Inversion of potential fields on nodes for large grids

Journal of Applied Geophysics 110 (2014) 90–97

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Inversion of potential fields on nodes for large grids Denis Marcotte ⁎, Pejman Shamsipour, Olivier Coutant, Michel Chouteau Département des Génies Civil, Géologique et des Mines, Polytechnique Montréal, C.P. 6079, Succursale Centre-Ville, Montreal, Quebec H3C 3A7, Canada

a r t i c l e

i n f o

Article history: Received 7 April 2014 Accepted 4 September 2014 Available online 16 September 2014 Keywords: Multi-scale Stochastic inversion Inversion on nodes Gravity data Potential field

a b s t r a c t The non-iterative direct inversion of potential field data by stochastic approach enables to incorporate in a coherent way a priori geological knowledge, the known densities on any support size and the gravity data. The weakness of the method is the necessary computation of the parameter covariance matrix. For a large mesh made of prisms, the matrix is simply too large to fit in memory. The new approach approximates the prism covariance matrix by a surrogate matrix computed from the covariance matrix of a reduced set of nodes aimed at representing the whole domain of inversion. Care is taken to preserve the properties of direct stochastic inversion on the whole set of prisms. Hence, the approach accounts in a consistent way for the support effect, the inversion remains exact in the absence of noise on data, point and block known densities are exactly reproduced, any set of linear constraints can be applied, and the inversion is non-iterative in all cases. It is shown on synthetic examples that the number of nodes needs not to be very large to ensure a good approximation of the parameter covariance matrix or to ensure similarity of the inversion solutions. An application to a gravity survey including borehole density data shows the applicability of the method for a large number of cells in the inversion domain. Even with as much as 10,000 nodes and one million prisms, the computing time remained acceptable at less than two hours on a workstation. The inverted solution obtained with the nodes approach is compared to a direct kriging of borehole density data and to direct inversion using only the gravity data. The solution combining both information is different from the inversion using only gravity, but only in the area where borehole data are numerous. Although shown for the gravity–density potential, the approach is general and can be extended to magnetic-susceptibility and joint inversion. The proposed approach helps solving the recurrent problem of the application of stochastic inversion to large grids. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Separate and joint inversions of gravity and magnetic potential fields by non-iterative stochastic methods using cokriging (Wackernagel, 2003) and conditional simulation have been described by Franklin (1970), Asli et al. (2000), Hansen et al. (2006), and Shamsipour et al. (2010, 2011a, 2011b, 2012). The linear inverse Gaussian theory assumes stationarity of the covariance although the stationarity hypothesis can be relaxed (Shamsipour et al., 2013). One advantage of the stochastic approach over the deterministic approach (Menke, 1989) is that it is based on the modeling of the parameter covariances from the observed data and available a priori geological knowledge. Moreover, the physical links between data and parameters are fully integrated in the computation of the data covariances and the data-parameter cross-covariances. As a result, the inverted models exactly reproduce the observed data except when the user chooses to include noise variance–covariance in the

⁎ Corresponding author. E-mail addresses: [email protected] (D. Marcotte), [email protected] (P. Shamsipour), [email protected] (O. Coutant), [email protected] (M. Chouteau).

http://dx.doi.org/10.1016/j.jappgeo.2014.09.003 0926-9851/© 2014 Elsevier B.V. All rights reserved.

data covariance model. In that case, the reconstructed gravity is exact within the gravity data uncertainty. Finally, the inclusion in the inversion of parameter observations (e.g. density or susceptibility in boreholes) is straightforward (Shamsipour et al., 2010, 2011a, 2011b, 2012) and done coherently as all these data are logically linked through their covariances. Inversions obtained by cokriging are smooth and tend to put the mass excess close to the surface. The larger the correlation range of the parameter covariance function, the larger in size the inverted mass anomalies are. Borehole density (or gravity) data help to improve resolution of the inverted field in their neighborhood. Also, depth weighting (Li and Oldenburg, 1998; Boulanger and Chouteau, 2001) can be incorporated in the cokriging approach as shown in Shamsipour et al. (2010). The inversion problem having an infinite number of solutions, non-smooth conditional simulations can be preferred to cokriging as described in Shamsipour et al. (2010). The exact reproduction of gravity anomalies (within gravity data error variance), is ensured for all realizations thanks to the applied post-conditioning by cokriging. Admittedly one important problem with linear Gaussian stochastic inversion (cokriging or cosimulation) is the size of the grid one can tackle. The stochastic approach implies that the explicit computation of the parameter covariance matrix Cρρ of size np × np where np is the number

