EP0535208A1

EP0535208A1 - Method for obtaining a representation adapted to the correlation of monodimensional recordings

Info

Publication number: EP0535208A1
Application number: EP19920910310
Authority: EP
Inventors: Jean-Luc Larsonneur
Original assignee: Societe National Elf Aquitaine; Societe Nationale Elf Aquitaine Production SA
Current assignee: Societe National Elf Aquitaine; Societe Nationale Elf Aquitaine Production SA
Priority date: 1991-04-19
Filing date: 1992-04-16
Publication date: 1993-04-07
Also published as: FR2675589A1; CA2085616A1; FR2675589B1; WO1992018941A1

Abstract

Représentation permettant la corrélation optique aisée de deux signaux caractérisée en ce que, après avoir choisi une famille d'ondelettes élémentaire, l'on représente le module, la phase, la partie réelle ou imaginaire de la transformée en ondelettes de l'un et de la transformée en ondelettes de l'autre, transposée, côte à côte.Representation allowing the easy optical correlation of two signals characterized in that, after having chosen an elementary wavelet family, one represents the module, the phase, the real or imaginary part of the wavelet transform of one and the wavelet transform of the other, transposed, side by side.

Description

PROCESS FOR OBTAINING A REPRESENTATION ADAPTED TO THE CORRELATION OF SINGLE-DIMENSIONAL RECORDS

The invention relates to a method for obtaining a representation particularly suitable for correlating signals in identical frequency bands and the application of this method in geophysics to stratigraphic calibrations and to calibrations between well data and seismic data. In many fields, it is necessary to quickly correlate two or more curves representing the variations of a first quantity y as a function of a second quantity x, for the purposes of comparison, adjustment, etc. The curves to be compared can be of the same nature, that is to say represent the variations of the same first quantity as a function of the same second quantity, or of different nature. It may for example be records of the same physical phenomenon but shifted in time or space, or records relating to different physical phenomena or even records relating to the same physical phenomenon recorded for example by methods different so that their frequency content is different. Correlations can be made digitally. The result obtained is generally global and unreliable if one does not make constraining assumptions on the signals, the method then consisting in choosing between several autocorrelation peaks. The correlation can be carried out visually, by manual shifting of one of the curves with respect to the other along the axis of the second quantity. We thus seek by successive shifts the optimal similarity on one or more portions of the curve. This method makes it possible to take knowledge into account a priori. It is that which is commonly used in geophysics for the depth or time calibration of seismic horizons or for the correlation of recordings made in a well and of seismic recordings. The main drawback of such a method resides in the difficulty that there is in comparing signals of forms which can be very different, for example if their frequency content is different. The method according to the invention allows the improvement of the reliability of the correlations carried out visually between at least two recordings thanks to 1 • introduction of time-frequency analyzes to compare the signals in identical frequency bands. There is a known method for analyzing a signal making it possible to decompose said signal as a sum of elementary functions _{a fa} or wavelets which each vibrate like sinusoids over a range whose position on the x axis is linked to the parameter b and whose the width is linked to the parameter a (center frequency), and which are very strongly damped outside this range. The decomposition of a signal using a family of these wavelets constitutes what is called a "time - frequency" analysis because the first and most common decompositions were carried out on records of variations of a first quantity as a function of time (the second quantity). In this case, the dimension of the parameter b is that of a time and the dimension of the parameter a is the dimension of the inverse of a time therefore of a temporal frequency.

For more information on the wavelet decomposition or "time - frequency" analyzes, refer to the article "wavelet analysis" by Yves MEYER et al., Published in "Pour la Science" from September 1987 , or to the book "Wavelets" by JN COMBES et al. in Springer-Verlag editions, article and work which will be included in this description.

Several types of functions can be used, making it possible to define many families of wavelets having different properties. These can for example be Gaussians, square or triangular functions, real or complex functions, orthogonal or not between them. We will refer to the article cited previously to find out the constraints applicable to these various functions and to others for generating wavelet families.

