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Geophysical Prospecting doi: 10.1111/j.1365-2478.2011.01046.x

Gravity inversion of 2D bedrock topography for heterogeneoussedimentary basins based on line integral and maximum differencereduction methods

Xiaobing Zhou∗Department of Geophysical Engineering, Montana Tech of the University of Montana, 1300 W Park Street, Butte, MT 59701

Received December 2010, revision accepted August 2011

ABSTRACTBased on the line integral (LI) and maximum difference reduction (MDR) methods,an automated iterative forward modelling scheme (LI-MDR algorithm) is developedfor the inversion of 2D bedrock topography from a gravity anomaly profile forheterogeneous sedimentary basins. The unknown basin topography can be smoothas for intracratonic basins or discontinuous as for rift and strike-slip basins. In casestudies using synthetic data, the new algorithm can invert the sedimentary basinsbedrock depth within a mean accuracy better than 5% when the gravity anomalydata have an accuracy of better than 0.5 mGal. The main characteristics of theinversion algorithm include: (1) the density contrast of sedimentary basins can beconstant or vary horizontally and/or vertically in a very broad but a priori knownmanner; (2) three inputs are required: the measured gravity anomaly, accuracy leveland the density contrast function, (3) the simplification that each gravity station hasonly one bedrock depth leads to an approach to perform rapid inversions using theforward modelling calculated by LI. The inversion process stops when the residualanomalies (the observed minus the calculated) falls within an ‘error envelope’ whoseamplitude is the input accuracy level. The inversion algorithm offers in many cases thepossibility of performing an agile 2D gravity inversion on basins with heterogeneoussediments. Both smooth and discontinuous bedrock topography with steep spatialgradients can be well recovered. Limitations include: (1) for each station position,there is only one corresponding point vertically down at the basement; and (2) thelargest error in inverting bedrock topography occurs at the deepest points.

Key words: Line integral, Maximum difference reduction, 2D gravity inversion, Het-erogeneous sedimentary basin.

INTRODUCTIO N

The sediment thickness and bedrock topography are im-portant parameters in mapping subsurface geology, investi-gating geotectonics, modelling groundwater flow, exploringpetroleum, studying ice stream flow and modelling ground-motion amplification during an earthquake in a sedimentary

∗E-mail: [email protected]

basin. The sediment thickness and bedrock topography arecritical to the understanding of a groundwater system in avalley and to determine a major faulting system. Such a fault-ing system can affect groundwater flow patterns on a regionalscale and can act as conduits for the migration of poor qualitygroundwater or as barriers to retard the movement of ground-water (Schaefer 1983; Annecchione, Chouteau and Keating2001). Knowledge of the discontinuous relief of a sedimentarybasin could help to locate oil structural traps, which might beassociated with the discontinuity (Silva, Oliveira and Barbosa

C© 2012 European Association of Geoscientists & Engineers 1

2 X. Zhou

2010). Because of the higher density (higher seismic velocity)of bedrocks than that of sedimental, alluvial, or volcanicdeposits, seismic waves can be trapped and thus amplified,resulting in disastrously large ground motion and extendedearthquake duration (Campillo et al. 1989; Abbott and Louie2000). Knowledge of subglacial sediment will help understandthe dynamic evolution of the ice streams in polar regions (Bellet al. 1999; Studinger et al. 2001).

Models for sedimentary basins are classified as either ho-mogeneous, where the density contrast of the sediments isa constant, or heterogeneous, where the density contrast ofsediments varies with space. All real sedimentary basins haveinternal density variations. Heterogeneity is often evidencedin most well cores; mass density varies with position and di-rection. The sediment thickness and bedrock topography canbe obtained through gravity inversion.

Gravity survey data and sediment thickness of a basin oftenindicate that the large negative gravity anomaly values are as-sociated with thick valley-fill low-density deposits overlyingthe bedrock (Schaefer 1983). Conventionally, when the den-sity contrast of the sediments of a sedimentary basin is con-stant, the non-linear gravity anomaly inversion for bedrockdepth starts with the initial estimate of the deposit thicknessat each measurement point as the thickness of a homoge-neous Bouguer slab of infinite lateral extent that producesthe gravity anomaly at the point. The source body is usuallyapproximated by a set of juxtaposed rectangular prisms andits gravitational field is then calculated iteratively, until thecalculated gravity anomaly curve fits well the observed fieldwithin some convergence criteria such as the residual gravityanomaly, which is defined as the observed minus the calcu-lated gravity anomaly, reaching a predefined value or as a spe-cific number of iterations that has been completed in eitherthe space domain (Bott 1960; Danes 1960; Corbato 1965;Tanner 1967; Barbosa, Silva and Medeiros 1999b) or theFourier domain (Parker 1973; Oldenburg 1974; Pilkingtonand Crossley 1986). This method of successive approxima-tions has been widely used to map bedrock topography of ho-mogeneous sedimentary basins (Bhaskara Rao 1986; Jachensand Moring 1990; Abbott and Louie 2000; Annecchione et al.2001).