D. Marcotte et al. / Journal of Applied Geophysics 110 (2014) 90–97

of elements (usually prisms) in the model. This number can be typically very large (say one million prisms or more), so the computation of Cρρ is expensive and quite demanding in memory. It constitutes the Achilles' heel of the method. To alleviate the problem, one avenue is to limit the parameter covariances to models having compact support (i.e. with finite correlation range). In the case where the range is small enough with respect to the field size, the Cρρ matrix becomes very sparse. However, one does not have control on the correlation range as it reflects the spatial continuity of the geological structures. Another approach benefits of the block–block Toeplitz structure of the parameter covariance matrix to save on storage and to speed up computation using the circulant embedding approach of Nowak et al. (2003). However, this simplification is not possible for the point–block covariance computation required when densities are measured in boreholes. In Shamsipour et al. (2010), the inversion is performed using the np prisms of the model, which limits its applicability. Shamsipour et al. (2011b) presents a three-step inversion where the initial inversion on large prisms from surface gravity is conditioned to a set of large prism densities obtained by kriging from borehole point densities. The initial inversion on large prisms is then down-scaled to smaller prisms using kriging. The main drawback of this approach is that the large prisms are not treated equally in the method as some are estimated only by kriging and others only from the gravity data conditional to the kriged densities of the former. In their method, the computed response from the densities estimated for the final small prisms does not replicate exactly the gravity data. The present article proposes to do the inversion by cokriging on a limited number of nodes that are related to the small prisms with a coherent model. The nodes include all the available point density data (if any) plus points aiming to sample fairly the inversion domain. It allows including, in a single step, all the available information about gravity and density. Although illustrated with the gravity–density, the method is general and applies equally well to the magnetic potential and to the joint inversion of gravity–magnetic. The idea of using nodes has been previously used by Ku (1977) for the forward modeling of gravity and magnetic anomalies. Also, Coutant et al. (2012) used nodes for the joint inversion of gravity and seismic data. Such model discretization is classically used in seismic traveltime inversion (Latorre et al., 2004; Zhang and Thurber, 2006). However, since the seismic problem is nonlinear, the inversion is solved iteratively using the formalism of Tarantola and Valette (1982) rather than by cokriging. Moreover, the nodes were used in allowing continuous variations of the parameters rather than reduction of the size of the inversion space. After revisiting the main equations for inversion by cokriging, we describe the new proposal and underline the modifications and advantages it brings compared to the direct inversion on prisms as done in Shamsipour et al. (2010). The results of gravity inversion directly on prisms and those obtained indirectly by inversion on nodes are compared on two synthetic models. A data set from the Mattagami area (Shamsipour et al., 2011b) is used for inversion over a million prisms represented by a few thousands nodes, proving the applicability of cokriging (and cosimulation) inversion on large grids.

2. Method First, we briefly recall the main equations for the geostatistical inversion of potential field with, for illustration, the gravity field (g)–density parameter (ρ) case. It is initially assumed that nd Bouguer anomaly gravity data are available so the objective is to estimate density contrasts relative to a mean density. The field is discretized by np prisms. Although not described here, extension to magnetic field-susceptibility parameter or to the joint inversion case is straightforward as only the kernel matrix changes.

91

2.1. Inversion on prisms The nd × np kernel matrix G relates the density contrasts to the gravity anomaly: g ¼ Gρ:

ð1Þ

The np × np covariance matrix Cρρ of the density parameters and G fully determines the gravity covariances (up to a possible noise covariance matrix) and the gravity–density cross-covariances required for cokriging: T

C gg ¼ GC ρρ G þ C 0

ð2Þ

C gρ ¼ GC ρρ

ð3Þ

where C0 is the nd × nd noise covariance matrix, usually assumed diagonal and constant (nugget effect). The cokriging estimate of density parameters is obtained by computing:
−1 T Λ ¼ GC ρρ G þ C 0 GC ρρ



