Radial gravity inversion constrained by total anomalous mass excess for retrieving 3D bodies Vanderlei Coelho Oliveira Junior Valéria C. F. Barbosa Observatório Nacional

Contents Objective Methodology Real Data Inversion Result Conclusions Synthetic Data Inversion Result 3D Gravity inversion method Do the gravity data have resolution to retrieve the 3D source?

y x N E z Depth 3D source Objective Estimate from gravity data the geometry of an isolated 3D source Gravity data Source’s top

Methodology

y x z y x Gravity observations g o N R  3D Source Depth S The 3D source has an unknown closed surface S. Methodology

x z y x Gravity observations g o N R  3D Source Depth Approximate the 3D source by a set of 3D juxtaposed prisms in the vertical direction. S y Methodology

x z y x Gravity observations g o N R  3D Source Depth We set the thicknesses and density contrasts of all prisms y Methodology

x z y x Gravity observations g o N R  3D Source Depth y Methodology The horizontal cross-section of each prism is described by an unknown polygon The polygon sides approximately describe the edges of a horizontal depth slice of the 3D source.

y x Gravity observations g o N R  Methodology The polygon sides of an ensemble of vertically stacked prisms represent a set of juxtaposed horizontal depth slices of the 3D source. x z 3D Source Depth S y

y x Gravity observations g o N R  Methodology The polygon sides of an ensemble of vertically stacked prisms represent a set of juxtaposed horizontal depth slices of the 3D source. x z Depth y

x z Depth y Methodology We expect that a set of juxtaposed estimated horizontal depth slices defines the geometry of a 3D source

x z The horizontal coordinates of the polygon vertices represent the edges of horizontal depth slices of the 3D source. MethodologyDepth y  jj yx,  jj yx,

The polygon vertices of each prism are described by polar coordinates Methodology x z Depth y with an arbitrary origin within the top of each prism. Arbitrary origin 1 1  r, () 2 2  r, () 3 3  r, () ( 4 4  r, ) ( 5 5  r, ) ( 6 6  r, ) ( 7 7  r, ) ( 8 8  r, )

x z Depth y 1 1  r, () 2 2  r, () 3 3  r, () ( 4 4  r, ) ( 5 5  r, ) ( 6 6  r, ) ( 7 7  r, ) ( 8 8  r, ) The vertical component of the gravity field produced by the k th prism at the Methodology ),,,,, ( k 1 k z dz k m,, i z i y i x f  k  1 k z i th observation point ( x i, y i, z i ) is given by (Plouff, 1976) dz T M k M kk   11 ][    Tkkk M k k yxrr m 001 ][ M)2(1    i g =

The gravity data produced by the set of L vertically stacked prisms at the i th observation point ( x i, y i, z i ) is given by Methodology   L k k f 1 i g = ),,,,, ( 1 k z dz k m,, i z i y i x  k  x z Depth y

Methodology THE INVERSE PROBLEM Estimate the radii associated with polygon vertices x z Depth y Arbitrary origin 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r and the horizontal Cartesian coordinates of the origin. ( x o, y o )

y x Gravity observations Methodology By estimating the radii associated with polygon vertices and the horizontal Cartesian coordinates of the arbitrary origin from gravity data, x z 3D Source Depth y (x o, y o ) we retrieve a set of vertically stacked prisms

The Inverse Problem   )(m Parameter vector The data-misfit function The constrained inversion obtains the geometry of 3D source by minimizing :  )( m  2 )(  m g g o 2 The constrained function )( m  =   The constrained function  ( m ) is defined as a sum of several constraints:

Smoothness constraint on the adjacent radii defining the horizontal section of each vertical prism The Inverse Problem The first-order Tikhonov regularization on the radii of horizontally adjacent prisms x z Depth y 2 r 3 r 4 r 5 r 6 r 7 r 8 r 1 r This constraint favors solutions composed by vertical prisms defined by approximately circular cross-sections. 1 r  2 r

k j r 1  k j r Inverse Problem Smoothness constraint on the adjacent radii of the vertically adjacent prisms x z Depth y The first-order Tikhonov regularization on the radii of vertically adjacent prisms This constraint favors solutions with a vertically cylindrical shape. j r  j r k k+1

Inverse Problem Smoothness constraint on the horizontal position of the arbitrary origins of the vertically adjacent prisms It imposes smooth horizontal displacement between all vertically adjacent prisms.  x z Depth y ), ( 00 kk yx ), ( 0 0 k +1 y x ), ( 00 kk yx ), ( 0 0 y x

Inverse Problem The estimation of the depth of the bottom of the geologic body x z Depth y dz zozo The interpretation model implicitly defines the maximum depth to the bottom (z max ) of the estimated body by L L. dz z   max o z z How do we choose z max ? Do the gravity data have resolution to retrieve the 3D source?

