based on the assumption that if two turnings lie on a three-dimensional seismic event, then other turnings lying on the same event surface are nearby. Specifically, this criterion is described as follows:

Let T1 and T2 be sets of turnings such that T1 contains the turnings from one seismic trace and T2 contains the turnings from an adjacent trace in any direction within the seismic data section. Let a closed loop of turnings be an ordered list of at least three turnings, (t0, t1, t2, . . . tn), such that the ti are distinct and the following pairs exist:

((t0, t1), (t1, t2) . . . (tn-1, (tn), tn, t0)). Then t1 in T1 and t2 in T2 may only form a pair when a closed loop of turnings exists that include t1 and t2.

Closed loops spanning just a few (e.g, four) traces in closely spaced three-dimensional seismic data can be assumed to bound a region of a surface that intersects those traces. Reference to FIGS. 8A and 8B should now be made where, in FIG. 8A, four traces Ta, Tb, Tc and Td are shown. Turning point pairs (a, b) (b, c) (c, d) and (d, a) are shown forming a closed loop. The four turning points, a-d, form four pairs of turnings as indicated and also form a closed loop. On the other hand, pairs of turnings (g, e) and (e, f) do not form a closed loop and therefore do not define any surface.

A top view of a closed loop concept is shown in FIG. 8B. It can be seen that areas connected together are areas of four turnings forming four pairs of turnings in a closed loop configuration. However, the pairs of turnings (n,q) (m, n) (p, q) and (m, p) do not form a closed loop. That is, there is no pairing between turning points m and p. Therefore, the pairing (n, q) is removed and only distinct horizons 73 and 74 are determined. As may be seen in the Appendix beginning on page 7 under "APPLY-3-D-CONTINUITY-CRITERION". In that implementation, four turnings forming four pairs of turnings provide three-dimensional continuity. A pair that does not satisfy the criterion is "isolated" and rejected.

After Steps 1 and 2 are completed, the horizons have been determined within the three-dimensional volume 40. To fully identify the horizon, however, entry is made at one of the turning points defining that horizon and then all of the remaining turning points that are directly or indirectly connected to the entry turning point are collected. Reference should be made to FIG. 10 to illustrate the criterion as a part of step 3. Turning point d2 is paired with turning point a2 which in turn is paired with turning point b2 which in turn is paired with turning point c2. Therefore, turning points c2 and d2 are in the same horizon. The horizon occurs therefore wherever turnings in a collection are related (directly or indirectly) by pairs. This criterion allows explicit determination of the informational content and three-dimensional extent of horizons.

This criterion is described as follows: Let the relation (pair) (ti, tj), indicate that ti and tj are turnings of the same horizon. This relation is transitive. t1 and t3 belong to the same horizon when pair (t1, t2) and pair (t2, t3) belong to the horizon, because pair (t1, t3) is implied. The transitive property is used to complete the pair relation. The specific implementation may be studied in the Appendix beginning on page 7 under "STEP 3: EXTRACT HORIZONS". This reformatting results in an array format. The array format is appropriate for use in the printer 30 and video display 20 of FIG. 2 for displaying the recommended horizons.

Also, it may be that one horizon is connected to another horizon. Step 3 eliminates connections between horizons. A horizon can contain only turnings such that no turnings are positioned with one directly above the other. This criterion prevents spiraling horizons and assists in determining separate horizons as follows:

Let the horizontal position of a turning, ti, be (xi, yi). Then turnings t1 and t2 can only be part of the same horizon if (x1, y1) is not (x2, y2), i.e., if at most one turning of the horizon lies in any given trace in the volume 40. Thus, each turning of a horizon is guaranteed to be "vertically unique". Implementation of vertical uniqueness is shown in the Appendix, on page 8 under "ADD-TURNING-TO-HORIZON".

FIG. 9 illustrates the application of this criterion. That is, a complete loop is not achieved because points a1 and e1 appear on the same trace Ta. A collection of turnings of a horizon may include a1 and d1, but turning e1 may not be included in a horizon that includes turning points a1, b1, c1 and d1.

The apparatus for both printing and CRT display are well known, as are the manner in which that apparatus operates. For a specific implementation, please refer to the Appendix beginning at page 8 under "STEP 4; DISPLAY HORIZONS".

While this invention has been described in specific steps and in specific hardware, it is contemplated that those skilled in the art may readily substitute hardware and method steps without departing from the scope of this invention which is limited only by the appended claims. ##SPC1##