T

ð4Þ

ρ ¼Λ g

ð5Þ


 2 T σk ¼ diag C ρρ −Λ C gρ

ð6Þ

where Λ is the nd × np cokriging weight matrix and σ2k is the vector with the np cokriging estimation error variances. When the number of prisms np is large, the matrix Cρρ, in Eqs. (2) and (4), becomes huge, therefore limiting the applicability of geostatistical inversion to small sets of large prisms. A possibility could be to replace Cρρ by a proxy matrix of lower rank that keeps most of the information available in Cρρ. Eigenvalue–eigenvector decomposition of Cρρ is not a realistic solution due to the size of this matrix. The idea we pursue herein is to invert on subsets of nr nodes that are related to the prisms in a coherent way. 2.2. Inversion on nodes Assume nr ≪ np nodes are implanted over the inversion domain. The nodes do not need to be on a regular grid. One assumes that the prism densities are linearly related to the densities on the nodes: ρ ¼ Ar

ð7Þ

where A is, for the moment, any real matrix of size np × nr and r is the nr vector of densities on the nodes. The physics remains the same, therefore: e g ¼ Gρ ¼ GAr ¼ Gr

ð8Þ

e ¼ GA. It immediately stems that the densities on nodes can be where G estimated by a cokriging similar to the one done on the prisms:
 e G e T þ C −1 GC e Λr ¼ GC rr 0 rr



T

ð9Þ

r ¼ Λr g

ð10Þ


 2 T σkr ¼ diag C rr −Λr C gr

ð11Þ

where Crr is the nr × nr covariance matrix between the nodes. In Eq. (9), the right hand side matrix requires nr ≥ nd to be invertible when C0 = 0.

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This matrix is much smaller than Cρρ, therefore allowing to realistically estimate r⁎. Note that the solutions (Eqs. (5) and (10)) both recover g exacte  ¼ g. When C0 ≠ 0, the reconstitution ly when C0 = 0. That is, Gρ ¼ Gr of g is no more exact, but compatible with the level of noise variance included in the model. Of course, one is not primarily interested to estimate r but rather ρ. The idea is simply to use the linear relation applied on r⁎: ρ



e ¼ Ar ¼ AΛr g ¼ AC rr G 

T

T




e G eT þ C GC rr 0

−1

g

ð12Þ

and it stems: Gρ




 e T GC e G e T þ C −1 g ¼ GAC rr G rr 0

e G e ¼ GC rr

T

ð13Þ


 e G e T þ C −1 g: GC rr 0

ð14Þ

Again, when C0 is nil, the inverted model ρ⁎⁎ recovers perfectly the e ¼ GA one data g. Comparing Eq. (14) with Eq. (4), and recalling that G can see that the inversion on nodes amounts to approximate the full rank (np) parameter covariance matrix Cρρ by the low rank (nr) covariance matrix ACrrAT. Finally, the estimation variance corresponding to the estimator ρ⁎⁎ can be expressed as:

  2 T T σk;ρ ¼ diag A C rr −Λr C gr A :

ð15Þ

2.3. Which A? The above equations are valid whatever the particular choice of A is. However, one reasonable option to preserve internal coherency of the model is to choose A to be the simple kriging weights obtained from the kriging of prisms' density ρ from nodes' density r. That is: −1

A¼C ρr C rr :

ð16Þ

The np × nr Cρr matrix represents the prism–node covariances. These block–point covariances are obtained by numerical integration of the point–point density covariances as routinely done in geostatistics

0.0001

a = 500 m a = 1000 m a = 2000 m a = 5000 m

(Journel and Huijbregts, 1978). The more nodes are used to represent the inversion domain, the closer is the approximating matrix: e ¼ C C −1 C C −1 C T ¼ C C −1 C T C ρr ρr ρρ ρr rr rr rr ρr rr

