Monte Carlo analysis of seismic reflections from Moho and the W reflector

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Monte Carlo analysis of seismic reflections from Moho and the W reflector
  JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B2, PAGES 2969-2981, FEBRUARY 10, 1997 Monte Carlo analysis of seismic reflections from Moho and the W reflector Klaus Mosegaard Dept. of Geophysics, Niels Bohr Institute for Astronomy, Physics and Geophysics, Copenhagen Denmark Satish Singh and David Snyder British Institutions Reflection Profiling Syndicate, Bullard Laboratories, University of Cambridge Cambridge, England Helle Wagner Dept. of Geophysics, Niels Bohr Institute for Astronomy, Physics and Geophysics, Copenhagen Denmark Abstractø Near-normal-incidence eflections have been used to image the Moho and the W reflector structure in the lithosphere, offshore northern Scotland. To determine the impedance variations at these reflectors, we use a Monte Carlo technique which allows ncorporation of geologically ealistic a priori information as well as an extensive exploration of the model space, after testing it on a synthetic data set. The method is based on Bayesian nversion theory. The modeled Moho consists of a series of layers with a total thickness of •- 2.4 _q- .3 km with an overall positive mpedance contrast. Inversion of the W reflector results n a model of five-seven ayers with a total thickness f about 3.7 _+ 0.6 km and mostly nonpositive impedance contrasts. The implied fine-scale mpedance structure of the Moho is consistent with the broader velocity structure determined rom previous wide-angle reflection/refraction rofiles. However, he overall nonpositive mpedance ontrast at the W reflector requires a structure which is overlain or underlain by a broad increase n velocity in order to match amplitudes of reflected phases observed at large offsets nterpreted previously o srcinate at a similar depth. Introduction In the last few decades, deep seismic eflection pro- filing has provided spectacular mages of the continen- tal lithosphere worldwide, of which subhorizontal eflec- tions from the lower crust, reflections rom the Moho, and reflections from the upper mantle have been par- ticularly notable. Some reflections have been associ- ated with known structures; for example, bright re- flections near the Moho depth have been associated with the crust-mantle transition zone, and dipping re- flections in the crust have been associated with near surface aultis nd known ubduction ones Klemperer and Hobbs, 1991; Clowes et al., 1992; BABEL Work- ing Group, 1993; Zhao et al., 1993; Clowes and Green, 1994]. Many features, mainly subhorizontal nes, e- main incompletely understood. Although various mod- els have been proposed or these eatures, one ultimately requires outcrops, drill holes, or the physical proper- Copyright 997 by the American Geophysical nion. Paper number 6JB02566. 0148-0227/97/96JB-02566509.00 ties (density nd seismic elocity) f these eflectors o choose rom among hese models. Deep seismic eflec- tions, as such, are incapable f providing direct nforma- tion on physical properties. Instead, one derives some estimates of the physical properties rom the seismic record using a modeling strategy. Conventional seismic data processing allows a com- paratively ast but rough analysis of large volumes of seismic data. It is essentially based on a linear model of the seismic race, the so-called onvolutional model, n which he seismic race s(t) is approximated y a con- volution of the subsurface eflectivity (t) and a source wavelet w(t): = ß According o the convolution heorem, one consequence of this approximate model s that only hose requencies that are present n the source wavelet can be retrieved from the reflectivity series of a recorded seismic race. Seismic sources used in conventional deep seismic ma- rine experiments ypically have 5 to 80-Hz bandwidths [Hobbs nd Snyder, 993]. Due o attenuation, seismic wavelet at a given wo-way ime will have ittle energy above 60 Hz, and the convolutional model alone cannot 2969  2970 MOSEGAARD ET AL.- MONTE CARLO ANALYSIS OF DEEP REFLECTIONS WEST o EAST • 40 6o 8O Figure 1. The DRUM seismic ection howing he Moho (M) and W (W) reflectors nd the location of the data used n the Monte Carlo nversion. The section was migrated using a two- dimensional velocity ield derived rom nearby efraction esults Snyder nd Flack, 1990] and then depth converted sing he same velocities. The Flannan reflector can also be observed, dipping rom 30-kin depths at the western edge of the profile o 80-kin depths n the east. provide any nformation bout he reflectivity at spatial frequencies quivalent o _• 60 Hz at that two-way ime. At frequencies ess han 5 Hz, reflected eismic nergy is normally absent or obscured by ship noise. Conse- quently, the convolutional model is unable to predict any low-degree rend or nonzero average value of the reflectivity. If one s interested n extracting nformation outside the limited passband f the source wavelet, t is neces- sary to incorporate a priori information about the sub- surface, bandon he convolutional odel and replace t with the correct relationship between seismic data and Earth model. In