For a heterogeneous alluvium basin where sediment com-paction is considered, the subsurface is divided into an N

homogeneous Bouguer slab of infinite lateral extent or cellswithin each of which the density is constant (Barbosa et al.1999b). It is a non-linear problem to find the shape of the dis-turbing body from its gravity anomaly. However, it can also besolved by linearizing the problem and then by linear inversion

(Blakely 1995). Stability of inversion is maintained throughapplication of various regularization algorithms. Based on thetheory of Tikhonov regularization, the goal of the selectionof various stabilizers is to suppress the instability of the in-verse problem and to choose the solution among the set ofequivalent solutions that has the desired properties such assmoothness, minimum deviation, etc., since different stabi-lizers have different properties (Tikhonov and Arsenin 1977;Lavrent’ev, Romanov and Shishatskii 1986; Zhdanov 2002).Therefore, various regularization algorithms have been de-veloped, based on the selection of stabilizing functionals,including Tikhonov regularization of various orders (Aster,Borchers and Thurber 2005), the maximum entropy regular-ization (Smith et al. 1991), the minimum entropy regulariza-tion (Ramos et al. 1999), total variation regularization (Rudin,Osher and Fatemi 1992; Vogel and Oman 1998), minimumgradient support regularization (Last and Kubik 1983; Port-niaguine and Zhdanov 1999), etc. The gravity anomaly at asurface station results from density contrast distribution andthe geometry of the source body. The inversion of the gravityanomaly for density distribution and the shape of an objectlead to a non-unique solution. Non-uniqueness is ubiquitousin gravimetry inversion even if the gravity anomaly is knownprecisely over the continuum set of the surface points. Thenon-uniqueness is even worse if an anomaly is known over adiscrete set of points, as is always the case in practice. Thus, asclaimed by Skeels (1947), “A unique interpretation, therefore,depends upon some information other than gravity”. Indeed,Leao et al. (1996) developed an algorithm to constrain in-verted basement relief using a few known depths. The densitydistribution of a subsurface with certain constraints such asminimum volume and smoothness distribution of density con-trast can also be used to map the bedrock topography as theiso-surfaces of the bedrock density (Last and Kubik 1983;Barbosa et al. 1999b).

The accuracy of the inverted bedrock depth depends on thequality of the input gravity survey data and the fidelity ofthe mass density model used for inversion. Overall, the infi-nite slab approximation works well when the density contrastfor the problem is constant and the depth is much less thanthe lateral dimension of the basin but not so when nearingthe basin edge, where this approximation fails. This methodproduces a smoothed basin-depth profile but depth calibra-tion using well-log information indicated that it underesti-mates basin depth for a given density contrast (Abbott andLouie 2000). On the other hand, when the density contrastat the bedrock-sediment interface becomes very small or verydeep, the prisms’ thicknesses may increase infinitely without

C© 2012 European Association of Geoscientists & Engineers, Geophysical Prospecting, 1–15

Gravity inversion of 2D bedrock topography 3

changing the fitted anomaly, causing unstableness of the al-gorithm. The intrinsic non-convergence makes it difficult toautomate the inversion procedure.

The interactive method is almost always on a trial and errorbasis; obviously, experience is extremely useful in both the de-sign of the original model and subsequent model adjustment(Corbato 1965). To obtain a stable and convergent inversionsolution, a smoothness constraint is usually superimposed forhomogeneous sedimentary basins (Bott 1960; Parker 1973;Oldenburg 1974; Pilkington and Crossley 1986; Leao et al.1996; Barbosa, Silva and Medeiros 1997, 1999a; Barbosaet al. 1999b). These inversion methods are then extended toinclude heterogeneous cases where the density contrast varieswith depth linearly (Reamer and Ferguson 1989), quadrat-ically (Bhaskara Rao 1986), exponentially (Chai and Hinze1988; Bhaskara Rao and Mohan Rao 1999), as a polyno-mial function of horizontal and vertical positions (Oliva andRavazzoli 1997; Zhou 2010), hyperbolically (Rao et al. 1994;Silva, Costa and Barbosa 2006; Silva et al. 2010), or paraboli-cally (Chakravarthi and Sundararajan 2004) with smoothnessconstraint imposed.

The recently developed line integral (LI) method (Zhou2009) circumvents the task of dividing the density structureinto cells or prisms for each candidate mass body. Calculationof the gravity anomaly is eventually converted to LIs alongthe contour of the mass body if the density contrast can bewritten as a function of space. An area with an order of N2

internally discretized elementary grids is expected to have anorder of N line segments. The number of integration steps canbe reduced from an order of N2 to an order of N and thus com-putation time is efficiently saved. The objective of this studyis to develop an iterative algorithm to invert accurately thebedrock topography for sedimentary basins based on forwardmodelling using the LI and the maximum difference reductionmethods (see below), taking advantage of the capacity of theLI method in handling complex density contrast structuresand fast calculation (Zhou 2009).

The inversion algorithm to be developed in this study isintended to reduce the constraints and interaction of computerand user intervention, making a way for the idea of automaticinterpretation of geophysical survey data from heterogeneousbasins where the density contrast can vary in a very broadfashion (see equation (1) below) and the bedrock topographycan be smooth or have steep spatial gradients. The abilityof the inversion algorithm to invert bedrock topography forsmooth intracratonic basins and discontinuous rift/strike-slipbasins with sequences of faults will be demonstrated withsynthetic and field data. The results show a close agreement

with known structures and published results (Barbosa et al.1999b; Bhaskara Rao and Mohan Rao 1999).