to the full Cρρ matrix. For example, Fig. 1 shows the correlations obtained with 18 × 18 × 18 cubes of edge 200 m, when represented by nr nodes on a regular grid within the inversion domain. A point density spherical covariance function is assumed with different correlation ranges. Clearly, the nodes need not to be numerous to provide a good approximation of the Cρρ matrix. For example, when the number of nodes represents 1% of the number of prisms, the approximation is already [90–99.97]% exact (measured as 100R2 between the elements of the two matrices). As expected, from Fig. 1 the greater the correlation range, the smaller the number of nodes can be set to obtain a given quality of approximation. Similar results (not shown) were obtained using as measure of proximity the Froebenius norm of the difference matrix. Note that when the range is very short, an option is to work directly on the prisms and make use of the sparsity of Cρρ. In Eqs. (9) and (12), the largest matrix involved is of size np × nr (Cρr in Eq. (16)). This represents a substantial gain compared to the np × np size of the Cρρ matrix required in Eq. (4). The gain factor (np / nr) can reach up to 2 to 3 orders of magnitude both in the memory and number of multiplications. More importantly, Cρr never has to be loaded in the memory. It can simply be computed, when needed, for small subsets of prisms (see Section 3.1). 2.4. Which nodes? The selection of nodes should be done so as to represent efficiently the whole inversion domain. Moreover, as extrapolations should be avoided, extremal points of the inversion domain are always included in the node set. All available (if any) borehole points where the density is known should also be included. The number of nodes should be kept at a reasonable level (a few thousands or less) for computational efficiency. When only surface gravity data are available, it is reasonable to put more nodes closer to the surface, where the sensitivity is maximal, and less at depth. The number of nodes should also be related to the correlation range of the density covariances, larger ranges requiring lesser nodes as shown in Fig. 1. 2.5. Case with borehole density It is simple to adapt the previous equations to the case where density data are observed in the boreholes. First, one splits the set of nodes in two parts: the nodes corresponding to borehole density data identified as r1 and the regular nodes identified as r2. Then, the cokriging system for estimation on nodes writes simply as:

0.001 

h

r ¼ g

T

T ; r1

i

h

Λr ¼ g

T

T ; r1

" i GC e G eT þ C rr 0g eT C G r1 r

2

100 (1−R )

ð17Þ

0.01

e GC rr1 C r1 r1 þ C 0r1

#−1 "

e GC rr C r1 r

#

ð18Þ where C0g and C 0r1 are the measurement error covariance matrices (usually modeled as nugget effect) on gravity data and borehole densities respectively. Then, the cokriging error variances on nodes can be computed as:

0.1

1

" 2 σkr

10 0.1

1

10

100

nr/np in % Fig. 1. Proportion of information lost, (measured as 100 (1 − R2)), in matrix Cρρ when approximated by matrix CρrCrrCTρr, for different correlation ranges of a spherical model for 183 = 5832 cubes of side 200 m.

¼ diag

T C rr −Λr

e GC rr C r1 r

#! ð19Þ

from which the error variances on the prism densities can be computed using a similar relation as Eq. (15). Recall that r1 is a subset of r. As cokriging is an exact interpolator, one will necessarily have r1⁎ = r1 and σ2kr1 ¼ 0, so only r2 really needs to be estimated.

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3. Adding linear equality constraints

4.1. Synthetic model 1

The geostatistical inversion formalism makes it easy to include sophisticated linear constraints in the inversion (Shamsipour et al., 2010; Marcotte et al., 2014). The constraints are exactly reproduced, in a single step, provided the constraints are fully integrated in the covariance computations. To illustrate this flexibility, we consider the forcing of an inverted field with an imposed difference d for the mean block densities of the first layer of the model with respect to the mean block densities of the last layer. One has nx × ny × nz prisms. The constraint writes:

The first synthetic model consists in an L-shaped block oriented along x coordinate with density contrast 500 kg/m3 and center located at y = 3; the subsurface is meshed with cubic cells of side 50 m (Fig. 2a). The corresponding gravity anomaly (in mgal) is given in Fig. 2b. A spherical model of covariance with range 200 m is used. The inverted model by cokriging directly on the prisms is shown in Fig. 2c. The inversion on nodes propagated to the prisms is illustrated in Fig. 2d. Note that despite 216 nodes only are used compared to 1331 prisms, the two solutions are similar as their Pearson correlation coefficient is r = 0.988. The Pearson correlation coefficient between Cρρ and e ρρ is 0.996. Note that both solutions do not recover well the lowest part C of the L-shaped block due to the poor sensitivity of gravity to density contrasts at depth. A well-known solution is to use depth weighting (Li and Oldenburg, 1998; Boulanger and Chouteau, 2001), but this is not used here.