this paper, we propose a flamework under which variations n impedance nside and outside the frequency passband f the seismic wavelet can be es- timated by nonlinear nversion using prior information on the model in terms of probabilities. A Monte Carlo inversion echnique Mosegaard nd Tarantola, 995] provides models which fit the observed data and esti- mates errors and resolution f these models. We apply this lamework o shot ecords rom he DRUM (Deep Reflections rom the Upper Mantle) reflection rofile from he north of Scotland o estimate mpedance aria- tions at the Moho and at the W reflector Figures and 2) and to provide estimates f the resolution f these impedances. The DRUM line was designed o investi- gate the reflectivity of the lower continental ithosphere and was recorded or 30 s two-way ravel ime (TWT), corresponding o about 110-km depth [McGeary and Warner, 1985]. Various ipping eflections re observed in the upper crust 0 to 5 s) and a series f subhorizon- tal reflections n the lower crust (6 to 9 s), the base of which, at -• 9 s TWT, coincides with the Moho defined by nearby efraction bservations Barton, 992]. Three strong reflectors occur in the mantle: a subhorizontal reflector round 13-15 s TWT (the W reflector), n east- erly dipping eflector the Flannan eflector) ecorded from 9 s down to at least 27 s and possibly 30 s, and a 15-kin-long anded one at 23 s (Figure 1) [McGeary and Warner, 1985]. Derivation f the physical roper- ties of these deep reflectors rom the seismic ecords s critical to understanding the formation and evolution of the continental lithosphere around the British Isles. A Probabilistic Formulation of the Problem Analysis of the resolution of subsurface structures from seismic data requires a probabilistic formulation of the inverse problem. Due to the often strongly non-  MOSEGAARD ET AL.' MONTE CARLO ANALYSIS OF DEEP REFLECTIONS 2971 (a) 13 Trace Number 10 20 30 40 50 60 15 Figure 2. The data used or inversion. a) The shot gather rom 13 to 15 s at shot point 3564 of the DRUM profile, a TWT zone for the W reflector. These 60 traces have increasing offset from eft (209 m) to right (3210 m). These races were summed n groups f six to produce 0 traces with enhanced ignal-to-noise atios for the inversion. b) The summed races over the time range of 8-10 s that contains he Moho reflection t about 8.9 s. (c) The summed races over he the time range of 13-15 s that contains he W reflection t about 13.8 s. (d) Amplitude spectrum f data with Moho reflection. e) Amplitude spectrum f data with W reflection. f) Estimated marginal noise distribution or a single data sample. Moho data). The solid curve s a Gaussian istribution. g) Estimated oise utocorrelation Moho data). linear relationship between the subsurface mpedance model and seismic data, the distribution of errors n the observed data is mapped into the model space as a com- plex error distribution. Among the important patholo- gies of this relationship s an inherent nonuniqueness f models hat fit the data. Further complexity s added to the model distribution when we, on geological grounds, must introduce complex, data-independent a priori in- formation to further weigh models based on their geo- logical feasibility. We use a Bayesian formulation of the inverse prob- lem. In this formulation, the state of information about the subsurface after incorporation of both a priori infor- mation and data information is completely described by the a postertort robability ensity (m) over he model space [Tarantola and Valette, 1982]. From c(m) it is possible o calculate the probability that the true model belongs o a given class 4 of models: P(m elongs o 4)-/A (m)dm, The a postertort probability density s the complete so- lution to the inverse roblem [Tarantola nd Valette, 1982]. It contains ll the available rior information such as the approximate sizes of reflection coefficients and layer thicknesses, nd all the data information, as "seen hrough the glasses" of the theoretical relation- ship between model and data in the form of the wave equation. In the Bayesian analysis, all the input infor- mation is preserved and no uncontrolled subjective bias is introduced. The a postertort probability distribution for the in- verse problem is given by a(m) - •'M mldm2"' (m) (m) (1) [Tarantola nd Valette, 982]. The a postertort roba- bility density a(m) equals except or a normalization constant) he a priori probability ensity p(m) times a likelihood unction L(m) measuring he fit between observed data and synthetic data calculated from the model m. The likelihood function is typically of the form L(m) = C exp -$(m)] where C is a constant nd $(m) is a misfit function. $(m) measures he difference etween he observed  2972 MOSEGAARD ET AL.' MONTE CARLO ANALYSIS OF DEEP REFLECTIONS (f) 2soJ 200 - 150 - 100 - 50- (b) (c) 8.0 8.5 9.0 9.5 10.0 13.0 13.5 14.0 14.5 15.0 (d) (e) _- _ I I I I I I 10 20 30 40 50 60 Frequency Hz] i I I Noise mplitude Arbitrary nit] 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.2 (g) 1 0 10 20 30 40 50 60 Frequency Hz] , i i i -o.1 o o.1 Time lag [s] Figure 2. (continued) 0.2  MOSEGAARD T AL.: MONTE CARLO ANALYSIS OF DEEP REFLECTIONS 2973 data and synthetic ata calculated rom he