ALGORITHM D EVELOPMENT

When the density-contrast function �ρ(x, z) of 2D space is a

priori known and expressible in the form

�ρ(x, z) = h(x) + v(z) +Nx∑j=1

Nz∑k=1

ξ j (x)ζk(z), (1)

where functions ξ j (x), ζk(z) are integrable in the sense ofequations (4)–(5) below, the gravity profile can be calcu-lated from the LI method developed by Zhou (2009). Theleading terms h(x), v(z) respectively describe purely horizon-tal (x) and vertical (z) spatial dependencies in the densitymodel. The third term describes a sum of cross-terms thatdepend on both horizontal and vertical positions. The indexj runs over 1, 2, ..., Nx, and k over 1, 2, ..., Nz. Nx and Nz arethe number of functions that depend only on x and z, respec-tively. The constant density contrast model (e.g., Bott 1960;Corbato 1965; Pohanka 1988; Ferguson et al. 1988; Litin-sky 1989), the homogeneous layered models (step functions)(e.g., Jachens and Moring 1990) and any depth-dependentdensity models discussed in the above section are special casesof equation (1) with specific forms of h, v, ξ , and ζ to bespecified. The vertical component of the gravity anomaly atstation position (xi , zi ) is given by line integrals as

�gz(xi , zi ) = G∮

C

⎧⎨⎩h(x) ln

[(x − xi )2 + (z − zi )2]

+2Nx∑j=1

Nz∑k=1

ξ j (x)Zk(x, z)

⎫⎬⎭ dx

−2G∮

C

[v(z) arctan

(x − xi

z − zi

)]dz,

(2)

or

�gz(xi , zi ) = G∮

Ch(x) ln[(x − xi )2 + (z − zi )2]dx

−2G∮

C

[v(z) arctan

(x − xi

z − zi

)

+Nx∑j=1

Nz∑k=1

� j (x, z)(z − zi )ζk(z)

⎤⎦ dz.

(3)

where G = 6.6732 × 10−11m3kg−1s−2 is the Newton’s gravi-tational constant and

� j (x, z) =∫

ξ j (x)

(x − xi )2 + (z − zi )

2 dx, (4)


4 X. Zhou

Figure 1 The coordinate setup and the 2D cross-section of a sedi-mentary basin, where C is the contour of the cross-section and anarbitrary observation station is denoted by (xi , zi ).

Zk (x, z) =∫

(z − zi ) ζk (z)

(x − xi )2 + (z − zi )

2 dz. (5)

As shown in Fig. 1, the line integration is counterclockwisealong the 2D contour C in the xz-plane. Equations (2) and (3)are the updated forward modelling equations for any station(xi , zi ) from equations (18) and (19) of Zhou (2009) thatapply to the transect along z = 0 in the x-axis direction. Nowequation (2) or (3) can be applied to any point in the 2Dspace outside the mass source. Equation (2) is used if Zk(x, z)is easier to obtain than � j (x, z), otherwise equation (3) is used.

The inversion algorithm developed here is based on thefollowing considerations.(1) For each survey-station position, it is assumed that thereis only one corresponding bedrock position, a common as-sumption in gravity inversion of bedrock topography (Barbosaet al. 1999b). Thus if there are N observation stations, thesupposed contour of the causative 2D mass source has 2N

vertices. If the top surface is flat, the N-1 segments on the topsurface should be integrated into one for the contour integra-tion to save computation time. Before inversion, the bottom N

vertices coincide with the corresponding top vertices (surveystations).(2) The inversion process consists in moving the bottom partof N vertices of the contour downward or upward followingthe sign of the residual anomaly. Each vertex of the movingpart of the contour moves automatically by a depth step thatis residual anomaly modulated and density-contrast weighed.The whole contour of the causative mass body is then updated.

(3) The gravity at each station is then calculated using theLI method along the updated contour of the basin when thedensity contrast is either constant or variable with space, untilthe specific criterion is met where all residual gravity anoma-lies are confined within the ‘error envelop’ whose amplitudeis the prescribed accuracy or any updating of the source con-tour does not change the gravity-anomaly profile any more.The criterion used for the inversion algorithm is that the max-imum difference between the modelled gravity anomaly andthe observed at any station does not exceed the accuracy ofthe gravimeters by which the measurement is taken, or theaccuracy with which the residual anomaly can be separatedfrom the observed gravity anomaly for regional-scale surveys,or the error level because of noise, whichever is larger. Thiscriterion is based on the philosophy that any interpretation ofgeophysical data is not better than the accuracy of the input.(4) For each iteration the depth increment for the station thathas the maximum absolute gravity anomaly is first estimatedaccording to

�n = max(

z0,Cn

max

Cn−1max + Cn

max

�n−1

), (6)

where �n is the depth increment for the nth iteration, z0 is theminimum depth increment (1–5 m) when �n decreases with it-erations as the contour of the candidate mass body approachesthe true contour. Cn

max is the maximum vector element of theabsolute residual gravity anomaly vector |�gobs − �gn

cal| ofall stations

Cnmax = max

(∣∣�gobs − �gncal

∣∣) , (7)

where �gobs is the observed gravity anomaly vector of N

stations and �gncal is the calculated gravity anomaly vec-

tor at the nth iteration. Please note here absolute opera-tor |...| acts on the elements of the vector �gobs − �gn

cal,so that |�gobs − �gn

cal| is a vector. Before iteration begins,the calculated gravity anomaly is zero at all stations, there-fore, C0

max = max(|�gobs|). Once Cnmax reaches the speci-

fied accuracy requirement, the execution of the algorithmwill stop. Therefore, in practice, the case that �goi −�gn

ciCn

max→ 0

0

will never happen. Equation (6) is to ensure that the maxi-mum expansion or shrinkage of the contour is not too smallwhen Cn

max

Cn−1max+Cn

max�n−1 becomes very small, otherwise, it may

take too long to converge to the input accuracy. The bestinitial value for �0 is that which produces a gravity anomalyprofile close to the observed. Test runs showed that �0 =(0–7) times the thickness of the infinite slab corresponding tothe maximum absolute value of the observed gravity anomalyis a good empirical selection, i.e., �0 = (0 − 7) max(|gobs |)