For a family of wavelets ψ _a ^ determined, o defines the "wavelet transform" in two dimensions and z associated with a recording along the x axis as l following the coefficients ^c _{a /} b each corresponding to the integral of the product of the record to be analyzed by the elementary wavelet _a ^ according to the values of b along the x axis and the values of a along a z axis. In the case where complex wavelets have been chosen to perform the time-frequency analysis of a recording or a signal, it becomes possible to define the real part, the imaginary part, the module or even the phase of transformed it into wavelets.

The method for obtaining a representation which is particularly suitable for correlating, in the same frequency band [fl, f2] along an axis z, of at least two one-dimensional records si and s2, variations of a first quantity ( yl and y2) function of a second quantity (xl and x2) varying along an axis x, is characterized in that:

- a family of analysis wavelets is defined by choosing a wavelet function y _a £, (*) depending on a parameter a representing the frequency of the wavelet and a parameter b defining the position of the wavelet on The x axis and by defining for the frequency has a sequence of m values going from i ^ to f ₂ , m being at least equal to the number of octaves included in the frequency range [f- ^, î_} ' ^e * _" for parameter b a sequence of n values,

- on the basis of this wavelet family, the transforms into corresponding wavelets SI and S2 are calculated for each of the records si and s2 with their respective nm coefficients Cl _ab and C2 _ab , the wavelet transform of a record s (x ), s (x) designating S _j or s ₂ , being defined as the double sequence of the nm coefficients C _a ^ _b such that C. ^ _b = / ε (x) _ab (x) dx, when the parameters a and b successively take the different values assigned to them,

- for each of the records, at least one matrix is constructed whose coefficients A _ab are a quantity, namely either the module, the phase, the real part or even the imaginary part of each of the coefficients C _a £ as a function of the values increasing from a and b,

- one transposes according to the frequency axis the matrix or matrices associated with one of the two recordings, and - one represents by pairs, for the purpose of optical correlation, the transposed matrix associated with one of said quantities and corresponding to one of the recordings and the non-transposed matrix associated with the same quantity and corresponding to the other record, by placing said transposed and non-transposed matrices side by side so that all the coefficients A _ab corresponding to the same value of b are in the same alignment and that when we describe each alignment we meet the coefficients of one of the matrices according to the decreasing values of a then the coefficients of the second matrix according to the increasing values of a or, preferably, we meet the coefficients of one of the matrices according to the increasing values of a then the coefficients of the second matrix according to the decreasing values of a. For the representation, one chooses for example a palette of colors or gray adapted to the values taken by the various A _a ^.

The wavelet function Ψ _a ^ (x) chosen to build the analysis wavelet family can be, for example, a Gaussian, a square or triangle function, this wavelet function can be real or complex and give rise to a family d 'wavelets orthogonal or not between them. The functions of the family are elementary wavelets deduced from each other by dilation. As indicated above, m is at least equal to the number of octaves included in the fixed analysis range [fl, f2]. The n values forming the sequence of values assigned to parameter b for the analysis correspond preferably at the coordinates of the center of the elementary ranges of analysis on the x axis.

Preferably, complex elementary wavelets will be chosen so as to represent the phase of transforms, for example so-called "MORLET" wavelets.

The method of the invention finds a particularly interesting application in the field of geophysics, in particular for so-called stratigraphic setting operations. The operations consist in correlating at least one seismic record (or seismic trace), located at the right or in the vicinity of a wellbore with a synthetic trace constructed most often from measurements carried out in the well, this in order to associate with precision to each of the main seismic markers a recognized geological level in the well. In general, one can in geophysics be brought to correlate between them recordings of different origins, for example two seismic traces associated with the same vertical at the crossing of two seismic profiles but recorded during two different seismic campaigns, or even a trace seismic and a recording made in a nearby well (log logs ....).

It is common in this area, to facilitate the optical correlation of the seismic and synthetic recordings during a stratigraphic setting, to duplicate the synthetic trace ten times for example and to insert the group of traces thus obtained between the two seismic recordings surrounding the location of the well.