model m. If, for nstance, he observational oise onsists f nde- pendent aussian rrors, S(m) is proportional o the sum of squared ifferences etween bserved nd cal- culated ata values the L• misfit), and he likelihood function s Gaussian. f, instead, he noise consists f independent aplace istributed rrors, $(m) is the Lx misfit where quares re replaced y absolute al- ues), and he likelihood s Laplacian. Monte Carlo Sampling of the A Posteriori Distribution Real, quantitative eological priori knowledge an- not be described y means of simple mathematical x- pressions or p(m). Such nowledge s often available as statistical information: histograms giving the oc- currence requency f, for instance, ertain ithologies or physical ock parameters, bserved n outcrops r nearby wells. Mosegaard nd Tarantola 1995] rovide method for Bayesian onte Carlo nversion hat overcomes his problem. This method has wo major advantages, s compared o previously ublished ethods e.g., toffa and Sen, 1991]. First of all, the method s exact n the sense hat it will provably ample he posterior roba- bility density. Second, t allows ncorporation f arbi- trarily complex tatistical priori nformation nto the inversion. The algorithm onsists ssentially f two nteracting parts. The irst part s an a priori model enerator hat is able o produce andom ubsurface odels, onsistent with the available priori knowledge. onsistency ere means hat the generated odels ave exactly he same statistical roperties s those we have obtained rom observations n the real Earth. The models produced by the a priori model enerator re ed nto he second part, an algorithm hat decides f the a priori model an pass a data fitting test. One iteration of the algorithm uns as follows: First, given he current model, new model s chosen y the a priori model generator by perturbing he current model), nd he probability or a given ew model o be chosen s proportional o its a priori probability. he new model mn•,v s now accepted r rejected ccording to the following rule: 1. If the value of the likelihood (mn•,v) of the new model s arger han or equal o the ikelihood (n•u•) of the current model, he model mn•w s accepted ith probability 1. 2. If the value of the likelihood (mn•,v) s smaller than the likelihood (mcu•), the model mn• is ac- cepted as he next "current odel") nly with proba- bility L(mn•) Paccept --- (mcu•) If the new model is rejected, he current model also becomes the next current model. The series of "current models" produced by this two- part algorithm re, asymptotically, amples rom he a posterJori istribution r(m) [Mosegaard nd Tarantola, 1995]. After a large umber f terations, he number f times a given model m occurs n the collection f "cur- rent models" s approximately roportional o or(m). A large collection f such amples rom or(m) provides the raw material from which various characteristics of the models can be obtained [Mosegaard nd Taran- tola, 1995]. A set of statistically ndependent amples of the accepted models allows structures n the sub- surface hat are well-resolved o be distinguished rom those that are ill-resolved. A well-resolved structure will appear n most of the accepted models, whereas an ill-resolved tructure will appear n only a few mod- els. The probability hat a certain structure exists s roughly roportional o its frequency f occurrence n the set of a posteriori samples. Therefore approximate a posteriori robability distributions or model parame- ters can be represented y normalized istograms f the parameters alues. The peak of the histogram ill be at the most probable model parameter, nd the deviation from this peak will provide a measure f uncertainty. Monte Carlo Sampling of Lithosphere Reflectivity We use he Bayesian Monte Carlo algorithm escribed above o generate eflectivity models or selected arts of the DRUM reflection rofile. We have concentrated our efforts nly at a location Figure ) where he Moho and W reflectors re brightest nd subhorizontal s he algorithm equires arge computation ime. We have analyzed total of 60 unprocessed races n the nterval 8.0 s - 10.0 s TWT bracketing he Moho and he nterval 13.0 15.0 s covering he "W reflector." We chose he 60 traces rom one shot gather 3564) with the best signal to noise atio among 0 neighboring hot ecords. he receiver roup pacing as 50 m along 3000-m-long streamer. The raw shot gather shows oherent nergy between 3.7 s and 14.2 s TWT for the W reflector Fig- ure 2a). To ncrease he signal o noise atio, we applied a correction or dip move out and stacked ix adjacent traces, ielding 0 traces or the nversion Figures b and 2c). The normalized requency pectra how om- inant nergy etween and 30 Hz (Figure d and 2e). Since he Moho nd W reflectors re Subhorizontal nd are at large depths, he incident eismic aves re es- sentially ertically raveling lane waves. t is therefore possible o perform rather areful alculation f syn- thetic seismograms sing fast one-dimensional ropa- gator matrix method Haskell, 953]. Wavelet Estimation The DRUM profile was shot by GECO Geophysical Company f Norway) sing n 8536-cubic-inch ir gun array at 7.5 m below he sea surface McGeary nd Warner, 985]. A simulated ar-field ource ignature
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