correlation of stratigraphic sections by continuous ... - Science Direct

Computers & Geosciences,Vol, 4, pp. 25%.?$0 Pergamon Press Ltd., 1978, Printed in Great Britain.

CORRELATION OF STRATIGRAPHIC SECTIONS BY CONTINUOUS VARIABLES JOHN R. SOUTHAMand WILLIAMW. HAY Division of Marine Geologyand Geophysics, Rosentiel Schoolof Marine and AtmosphericScience, Universityof Miami, 4600 Rickenbacker Causeway,Miami, Florida 33149,U.S.A. (Received 28 October 1977)

Abstract--A technique is presented to obtain greater stratigraphic resolution than methods based on unique events. The information content of the stratigraphic record between unique events in any two sections is used to calculate a cross-correlation coefficient. Stratigraphic correlation is obtained by varying the stretching or shrinking of the records to maximizethe cross-correlationcoefficient.This procedure yields the relative sedimentation rates of the two sections being compared. Key words: Continuous variables, Cross-correlation coefficient, Fourier sine series, Marker events, Relative

sedimentation rate, Stratigraphic correlation, Well logs.

INTRODUCTION

Correlation of stratigraphic sections is usually achieved by one of three methods: (1) correlation by similarity of lithologic characteristics or fossil assemblage; (2) correlation by unique events which can be used to define reference points in the sections, the most usual being examples of biostratigraphic correlations based on the end points of the ranges of species; and (3) correlation by continuous variables which can be used to generate curves describing some parameter of the sections, the most familiar example being electric logging of wells. Classically, these techniques have suffered from one primary fault: stratigraphic correlations were absolute and could not be expressed in terms of relative likelihood of being correct. Consequently, comparison of different correlation schemes could not be carried out in an objective manner, nor could the relative reliability of alternative correlation schemes be expressed quantitatively. Multiple working hypotheses are the heart of scientific investigation; the impossibility of ranking alternative schemes of stratigraphic correlation in terms of likelihood constituted a severe impediment to stratigraphic interpretation. The first of the methods of correlation, by similarity, was quantified in the earlier decades of this century using standard statistical techniques for comparing two sets of observations. Using lithology as a base, statistical methods of correlation were developed using time-series analysis (Burnaby, 1953; Anderson and Koopmans, 1963; Schwarzacher, 1964), Markov chains (Krumbein and Dacey, 1969; Schwarzacher, 1969; Lumsden, 1971), or similar methods (Selley, 1970; Merriam, 1972). These are especially well suited to complex lithic cycles such as Pennsyivanian cyclothems but Markov-chain analysis requires that the sedimentary sequence have a "memory", that is that sequences of lithic units be repetitive. Thorough discussions of quantitative techniques of lithologic correlation have been presented by Lafitte (1972), Agterberg (1974), and Schwarzacher 0975). Recently there has been interest in developing tech-

niques to evaluate the degree of likelihood of stratigraphic correlation. Hazel (1970, 1977) has explored the use of multivariate and other techniques in correlation by assemblages of fossils. McCammon (1970) has developed a method for determining the relative biostratigraphic value of fossils. Hay (1972) suggested probabilistic techniques that might be used to express the degree of likelihood of stratigraphic correlation. This has been pursued and developed by Hay and Steinmetz (1973), Worsley and others (1973), and Southam, Hay, and Worsley (1975). Developments in this field have been reviewed by Hay and Southam (1978). CORRELATIONBYCONTINUOUSVARIABLE The continuous variables most familiar to stratigraphers are the various types of electric and other logs run on wells, techniques which have been summarized recently by Serra (1972). These are generated curves representing some parameter of the rocks comprising the stratigraphic section. Correlation of such logs is usually by the unique event method, that is a "kick" on one log is correlated with a "kick" on another log. The points actually correlated are usually the extreme values. Between these readily recognizable points there may exist a wealth of other detail on the logs which is not so easily correlated. There are a number of other types of curves representing continuous variables which also have been used for correlation in stratigraphy: heavy-mineral curves, component-ratio curves, magnetic signature curves, etc. Pleistocene geology makes great use of t6Ol~80 ratio curves. ,The methods of correlating these curves almost always falls back on the unique or marker event method: a recognizable extreme point on one curve is correlated with a recognizable extreme point on another curve. Paleontologic data can be used to construct similar curves, as has been illustrated by Parker (1958), developed from percentage ratios of groups of. planktonic Foraminifera, and by Lidz (1966), who demonstrated that curves developed from variations in

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J. R. SOUTHAMand W. W. HAY

abundance between groups of. planktonic Foraminifera could reproduce closely the ratio curve in a Caribbean Pleistocene core. Berger (1971) developed an expression, "stratigraphic effort", which could be determined quantitatively and used to express the amOunt of sliding and distortion required to adjust cores correlated by stratigraphic markers or by eye so that the intervals between correlation points are equal; however, his technique did not use the continuously variable data per se for the purpose of correlation. To increase stratigraphic resolution beyond the lower limit obtained by methods which consider the relations between occurrences of marker events, the information content of the interval between adjacent marker events must be used. The natural method to accomplish this is to treat the stratigraphic data as a continuous function of depth in the core. Cross-correlation techniques for comparing continuous signals have been developed by electrical engineers and applied to communication theory. Correlation between two signals is achieved by shifting the signals relative to each other to obtain a maximum "overlap". This "overlap" is expressed by the cross-correlation coefficient. The technique is used widely in other physical sciences such as radio astronomy, where it is employed to extract the information content of detected radio signals. A small number of investigators have applied crosscorrelation techniques to problems arising in the earth science. Morgan and Loomis (1971) have used this technique to compare observed magnetic anomaly profiles to model profiles generated from a paleomagnetic time scale. Neidell (1969) discussed the comparison of two nearby well logs and introduced a special variant, the ambiguity function method. Dean and Anderson (1974) applied the shifting correlation technique to permit detailed stratigraphic correlation for entire basins. The technique enabled them to establish precise correlation where correlation is known to exist but difficult to establish visually. Two difficulties are encountered in all earth-science application of cross-correlation techniques: (1) the problem of determining unique points common to both records, and (2) the problem of shrinking or stretching of the two records due to relative variations in sedimentation rates. Thus, for geoscience applications, two special techniques must be used to maximize the "overlap" or cross-correlation coefficient: (1) shift the records relative to each other, and (2) perform a transformation to shrink or stretch the data. Figure 1 represents the typical situation with stratigraphic data treated as a continuous function of depth in the core. X~(¢t) and X2(¢2) represent any type of stratigraphic data: species abundance, well-log resistivity data, oxygen isotope ratios, etc. plotted against depths in the core, ¢1 and ¢2. The depths ~ and ¢2 depend on local sedimentation rates and are a function of time since burial, that is

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T, = fl(t)

a n d ~2 =

fe(t).

The functions f~ and f2 express the shrinking and stretching of the record as a consequence of variations in

X

[t,]

t,'

X~{t2)

t,"

t 2'

t,

12"

t~

xt(t,)andx2(h)

Figure 1. Continuous functions representing any type of stratigraphic data-species abundance, oxygen isotope ratios, etc. and plotted against depths h and h. Stretching of record 2 relative to 1 is consequence of differences in sedimentation rate. sedimentation rate. For the special situation where the end points of the interval are determined by well-defined unique events such as key beds, the thicknesses TI = r l - ¢ , are deposited in the interval Correlation between cores is based on the intuitive concept of "overlap" expressed by the cross-correlation coefficient

t'