2πGρ, where



ρ is the reduction density. If we select �0 = 0, iteration startsfrom zero depth for all stations. Selection of the initial �0 hasno effect on the final accuracy but affects the overall compu-tation time.(5) The depth increment or decrease of the vertex on the bot-tom part of the contour corresponding to the ith survey stationfor the (n+1)th iteration is calculated according to:

�zn+1i =

∣∣�gobs,i − �gncal,i

∣∣Cn

max

∣∣∣∣ρ0

ρni

∣∣∣∣ �n, (8)

where (�gobs,i − �gncal,i ) is the residual gravity anomaly at the

ith station of the nth iteration. The absolute value bars inequation (8) are used so that �zn+1

i has the same sign as�n. ρn

i is the density contrast at the bottom vertex of the ithstation for the nth iteration, ρ0 is a reference density con-trast that has the same unit as ρn

i . In the following discussion,the unit for ρn

i is gcm−3 , thus ρ0 = 1gcm−3 in equation(8) is chosen. For each iteration, the depth increment only atthe station that has the maximum absolute residual gravityanomaly is specified, the depth increment at any other stationis automatically updated and is proportional to the ratio of itsabsolute residual gravity anomalies normalized by the max-imum absolute residual and then normalized by the densitycontrast. Equation (8) is an equation controlling the updateof the whole contour of the mass body (contour of the sed-iment basin) rather than just at one point for each iteration.The shift between two consecutive iterations is an expansionor shrink of the whole contour of the mass body. The rationalefor the choice of the iterative scheme (equation (8)) is that theexpansion or shrink of the contour follows the general trendof the residual gravity anomaly profile – where there is a big-ger difference in the residual profile, where there is a largerexpansion/shrink step on the contour. This method is referredto as the maximum difference reduction (MDR) method. TheMDR method has physical inheritance. For instance the heatflows optimally along the direction of the maximum temper-ature gradient in thermodynamics or water flows optimallyalong the maximum elevation gradient in surface water hy-drology. If we analogize the converging process to the waterflow or heat flow, the expansion or shrink of the shape ofthe anomalous body follows the residual gravity pattern – themaximum contour-change rate is along the maximum residualdirection. Once the depth increment at the station that has thelargest residual gravity anomalies is specified, the depth incre-ments at other stations or the whole contour of the mass bodyis automatically updated according to equation (8). Allowingthe contour of the mass body to enlarge or shrink along one

direction of the MDR of the gravity anomaly profile is forstabilization and also for efficient computation.

Equation (8) makes sure that the source contour ex-pands/shrinks not only vertically (

�gobs,i −�gncal,i

Cnmax

) but also hori-zontally ( ρ0

ρni) when the density contrast depends on the hor-

izontal direction. Inclusion of the ρ0ρn

iterm in equation (8) is

designed to make sure of the approaching of the contour of acandidate mass body to the true contour along the directionof the largest residual gravity for all stations when the densitycontrast is also dependent on x. To illustrate this, let us con-sider the density contrast as a function of both x and z andthere are two separate stations A and B of the same elevationbut different x. Assume after some iteration, the amplitude ofthe residual gravity anomaly is larger at station B than stationA because the density contrast ρn

A > ρnB. For the next itera-

tion, the depth increment for station B is larger than stationA because of 1

ρni, increasing the calculated gravity anomaly at

station B more than at station A. In this way, the approachof the contour is guaranteed to be along the direction of thelargest gravity difference to maximally reduce the difference.This is the essence of the MDR method. If the density contrastis only a function of z or constant, it is not necessary to includethe ρ0

ρni

term but inclusion of it does not impact the convergingprocess.

As the calculated gravity anomaly profile approaches theobserved profile, the sign of (�gobs,i − �gcal,i ) is tracked downat each station. The calculated gravity anomaly at each stationis always initialized to zero, the initial sign of (�gobs,i − �gcal,i )is thus the same as the sign of �gobs,i , which can be eitherpositive or negative. When the sign of (�gobs,i − �gcal,i ) is dif-ferent from that of �gobs,i , the corresponding bedrock depth isscaled down to a portion of the depth of the previous iterationaccording to

zn+1i = S, zn

i , (9)

where zni is the depth of bedrock at the ith station at the

nth iteration. The scaling factor S = (1 − 0.5|�gobs,i −�gn

cal,i |Cn

max) is

chosen. This choice makes sure that for the station wherethe difference between the calculated and observed gravityanomalies is small, the new depth is closer to that of theprevious iteration. If the sign does not change and the accuracyis not reached, then

zn+1i = zn

i + �zni . (10)

The strategy of tracking down the sign change of(�gobs,i − �gcal,i ) is to force the matching of the calculatedgravity anomaly profile to the observed profile from a single


6 X. Zhou

direction, avoiding intrinsic divergence in gravity inversion orinstability of the iteration scheme.

The input survey data include the x-coordinate along whichthe gravity anomaly is measured, the elevation of each sta-tion, the observed gravity anomaly after data reduction andthe accuracy level of the input gravity anomaly data in unitsof mGal. The accuracy of the input data is the maximum en-durable difference between the calculated and observed grav-ity anomalies for any station, chosen as the overall accuracylevel of the input data. The bedrock depth and the calculatedgravity anomaly at each station are initialized to zero. Thesign of (�gobs,i − �gcal,i ) is compared with that of �gobs,i , ifit changes and the hard criterion has not yet been reached,the bedrock depth at the station is scaled down according toequation (9); otherwise, the bedrock depth is increased or de-creased by a step determined by equation (10). The contourof the causative mass body is updated. Forward gravity mod-elling using the LI method is employed to calculate the newprofile of the gravity anomaly and the maximum amplitudeof the residual gravity anomaly is calculated and comparedwith the accuracy requirement. Updating of the bottom partof the contour and calculation of the new gravity anomalyprofile using the LI method are repeated until the criterion ofthe match is met.