Inverse Problem The depth-to-the-bottom estimate of the geologic body 1) We assign a small value to z max, setting up the first interpretation model. g Observed gravity data zozo z max 1 z x 2) We run our inversion method to estimate a stable solution Fitted gravity data s mt z max 1 mt X s-curve 5) We repeat this procedure for increasingly larger values of z max of the interpretation model z max 2 g Observed gravity data zozo z max 2 z x Fitted gravity data g Observed gravity data zozo z max 3 z x Fitted gravity data z max 3 g Observed gravity data zozo z max 4 z x Fitted gravity data 4) We plot a point of the mt X s - curve 3) We compute the L1-norm of the data misfit ( s ) and the estimated total-anomalous mass ( mt ) z max 4 Optimum depth-to-bottom estimate

Inverse Problem Do the gravity data have resolution to retrieve the 3D source? Correct depth-to-bottom estimate mt X s-curve L1-norm of the data misfit Estimated total-anomalous mass s ( mGal ) mt g Observed gravity data zozo z z x Fitted gravity data The gravity data are able to resolve the source’s bottom. z

Inverse Problem Do the gravity data have resolution to retrieve the 3D source? Minimum depth-to-bottom estimate s ( mGal ) mt mt X s-curve L1-norm of the data misfit Estimated total-anomalous mass g Observed gravity data zozo z z x Fitted gravity data The gravity data are unable to resolve the source’s bottom. z

INVERSION OF SYNTHETIC GRAVITY DATA

Synthetic Tests Two outcropping dipping bodies with density contrast of 0.5 g/cm³. Simulated shallow-bottomed body Simulated deep-bottomed body Maximum bottom depth of 3 km Maximum bottom depth of 9 km

Synthetic Tests Shallow-bottomed dipping body (true depth to the bottom is 3 km) The estimated total-anomalous mass ( mt ) x the L1-norm of the data misfit ( s ) s (mGal) L1-norm of the data misfit mt (kg x ) Estimated total-anomalous mass mt X s - curve z max = 1.0 km z max = 11.0 km z max = 2.0 km z max = 3.0 km

s (mGal) L1-norm of the data misfit mt (kg x ) Estimated total-anomalous mass z max = 1.0 km z max = 11.0 km z max = 2.0 km z max = 3.0 km Synthetic Tests Shallow-bottomed dipping body (true depth to the bottom is 3 km) Depth (km) y( km ) x( km ) Initial guess True Body mt X s - curve

Synthetic Tests Shallow-bottomed dipping body (true depth to the bottom is 3 km) True Body Estimated body Depth (km) x( km ) y( km )

Synthetic Tests Deep-bottomed dipping body ( true depth to the bottom is 9.0 km ) The estimated total-anomalous mass ( mt ) x the L1-norm of the data misfit ( s ) s (mGal) L1-norm of the data misfit mt (kg x ) Estimated total-anomalous mass z max = 6.0 km mt X s - curve True depth to the bottom Lower bound estimate of the depth to the bottom z max = 9.0 km

Depth (km) x( km ) y( km ) True Body Synthetic Tests Deep-bottomed dipping body ( true depth to the bottom is 9.0 km ) y( km ) x( km ) By assuming two interpretation models with maximum bottom depths of 6 km and 9 km Initial guess (6 km)Initial guess (9 km) True Body Depth (km) Lower bound estimate of the depth to the bottom (6 km)True depth to the bottom (9 km) 6 km 9 km

Synthetic Tests Deep-bottomed dipping body ( true depth to the bottom is 9.0 km ) By assuming two interpretation models with maximum bottom depths of 6 km and 9 km Depth (km) y( km ) x( km ) True Body x( km ) y( km ) True Body Estimated body 6 km Lower bound estimate of the depth to the bottom (6 km)True depth to the bottom (9 km) 9 km 6 km

INVERSION OF REAL GRAVITY DATA

Application to Real Data Real gravity-data set over greenstone rocks in Matsitama, Botswana. Study area

Simplified geologic map of greenstone rocks in Matsitama, Botswana. (see Reeves, 1985) Application to Real Data

Northing (km) Easting (km) Gravity-data set over greenstone rocks in Matsitama (Botswana). Application to Real Data A B

The estimated total-anomalous mass ( mt ) x the L1-norm of the data misfit ( s ) Application to Real Data s (mGal) L1-norm of the data misfit mt (kg x ) Estimated total-anomalous mass mt X s - curve z max = 3.0 km z max = 10 km z max = 8 km

Estimated greenstone rocks in Matsitama (Botswana). Application to Real Data Depth (km) Estimated Body Initial guess Northing (km) N Easting (km)

Application to Real Data Northing (km) Easting (km) Northing (km) Easting (km) Estimated greenstone rocks in Matsitama (Botswana).

Application to Real Data Northing (km) Easting (km) The fitted gravity anomaly produced by the estimated greenstone rock in Matisitama

Conclusions

Conclusions The proposed gravity-inversion method Estimates the 3D geometry of isolated source Introduces homogeneity and compactness constraints via the interpretation model To reduce the class of possible solutions, we use a criterion based on the curve of the estimated total-anomalous mass (mt) versus data-misfit measure (s). The solution depends on the maximum depth to the bottom assumed for the interpretation model. The correct depth-to-bottom estimate of the source is obtained if the minimum of s on the mt × s curve is well defined Otherwise this criterion provides just a lower bound estimate of the source’s depth to the bottom. Depth (km) s (mGal) L1-norm of the data misfit mt X s - curve mt (kg x ) Estimated total-anomalous mass 0 z max = 3.0 km Depth (km) mt (kg x ) Estimated total-anomalous mass s (mGal) L1-norm of the data misfit z max = 6 km Lower bound estimate of the depth to the bottom mt X s - curve

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