Δ¼

ny nx X X

ρi; j;nz −ρi; j;1 ¼ d

ð20Þ

i¼1 j¼1

Δ is treated as an additional information with value d. The only modification to the cokriging systems (Eqs. (4), (9) or (18)) is to add one column and one line in the left hand matrix and one line in the right member. More specifically, the constraint can be written in matrix form as Δ = bTρ where b is the np × 1 vector with weights 1, 0 or −1, whether the prisms belong to the first layer (−1), the last layer (1) or any other layer (0). Then, for Eq. (4) as an example: 

Λ λΔ

"

 ¼

T

GC ρρ G þ C 0

GC ρρ b

b C ρρ G

b C ρρ b

T

T

T

#−1 "

GC ρρ

#

T

b C ρρ

:

ð21Þ

Of course, the above generalizes immediately to any set of linear equality constraints as described by the B matrix of size np × nc where nc is the number of added constraints. Vector b is simply replaced by B in Eq. (21). The condition to fulfill is that the right side matrix in Eq. (21) remains non-singular. Therefore, B must be full rank and linearly independent of the GCρρGT + C0 matrix. Eq. (21) applies equally well e ρρ ¼ C ρr C −1 C T instead of Cρρ. using the surrogate matrix C rr ρr 3.1. Some remarks on the computational aspect For a large mesh made of prisms (see Section 4.3) the matrix Cρr can be too large to fit in the memory. However, it is possible to compute the matrix sequentially for small subsets of prisms as the full matrix is never required at once. Moreover, when cokriging variances are not needed, dual cokriging can be used (Chilès and Delfiner, 1999; Rivest et al., 2008) to speed up the computations. Hence, one can write for the case with gravity data and densities measured on cores: " # e  T GC rr r ¼f ð22Þ C r1 r with " f ¼

eT þ C e G GC rr 0g eT C G r1 r

e GC rr1 C r1 r1 þ C 0r1

#−1   g : r1

ð23Þ

The most demanding computation is indeed the prism–node covarie ¼ GA ¼ GC ρr C −1 . To speed up the computations, a ance matrix Cρr in G rr look-up table was used. Point–block covariance was computed on a grid along the three coordinates for distances up to the correlation range and then interpolated by the nearest-neighbor method. This allowed computing directly the position in the look-up table of all point–prism covariances, allowing a substantial reduction in CPU time. 4. Results The inversion results obtained on nodes are compared to the inversion obtained directly on the prisms with two synthetic models. Then, an application to a real data set illustrates the applicability of the method for inversion on a large number of prisms (N one million prisms).

4.2. Synthetic model 2 The second synthetic model is a stochastic simulation of density with spherical variogram, correlation range of 200 m (i.e. 4 cubes of side 50 m and a sill of 50,000 (kg/m3)2. Again, the inversions performed on prisms and on nodes are quite similar as their Pearson correlation coefficient reaches r = 0.994. The Pearson correlation coefficient between e ρρ is 0.996. Cρρ and C The inversion on nodes is further constrained to impose that the inverted mean density at the bottom layer is 164 kg/m3 less than the inverted mean density on the first layer of prisms. Fig. 3 shows the results obtained with and without the constraint. The correlation between the inversion directly on prisms and the inversion on nodes remains 0.994 after the constraint is imposed. Fig. 4 shows the mean layer density for both inversions with and without constraint. The effect of the constraint is felt over all layers but is less pronounced in the middle layers. The difference between the first layer and the last layer is exactly 164 kg/m3 as requested. 4.3. A large grid for the Matagami data set To show the applicability of the proposed approach, the Matagami data set already studied in Shamsipour et al. (2011b) is used. In addition to the gravity anomaly defined on a regular grid consisting in 253 data at ground surface, the densities are measured on 1274 cores from 277 different boreholes found in the northern part (584 m b Northing b 1150 m) of the area covered by the gravity survey, see Fig. 5. The inversion is done by dual cokriging on 10,294 nodes including the 1274 core nodes. The 9020 extra nodes are taken on a grid with equal spacing in the horizontal plane and logarithmic spacing along the vertical. No attempt was made to optimize their location and number as the interest bears mainly on the computational aspect and applicability of the method. The prisms are small cubes of 7-m side. There are 87 × 163 × 71 = 1,006,851 prisms over the inversion domain. The point density covariance model was estimated from the densities measured in boreholes using variography and cross-validation (Marcotte, 1995). An isotropic spherical model with a range of 80 m was retained. The point– block covariances are computed from a look-up table by computing the location of the nearest-neighbor for all the node–prism combinations (over 10 billion combinations). The inversion took approximately 1.9 h on a Xeon workstation. Almost 87% of the time was devoted to the computation of Cρr. This proportion is inversely related to the number of nodes for a given number of prisms. Hence, with 15,000 nodes instead of 10,000 and the same number of prisms, the proportion of total computation time devoted to Cρr decreases to 76%. Fig. 6 shows three horizontal sections of the inverted densities using gravity and density (left column). For comparison, the same sections obtained by gravity inversion alone (middle column) and the kriging