Accuracy of the gravity data to be inverted is one of themain input parameters for the present LI-MDR algorithm.The input value for this parameter needs to be consistent withthe real accuracy of measurement or the overall accuracy ofthe data set that can be corrupted by noise. The measurementerror of gravity is mainly associated with errors in gravityand position measurement, especially elevation. The accuracyof most gravity surveys lies in-between 0.5–0.05 mGal de-pending on the survey scale and the equipment used. In thefollowing case studies, 0.50 mGal and 0.05 mGal will be cho-sen as the accuracy controls for demonstration. The actualaccuracy level for a specific gravity survey data set is depen-dent on the gravimeter used, noise level and accuracy for thedata reduction. In the following discussion, the accuracy con-trol is the maximum absolute difference between the observedand calculated gravity anomaly of all stations. The iterationprocess stops when it reaches the input accuracy.

APPLICATIONS

For the first case, we apply the LI-MDR algorithm developedabove to a synthetic homogeneous sedimentary basin since it isa special case of heterogeneous sedimentary basins. Figure 2(a)is the gravity anomaly profile at a horizontal interval of

0.250 km on the surface at 81 stations of a simulated in-tracratonic or marginal basin, which is characterized bysmooth bedrock topography with a constant density contrast�ρ = −0.25gcm−3. The true bedrock topography is shown inFig. 2(b) (black solid line). The maximum depth of the basinis 3819 m. The surface is exhumed and is thus not uniform.As a flat surface is a special case of non-uniform surface, thisrepresents a more general case. Intracratonic and marginalbasins are formed due to broad downwarps of the continen-tal crustal basement (shield or craton), which are filled bycontinental lake or river deposits, or with marine sediments.These basins are important for their resources in groundwa-ter, natural gas, oil, coal, limestone and evaporites such assalt, potash and gypsum. The gravity anomaly profile versushorizontal distance at the 81 stations on the surface is inputto the inversion algorithm for the bedrock topography in-version. The inverted bedrock topography using the LI-MDRalgorithm, with the controlling accuracy being 0.50 mGal,is shown in Fig. 2(b) (filled squares). The inversion is auto-matically terminated when the iteration number N = 1537.The root mean square error (RMSE) for the fitting of theinput gravity anomaly profile is 0.26 mGal. The maximumdifference between the inverted and the original bedrock to-pography is 232 m, occurring at a horizontal distance of x =8.750 km. The RMSE of the inverted depth from the truebedrock depth is 71 m, corresponding to a normalized rootmean squared error (NRMSE) of 1.86%.

When the inversion modelling is rerun with a controllingaccuracy of 0.05 mGal, the RMSE for the fitting of the in-put gravity anomaly profile becomes 0.02 mGal. The invertedbedrock profile (open triangles in Fig. 2b) is closer to the truebedrock topography, with a maximum difference of 131 m,occurring at x = 9.550 km. For this run, we have RMSE =35 m, iteration number N = 1710 and NRMSE = 0.90%.Compared with the 0.50 mGal run, we can see that the higherthe data quality is, the higher accuracy of the inverted topog-raphy will be, while the iteration number increases only by11% when the NRMSE is reduced to 48% in the case of the0.50 mGal run. For both runs, even though at the beginningof the inversion, each gravity station is assumed to have adepth point at the bedrock interface, as a result of the correctinversion, the position of the sedimental basin is accurately lo-cated: the bedrock depth is approximately zero for any stationthat is x ≤ 0 km or x ≥ 10 km though the gravity anomaliesat these stations are not zero.

To compare the present LI-MDR algorithm with aMarqardt inversion algorithm (Marqardt 1963; Webring1985) implemented in the GM-SYS inversion routine



Figure 2 (a) The gravity anomaly versus horizontal distance x (km) on the surface of a synthetic intracratonic basin with density contrast�ρ = −0.25gcm−3. (b) The surface and true and inverted bedrock topographies using the gravity anomaly in (a) as inputs with an accuracycontrol of 0.5 mGal and 0.05 mGal, respectively, using the present LI-MDR method. The results from the GM-SYS inversion routine are alsoshown.

(Northwest Geophysical Associates, Inc. 2004), the invertedresults from the GM-SYS routine are also shown in Fig. 2(b)(open circles). In GM-SYS, a geological subsurface model isapproached as one or more blocks, each of which has a con-stant density contrast. For sedimentary basins, it deals withhomogeneous cases; otherwise the sedimentary basins need tobe divided into numerous sub-blocks, each with a differentconstant density, especially at places where the spatial gradi-ents of mass density contrast are large. The inversion processby iterations in GM-SYS is interactive. A constant usuallymust be subtracted from the calculated gravity data to matchthe observed data in one of three ways: the constant auto-matically calculated to minimize the RMS error; selecting a

point at which the calculated and observed values are forcedto match, or manual entrance of the constant. The invertedresults from GM-SYS shown in Fig. 2(b) were obtained byforcing the calculated and ‘observed’ gravity anomaly valueat the leftmost station at x = −5 km in Fig. 2(a) to match. Theaccuracy of the inverted results from GM-SYS lies between the0.5–0.05 mGal runs using the LI-MDR algorithm.