The representation method according to the invention makes it possible to further improve the optical perception and complete it by providing additional information due to the transform.

Other characteristics and advantages of the invention will appear on reading the following description of an example of a particular application, taken from the geophysical field, with reference to the appended drawings, in which: - Figure 1 represents a seismic trace T ₁ (t) of a seismic profile representing the ground accelerations at a location M as a function of time t for times varying between 200 and 3800 milliseconds as well as the phase and the modulus of its time - frequency transform, in 4 ms steps, with as many pixels as samples,

FIG. 2 represents another trace T2 (t) representing the accelerations of the subsoil at location M and belonging to a second seismic profile crossing the first profile at location M, as well as the phase and the module of its time-frequency transform,

- Figure 3 shows a representation according to the invention allowing easy optical correlation of the two traces T ^ (t) and T ₂ (t), from the use of the phase d their time - frequency transforms. The representation is limited to 2400 ms and the number of frequencies reduced.

The function used here for the wavelet decomposition of the traces is a so-called "MORLET" wavelet of the form + 2îrit ~ t ² Ln (2) j (t) = exe

In this formula, i is the imaginary number whose square is equal to -1, e is the base of the natural logarithms and Ln (2) is the natural logarithm of 2. This function allows to generate a whole family of elementary or wavelet functions ψ _a £ where a is the frequency of the elementary function and b the position on the t axis where said elementary function is centered.

For a given frequency, the coefficients ^c ab ^) ^{of the} wavelet transform are the result of the continuous sum of the product of the record T (t) representing the continuation of the elementary functions

It is common to choose a logarithmic scale for the frequency axis, then to decompose the frequency range studied in octaves (this however is not mandatory, the scale can just as easily be linear, logarithmic or other). In this case, the study was made here for the same frequency range between two frequencies f- ^ and f ₂ delimiting 4 octaves each comprising 16 samples, which made it possible to define 65 analysis frequencies a .; defined as follows, with f = 1000 / 16x16 Hz and f ₂ = 1000/16 Hz:

(j-l) / 16 a_: = f ^ _x2

At each of the m frequencies a .; (m = 65 in this example) is associated with the sequence of coefficients C _a v _j (t) calculated as indicated above.

The sequence of n analysis points b has been defined as identical following the sampling times of the traces ^ and T ₂ (every 4 ms).

The result of the time-frequency analysis of a recording T (l) is therefore here made up of the series of 65 time variation curves C _ab , each of the 65 curves being associated with one of the analysis frequencies and consisting in a sequence of n coefficients C _ab . The elementary functions used here being complex, it is advisable to represent either the module, or the phase, or the real part, or even the imaginary part of the transform. We have chosen to represent the phase and the module. Each of the monodi ensional traces T (t) of axis t thus gives after analysis two two-dimensional representations, image of two matrices A (TI), P (TI) one of whose dimensions is the same as that of the trace analyzed (axis t ) and whose other dimension is a frequency axis f. The matrix A is representative of an amplitude and the matrix P of a phase, for each point (t, f) considered.

To correlate the traces T ^ and T ₂ , we will look in the two matrices A ^ - ^) and A ₂ (T ₂ ) for the frequency ranges for which the two traces have a minimum amplitude so as to filter the image for work only on the part of the significant matrix (presence of signal).

Then, we will optically correlate the traces thanks to the matrices Pι () and P ₂ (T ₂ ) starting with the low frequencies and ending with the highest frequencies. The correlation stops in the highest frequency band where the signals can be compared in a coherent way.

The most efficient graphical representation to perform this optical correlation consists in transposing one of the matrices associated with - ^ along the axis f and in juxtaposing it along the axis t with the corresponding matrix associated with T ₂ so that the coefficients of said matrices corresponding to the same t are on the same alignment and when we describe each alignment we meet the coefficients of one of the matrices according to the increasing values of the frequency then the coefficients of the second matrix according to the decreasing values of the frequency. This takes advantage of the symmetry detection power of the human eye.