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D. Marcotte et al. / Journal of Applied Geophysics 110 (2014) 90–97

450

2

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350

mgal

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b)

500 1

kg/m

a)

100

10

50

11 2

4

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8

10

0

2

4

6 Y (prisms)

8

10

0

Fig. 2. First synthetic density contrasts model (kg/m3) a), gravity anomaly (mgal) b), inversion on prisms c), inversion on nodes d). Sections a), c) and d) taken at y = 4; x, y and z units are in prism indices, prisms are cubes of side 50 m.

of prism densities from borehole densities (right column) are given. The scatterplot of calculated vs observed gravity for the three modes of inversion is shown in Fig. 7. The two inversions obtained by cokriging with gravity reproduce perfectly the observed data, but not the kriging solution because it does not incorporate the relation between density and gravity variables. The borehole data and the kriged map show rather small contrasts of density compared to those estimated from the gravity field alone. This is somewhat unexpected as point densities should in theory be more variable than block densities. One possible explanation is that boreholes only sample the northern part of the study field where the gravity anomaly is smaller, hence where the density is more homogeneous. Another possible explanation is that the variations seen on the gravity map are caused by sources located below the depth reached by the boreholes. Comparing the different inverted solutions in Fig. 6 reveals interesting differences. First, the kriging of densities (right column) shows weak density contrasts, insufficient to explain the observed gravity field. Second, despite the weak density contrasts observed in boreholes, the inversion using gravity and density in borehole (first column) is different from the inversion using only gravity data (middle column) especially in the northern area where borehole data are numerous. In the first two rows for example, the maximum density appears where no borehole data are present (around Easting = 250, Northing = 800). This is because the borehole data did not show high density contrast at this location, preventing to invert high densities in prisms informed by boreholes. However, as something must cause the gravity anomaly, the excess mass is placed in the neighbor prisms, less informed by borehole data. On the contrary, the inversion based only on gravity data shows positive density contrasts that contradict the borehole density measurements.

5. Discussion The proposed approach applies to linear inversion of Gaussian fields only. The parameter covariance function was assumed stationary, but this condition can be relaxed following Shamsipour et al. (2013). Although illustrated for the simple case of gravity data–density parameters, the approach can be easily adapted to the inversion of magnetic data-susceptibility parameters or to the joint inversion of both data. In addition, any point (or block) measurement of parameter can be directly included in the inversions, thus avoiding the downscaling step to small prisms. In particular, the point observations are natural nodes to include in addition to the nodes covering the whole inversion domain. This is an important improvement over the approach proposed by Shamsipour et al. (2011b). Clearly, the approach extends as well to the conditional simulation using conditioning by cokriging as in Shamsipour et al. (2010). When density data is unavailable, the density covariance matrix can be estimated by the V−V plot method (Asli et al., 2000; Gloaguen et al., 2005). The most demanding step in the approach with nodes is undoubtedly the computation of the Cρr covariance matrix. However, this matrix is never required at once in the memory, so its computation can be done sequentially on small subsets of prisms. The computation time scales up linearly with the number of prisms over the whole domain. For one million prisms with 10,000 nodes, the time was approximately 1.9 h on a Xeon workstation. So, still with 10,000 nodes, realistic computation time is obtainable even for a few million prisms. Admittedly, one may wonder when such level of details is justified by the data at hand. Typically, many densities measured in boreholes would be required to justify such a fine description. One possibility to save on computation time is to define matrix A in Eq. (16) easier to compute. This can be achieved for example by triangulating the nodes and interpolating within the triangles. Although the