Figure 3(a) shows a field gravity anomaly profile over theGodavari basin, India (digitized directly from Fig. 7, BhaskaraRao and Mohan Rao 1999). The gravity anomaly was sam-pled at an interval of 2500 m with a total profile length of37500 m. The log data from a borewell at a horizontal dis-tance of about 21000 m on the profile show that the density


8 X. Zhou

Figure 3 (a) The profile of the gravity anomaly versus horizontal distance x (km) on the surface over the Godavari basin, India (BhaskaraRao and Mohan Rao 1999) with density contrast �ρ(z) = −0.4692e−4.078×10−4z g/cm3. (b) The inverted bedrock topography using the gravityanomaly in (a) as input using the present algorithm. The interpreted results by Bhaskara Rao and Mohan Rao (1999) are also shown as BR&MR,1999.

contrast decays exponentially with depth (Bhaskara Rao andMohan Rao 1999):

�ρ (z) = −0.4692 exp(−4.078 × 10−4z), (11)

where �ρ(z) is in g/cm3 and z in m. The inverted bedrockdepths at all the 16 stations with an accuracy control of0.50 mGal are shown in Fig. 3(b) (dotted-circled line). Theiteration number for the run is N = 262. For comparison,depths inverted by Bhaskara Rao and Mohan Rao (1999)are also shown in Fig. 3(b) (stepped solid line). The maximumbedrock depth at 20.0 km inverted using the present algorithmwith 0.50 mGal accuracy control is 2958 m, comparable wellwith 2935 m at the borehole site at a horizontal distance ofabout 21.0 km and 3210 m at the same station inverted byBhaskara Rao and Mohan Rao (1999).

Figure 4(a) shows a synthetic gravity anomaly profile alongthe x-axis (z = 0) starting at point (−2.0, 0) km to point (12.0,0) km at an interval of 50 m of a simulated sedimentary basinof two structural lows and one terrace high in the middle,

simulating a laccolith intruded into the sediments. The densitycontrast of the sediments varies with depth quadratically as inthe Sebastian Vizcaıno Basin, Mexico (Garcıa-Abdeslem et al.2005):

�ρ (z) = −0.7 + 2.548 × 10−4z − 2.73 × 10−8z2. (12)

The profile corresponds to a synthetic gravity survey at 281stations. The contour of the mass body that consists in 11 linesegments and 11 vertices is shown in Fig. 4(b) (thick solid line).Using these synthetic gravity anomaly data as input to thealgorithm to invert the geometry of the source body, the resultswith the 0.50 mGal accuracy control (dotted line) are shownin Fig. 4(b). The RMSE for the fitting of the input gravityanomaly profile is 0.18 mGal. The iteration number is N =282. The maximum difference between the inverted and thetrue bedrock topography is 106 m, occurring at a horizontaldistance of x = 6.0 km, corresponding to the position ofthe deepest point. The RMSE of the inverted depth from the



Figure 4 (a) The gravity anomaly profile produced from a simulated asymmetrical faulted sedimentary basin. (b) The true and inverted bedrocktopographies with two runs with accuracy = 0.50 mGal and 0.05 mGal, respectively.

true bedrock depth is 26 m, corresponding to an NRMSE of2.64%.

When the inversion modelling is rerun with a controlling ac-curacy of 0.05 mGal, the inverted bedrock topography is alsoshown in Fig. 4(b) as a thin solid line. The RMSE for the fittingof the input gravity anomaly profile becomes 0.02 mGal. Theinverted bedrock profile is closer to the true bedrock topog-raphy, with a maximum difference of 52 m, also occurringat x = 6.0 km; RMSE = 10 m, N = 316 and NRMSE =

0.98%. In this case, when the NRMSE is reduced to 37%in the case of the 0.50 mGal run, the iteration number onlyincreases by 12%. For both runs, the maximum error in thedepth inversion occurs at the deepest point. Since the match ofthe calculated gravity anomaly profile to the observed profileis controlled by the accuracy parameter as input, the maxi-mum difference between the two data sets does not exceed0.50 mGal and 0.05 mGal, respectively, at any station. Forthe accuracy = 0.50 mGal run, the maximum difference is


10 X. Zhou

0.49 mGal. For the accuracy = 0.05 mGal run, the maximumdifference is 0.04 mGal. From this synthetic example, we cansee that the algorithm works well in inverting the topographyof the bedrock with a high accuracy when the accuracy ofthe survey data is high. Just from the gravity anomaly pro-file alone measured at the surface, such as Fig. 4(a), we cannot judge intuitively the starting and ending positions of theanomalous mass body (sediment basin) at the surface area,though we can estimate there is a structural high betweenthe two lows. However, the inversion results do show clearlywhere the basin is and how wide it is.

To demonstrate the capability of the inversion algorithm toinvert the bedrock topography of a faulted basin with densitycontrast variable in both horizontal and vertical dimensions,let us consider Fig. 5(b) that shows the geometry of an asym-metric fault-bounded sedimentary basin of strike-slip type andthe more sophisticated spatial distribution of density contrastof the sediment infill of 3.0 km thickness within the basin.The asymmetric basin has a flat bottom and steep borderswith high-angle strike-slip faults. The spatial variation of thedensity contrast varies with x and z directions and can bedescribed as

�ρ (x, z) = − 0.7 − 8.7 × 10−6x + 3.6 × 10−4z − 3.6

×10−8xz + 1.2 × 10−8z2 − 7.8 × 10−8e−1.88z,(13)

where �ρ(z) is in g/cm3. Figure 5(a) shows the synthetic grav-ity anomaly profile along the horizontal distance on the sur-face. The inverted bedrock topographies with an accuracy of0.50 mGal (diamond symbols) and 0.05 mGal (circle sym-bols), respectively, are shown in Fig. 5(b). For the 0.50 mGalrun, the iteration number N = 530, RMSD = 110m andNRMSE = 3.68%. The maximum depth difference from thetrue topography is 574 m, occurring at the deepest point atx = 10 km. For the 0.05 mGal run, the iteration number N =651, RMSD = 83 m and NRMSE = 2.77%. In this case, whenthe NRMSE is reduced to 75% of the case of the 0.50 mGalrun, the iteration number increases by 23%. The maximumdepth difference from the true topography is 371 m, occur-ring also at the deepest point at x = 10.0 km. Compared withthe cases of a simpler structure of density contrast discussedabove, the inversion accuracy of the bedrock topography isgenerally more difficult to reach. However, an NRMSE <5%can still be achieved.