In the example treated, the axis t is vertical and the various axes f are horizontal. In FIG. 3, there are two representations according to the invention. On the first image on the left (title: PHASE), the two matrices V_ ^ {T_) and P ₂ (T) have been juxtaposed, Pη ^ T- _j ^ appearing on the left of the image (low frequencies on the left and high frequencies on the right), and P ₂ (T) appearing on the right

(high frequencies on the left, low frequencies on the right). It can be interesting to leave a blank between the images of these two matrices and even to insert between these two images at least one of the traces that one seeks to correlate with 1 • other.

In the second image (title: ENERGY), these are the matrices and A ₂ (T ₂ ) which have been represented, according to the same arrangement.

The image of the energies makes it possible to identify at a glance the frequency zones where there is energy in the two signals. The image of the phases allows, by symmetry, to distinguish common configurations, first in the low frequencies by comparing the extreme left and right parts of the representation, then to refine in the high frequencies by comparing the central parts of the representation.

The representation method according to the invention can be automatically executed using a microcomputer or station and software allowing the calculation of the time-frequency transform of a signal, and equipped with a high definition graphic screen allowing the visualization of matrices and curves. In the present case, the calculation was carried out by software written in F77 and C language on a computer of the HP 9000 s 300 type and the data were viewed on the same computer equipped with a CH screen (HP standard) on which was running an X Windows server (XII, R ₃ ).

The invention is not limited to the example described and shown, various modifications that can be made without departing from the scope of the invention. The method applies in particular to the correlation of all types of record, real or complex, whatever the variable.

Claims

Method for obtaining a representation particularly suitable for correlation, in the same frequency band (f _lf f ₂ ) of axis z of at least two one-dimensional records s- ^ and s ₂ of the variations of a first quantity function of a second quantity (x- ^ and x ₂ ) varying along an axis x, characterized in that - a family of analysis wavelets is defined by choosing a wavelet functionψ _a , _j3 depending on a parameter a representing the frequency of the wavelet and a parameter b defining the position of the wavelet on the x axis and defining for the frequency a a sequence of m values ranging from f- _{j ^} to f ₂ , m being at least equal to the number of octaves included in the frequency range [f _] _, ^ 2 ^ _r ^and for the parameter b a series of n values, - on the basis of this family of wavelets, we calculate for each of records si and s2 transform them into corresponding wavelets SI and S2 with their nm coeffici respective ents Cl _ab and C2 _ab , the wavelet transform of a record s (x), s (x) designating s- ^ or s ₂ , being defined as the double sequence of the nm coefficients C _ab such that C _ab = 1 s (x) ψa _b ( ^χ ) ^ ^x 'when the parameters a and b successively take on the different values assigned to them,

- for each of the records, at least one matrix is constructed whose coefficients A _ab are a quantity, namely either the module, the phase, the real part or even the imaginary part of each of the coefficients C _ab as a function of increasing values from a and b, - the matrix (s) associated with one of the two records are transposed along the frequency axis, and

- We represent in pairs, for optical correlation, the transposed matrix associated with one of said quantities and corresponding to one of the records e the non-transposed matrix associated with the same quantity e corresponding to the other record, by placing said transposed and non-transposed matrices side by side so that all the coefficients A _aj _. corresponding to the same value of b are on the same alignment and when we describe each alignment we meet the coefficients of one of the matrices according to the decreasing values of a then the coefficients of the second matrix according to the increasing values of a or, preferably, one meets the coefficients of one of the matrices according to the increasing values of a then the coefficients of the second matrix according to the decreasing values of a.

- Method according to claim 1 characterized in that the wavelets Ψ _ab are complex.

- Method according to claim 2 characterized in that the wavelets are wavelets called "MORLET".

- Application of the method according to one of claims 1 to 3 in the field of geophysics, in particular for the correlation of seismic data and well data, for example stratigraphic calibration.