D. Marcotte et al. / Journal of Applied Geophysics 110 (2014) 90–97

a)

b)

95

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0 −1 −2 −3 12

3

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Fig. 3. Second synthetic model: gravity anomaly (mgal) a), inversion on nodes without constraint b), inversion on nodes with constraint c), inversion on prisms with constraint d). The constraint imposes the bottom layer mean density to be 164 kg/m3 lower than the first layer mean density. Sections b), c) and d) are taken at y = 5; x, y and z units are in prism indices, prisms are cubes of side 50 m.

reconstruction of g would still be exact, the internal coherency of the model would be lost. Hence, the change of support from point support (nodes) to block support (prisms) would not be well accounted for. eρρ matrix would be less similar to the Cρρ than with our The resulting C approach. As a result the inversion on nodes would show more differences with the direct inversion on prisms. It was shown that the number of nodes needs not to be high to recover inversion results quite close to those obtained with a much larger

number of prisms. In the small examples tested a ratio of number of nodes/prisms of 0.05 was enough to produce correlations of 0.99 between the two inverted solutions. The ratio required to obtain a given correlation decreases with the number of prisms as the quality of approximation provided by nodes is foremost a function of the absolute number of nodes rather than a function of its relative number. Moree ρρ , the ratio over, for a fixed level of correlation between Cρρ and C

0.3 1

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mgal

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y (m)

Layer

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−0.2 8

400 −0.3

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decreases with the increase of the correlation range. For small correlation range, which requires more nodes, an alternative is to rely on the sparsity of Cρρ and do the inversion directly on prisms. However, with the nodes approach the Cρr matrix also becomes sparse. Using the sparsity of Cρr should likely keep the computational advantage of the nodes approach over the direct inversion on prisms. The inversion obtained on the Matagami data with borehole density and gravity is similar to those obtained with only gravity

in the zone not sampled by the boreholes. In the zone where boreholes are present, visible differences were noted likely created by the light density contrasts observed in boreholes that are not compatible with the stronger gravity anomaly. Both inversions reproduce perfectly the gravity data. On the contrary, the direct kriging of densities fails completely to reproduce the gravity data as it does not incorporate the physical relation linking density and gravity.

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6. Conclusion An inversion approach based on a small set of nodes was proven to recover similar estimated prism density to direct inversion on prisms. This allows using a large grid of prisms, up to a few millions, something impossible by direct inversion on prisms. The node approach also enables to treat all known borehole density data simply as nodes with known value. A solution with over 10,000 nodes and one million prisms was obtained for 1.9 h on a workstation. It was shown that any linear equality constraints on the inverted solution can be easily incorporated in the solution. Acknowledgments This research was funded by the NSERC (105603-2010) Discovery research grant of the main author (D.M.). References Asli, M., Marcotte, D., Chouteau, M., 2000. Direct Inversion of Gravity Data by Cokriging. In: Kleingeld, W., Krige, D. (Eds.), Geostats 2000: Cape Town, Proceedings of the 6th International Geostatistics Congress, pp. 64–73. Boulanger, O., Chouteau, M., 2001. Constraints in 3D gravity inversion. Geophys. Prospect. 49, 265–280. Chilès, J.P., Delfiner, P., 1999. Geostatistics: Modeling Spatial Uncertainty. Wiley. Coutant, O., Bernard, M., Beauducel, F., Nicollin, F., Bouin, M., Roussel, S., 2012. Joint inversion of p-wave velocity and density, application to La Soufrière of Guadeloupe hydrothermal system. Geophys. J. Int. 191, 723–742. Franklin, J.N., 1970. Well posed stochastic extensions of ill posed linear problems. J. Inst. Math. Appl. 31, 682–716. Gloaguen, E., Marcotte, D., Chouteau, M., Perroud, H., 2005. Borehole radar velocity inversion using cokriging and cosimulation. J. Appl. Geophys. 57, 242–259. Hansen, T., Journel, A., Tarantola, A., Mosegaard, K., 2006. Linear inverse Gaussian theory and geostatistics. Geophysics 71, R101–R111.

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