To compare the present method with other approaches,Fig. 6(a) shows the 215 equi-spaced raw Bouguer anomalies(solid circle, Fig. 3a, Barbosa et al. 1999b) and the mod-elled gravity anomalies using the LI method (Zhou 2009) of

a synthetic homogeneous 2D sedimentary basin with com-plex faulted bedrock topography that is shown in Fig. 6(b)(dark and thick solid line). This basin is a model for theReconcavo basin in Brazil (Barbosa et al. 1999b). The var-ious steep discontinuities with different step heights simulatedip-slip faults. The density contrast is �ρ = −0.30gcm−3. Thenoise-corrupted gravity data shown in Fig. 6(a) were invertedfor bedrock topography by Barbosa et al. (1999b, Fig. 2) andused in their regularized stable algorithm through subdividingthe gravity anomaly profile into several data windows. Thesubsurface region with pre-defined depth containing the basinis discretized into juxtaposed 2D rectangular prisms whosedensity contrasts are to be determined in percentage of thetrue density contrast (−0.30gcm−3). The bedrock relief is de-lineated by the interface between cells of null and those ofnonnull estimated density contrasts. Stablization conditionsinclude: (1) any vertical line through a surface gravity stationintersects the bedrock-sediment interface only once, same asthe present algorithm; (2) the source for the gravity anomalyis concentrated close to the surface; (3) the density contrast ofeach prism may be represented as a percentage of the assumedvalue but constrained to assume a value close to specifiedlower or upper bounds.

The raw gravity anomaly data were noise-corrupted andprocessed with a low-pass filter. The error level is estimated tobe 1.50 mGal. The inverted bedrock topography at all the 215stations using the present algorithm is shown in Fig. 6(b) (filleddiamonds). Compared with the true topography, it is foundthat N = 238, RMSE = 210 m and NRMSE = 3.51%. Wecan see that the overall bedrock topography is recovered, withNRMSE being below 5%. Similar to the cases shown in Figs 4and 5, Fig. 6 shows that the major steep discontinuities withsteep spatial gradients are well recovered even though thereare errors imbedded in the filtered data. Compared with theregularized stable algorithms discussed above, the inversionto bedrock depth is direct so that the inversion accuracy forsynthetic cases can be assessed and stablization conditions (2)and (3) are not needed. Since the present algorithm is forwardmodelling based on the non-linear inversion method, artificialparameters for regularization such as the Lagrange multiplieris not required as input.

The inverted results from the GM-SYS routine using thesame filtered data are also shown in Fig. 6(b) (open circles),which were obtained by forcing the calculated to match the‘observed’ gravity anomaly value first at the leftmost stationat x = 100 km, then at the rightmost station x = 314 kmand finally at the x = 100 km station again in Fig. 6(a). If justusing the matching at the leftmost station at x = 100 km once,



Figure 5 (a) The gravity anomaly profile of a simulated asymmetric sedimentary basin of strike-slip type. (b) The true geometry of an asymmetricsedimentary basin of strike-slip type and the spatial distribution of density contrast of sediments within the basin. The inverted bedrocktopographies with two runs of accuracy = 0.5 mGal and 0.05 mGal, respectively.

the results are not as good as shown. The results show thatboth the GM-SYS routine and the LI-MDR algorithm workfine, though LI-MDR works better for this case in recoveringthe bedrock topography from the noise-corrupted data.

DISCUSS ION AN D C ON C L USI ON S

In this paper, we constrain the density contrast and try tofind the depth of bedrock. Based on forward modelling us-ing the LI and the MDR methods, a 2D stable and automatic

iterative LI-MDR algorithm for non-linear gravity inversionfor bedrock topography is developed for heterogeneous sed-imentary basins. Iteration starts from zero depth or at anestimated depth using an infinite slab approximation and thecontour of the mass source is iteratively updated. The in-crease of depth at each station is proportional to the ratio ofthe difference between the observed and the calculated grav-ity normalized to the maximum value of this difference foundamong all stations in each iteration and inversely proportionalto the density contrast of the corresponding bedrock-sediment


12 X. Zhou

Figure 6 (a) The profiles of gravity anomaly versus horizontal distance x (km) on the surface by Barbosa et al. (1999b) and modelled using the LImethod (Zhou 2009) of a synthetic intracratonic basin with density contrast �ρ = −0.3gcm−3. (b) The true and inverted bedrock topographiesusing the LI-MDR algorithm (filled diamonds) and GM-SYS inversion routine (open circles).

interface. This strategy makes it possible for the source con-tour to expand/shrink vertically and horizontally when thedensity contrast varies vertically and horizontally along thedirection of the maximum residual gravity anomaly. Thismethod is referred to as the maximum difference reduction(MDR) method. The stabilization is implemented by track-ing down the sign change of the residual gravity anomaly ateach station for each update of the contour of mass source toforce the convergence of the inverted contour to the true con-tour from a single direction. Four cases were studied in whichthree are synthetic (Fig. 2 and Figs. 4–6) and one from field

data (Fig. 3). These case studies show that the bedrock topog-raphy can be inverted accurately enough using the LI-MDRalgorithm if the gravity anomaly profile has enough accuracy.

Comparison of the present LI-MDR algorithm with aMarqardt inversion algorithm (Marqardt 1963; Webring1985) implemented in the GM-SYS inversion routine (North-west Geophysical Associates, Inc. 2004) for two homogeneoussedimentary basins shows that LI-MDR performs similarlywell as or better than GM-SYS. Compared with the GM-SYSinversion routine, (1) the LI-MDR algorithm can deal with aheterogeneous basin as a single block easily when the density



contrast function varies with space and is known. If GM-SYSis used for such a case, the sedimentary basin needs to besubdivided into numerous blocks, each with a different con-stant density to account for the spatial variability of the massdensity contrast. But manual manipulation of adjusting pa-rameters for fitting the observation is very time-consuming.(2) The inversion by iterations is automatic rather than inter-active once the accuracy of the data is entered.

The characteristics of the present inversion algorithm in-clude: (1) input data to the algorithm include the measuredgravity anomaly profile, measurement accuracy and the den-sity contrast distribution. The algorithm applies to any casewhere the density contrast function satisfies the general den-sity model equation (1). In cases where density contrast in-formation is not available, either geological or well-loggingdata can be used to obtain the density information before theinversion is performed. The final inverted results will dependon the quality of the gravity anomaly data and subsurfacedensity modelling. ‘Geological’ noise (i.e., a signal created bythe unknown density inhomogeneities other than the inter-face position), resulted from inappropriate modelling of thedensity contrast function, could be large and can also impactthe accuracy of the final inverted results. How to model thesubsurface density itself is a hard task and deserves furtherresearch. High-resolution borehole loggings may be the bestway to solve the problem. (2) The gravity anomaly is calcu-lated from the LI of the contour of the causative mass bodythat is updated in each iteration so that the basin geometry canbe recovered with high accuracy. Compared with schemes inwhich the mass body is divided into rectangular prisms, the to-pography that is inverted by the presented algorithm is smoothrather than step-like, which should be more reasonable geo-logically (Oldenburg 1974). Quantitative assessment of theinversion accuracy for bedrock depth using the linear inver-sion method is usually difficult and thus no such assessmenthas been given. The non-linear inversion method presentedhere makes it possible to quantitatively assess the accuracy ofinversion through synthetic and field data. For a sedimentarybasin of simple structure of density contrast, a NRMSE ofbelow 1% can be achieved if the input gravity measurementhas an accuracy as high as 0.05 mGal. This means that fora basin of maximum depth of 1000 m, 10 m RMSD can bereachable. Even for a faulted basin of sophisticated structureof density contrast, a NRMSE of below 5% can be achieved.For comparison, inversion of bedrock topography using aninfinite slab approximation can have a depth uncertainty of250 m for a basin where the maximum depth is estimated toabout 1160 m (Abbott and Louie 2000). This corresponds to

an NRMSE of 21.55%. (3) The inversion algorithm is basedon a simplified assumption that the lower vertices of a polygo-nal body are located vertically, one below each measurementstation. In each iteration, gravity is calculated using the LImethod and thus does not require partial derivatives as in theinversion schemes with regularization. Partial derivatives inthe inversion scheme complicate the algorithm by the intro-duction of numerical integration. This simplification and LImethod lead to an algorithm for rapid inversions. The inver-sion process stops when the difference between observed andcalculated anomalies falls within an ‘error envelope’ whoseamplitude is defined a priori as the measurement accuracy,noise level, or the accuracy of the gravity data after data re-duction, which ever is the largest. This simplified model of-fers in many cases the possibility of performing an agile 2Dgravity inversion on basins with heterogeneous sediments. (4)The usual requirements for stabilization such as appropriatedesign of objective functions and that horizontal dimensionsbe much greater than its largest vertical dimension are notneeded. Since the present algorithm is a forward modellingbased non-linear inversion method, discontinuities with steepspatial gradients can be well recovered without being severelysmoothed (Figs 4–6). There is no physical need to estimate theinitial model parameters such as the depth of the subsurfaceregion containing the basin in the linear inversion. However,appropriate initial depth can speed up computation.

Limitation of the present algorithm includes: (1) for eachstation position, there is only one corresponding point ver-tically down at the basement. Thus, it does not apply forinverting bedrock topography for folded basins where a ver-tical line can intersect the bedrock interface by multiple (≥2)times. (2) The largest error in the inverted bedrock topogra-phy occurs at the deepest points where the density contrastcould also be smaller than at shallow places. This is intrinsicfor potential field inversion because the sensitivity of gravityat the surface to the depth change at the deepest place is theleast. (3) The algorithm developed here is 2D; it applies onlyto the cross-section of linear sedimentary basins. (4) The up-per part of the mass contour coincides with the observationalsurface. For air-borne gravity surveys, data reduction to theground surface is needed before the model run.

ACKNOWLEDGEMENTS

I am very grateful to Juan Garcıa Abdeslem, Fernando Guspıand Horst Holstein for their helpful comments and sugges-tions that improved the quality of the manuscript. Thanks toValeria C.F. Barbosa for providing the raw gravity anomaly


14 X. Zhou

and true topography data for Fig. 6. The very helpful com-ments and critiques by two anonymous reviewers, AssociateEditor Irina S. Elysseieva and Editor Alan Reid are very muchappreciated.

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