Full Tensor Gravity Gradiometry(FTG): an Overview

Hendra GrandisTeknik Geofisika - FTTM ITB

Hendra Grandis

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• Introduction

• Basic Concepts

• Simulation

• Advanced Processing

• Applications

• Summary

Outline

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• What is FTG stands for?

→ Full Tensor Gradiometry

→ Full Tensor Gradiometry of Gravity

→ Full Tensor of Gravity Gradiometry

• Tensor is just another name for ‘physical quantity’ described by defining a coordinate system, represented as matrix

→ Scalar ~ 0th order tensor, matrix 1 × 1

→ Vector ~ 1st order tensor, matrix 1 × 3

→ Tensor ~ 2nd order tensor, matrix 3 × 3

Introduction

in Cartesian coordinate system

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• Gradiometry ~ Gradient measurement(airborne or shipborne)

• Gradient

→ Spatial rate of change of anomaly

→ Enhances anomalous source boundaries

→ Enhances high-frequency anomalies, i.e.shallow sources, noise

→ Mathematically represented by derivatives

∂gz ∂gz ∂gzꟷꟷꟷ ; ꟷꟷꟷ ; ꟷꟷꟷ …∂x ∂y ∂z

Introduction

(Grandis et al, 2018)

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• Gradiometry ~ Gradient measurement(airborne or shipborne)

• Gradient

→ Spatial rate of change of anomaly

→ Enhances anomalous source boundaries

→ Enhances high-frequency anomalies, i.e.shallow sources, noise

→ Mathematically represented by derivatives

∂gz ∂gz ∂gzꟷꟷꟷ ; ꟷꟷꟷ ; ꟷꟷꟷ …∂x ∂y ∂z

Introduction

(Grandis et al, 2018)

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• Gradiometry ~ Gradient measurement(airborne or shipborne)

• Gradient

→ Spatial rate of change of anomaly

→ Enhances anomalous source boundaries

→ Enhances high-frequency anomalies, i.e.shallow sources, noise

→ Mathematically represented by derivatives

∂gz ∂gz ∂gzꟷꟷꟷ ; ꟷꟷꟷ ; ꟷꟷꟷ …∂x ∂y ∂z

Introduction

(Grandis et al, 2018)

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• Loránd Eötvös and torsion balance (1890)

Introduction

(Szabó, 2016)

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• Loránd Eötvös and torsion balance (1890)

Introduction

(Szabó, 2016)

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• Loránd Eötvös and torsion balance (1890)

Introduction

Szabo´

(Szabó, 2016)

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• Conventional gravity measures only one vertical component (Gz)

• Full tensor gravity gradiometry measurement results in all curvature components of G

• Only 5 components of G are independent

→ Gxy = Gyx, Gxz = Gzx, Gyz = Gzy

due to G = 0

→ Gxx, Gyy or Gxx, Gzz or Gzz, Gyy

due to Laplace's equation for potential field, Gxx + Gyy + Gxx = 0

Basic Concepts

Gz

Gx

Gy

Gzz

Gzy

Gzx

Gyz

Gyy

GyxGxz

Gxy

Gxx

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• Gravity

→ 1 Gal = 1 cm/sec2

→ 1 mGal = 10-3 cm/sec2 = 10-5 m/sec2 ~ 10-6 g

• Gravity gradient

→ 1 Eötvös = 10-9/sec2

→ 1 Eötvös = 10-1 mGal/km

• Gravity and gravity gradient are very small quantities

→ Difficult to observe or to measure

→ Interpretation should be done with care

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Basic Concepts

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Basic Concepts

(Evstifeev, 2017)

• Application of Gravity Gradiometry

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• Airborne FTG by Bell Aerospace

Basic Concepts

(Veryaskin, 2018)

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• Gravity Field and Ocean Circulation Explorer (GOCE) satellite launched in 2009 by European Space Agency (ESA)

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Basic Concepts

(Veryaskin, 2018)

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• Tensor of gravity gradient

∂Gx ∂Gx ∂Gxꟷꟷꟷ ꟷꟷꟷ ꟷꟷꟷ∂x ∂y ∂z

∂Gy ∂Gy ∂GyG = [Gij ] = ꟷꟷꟷ ꟷꟷꟷ ꟷꟷꟷ

∂x ∂y ∂z

∂Gz ∂Gz ∂Gzꟷꟷꟷ ꟷꟷꟷ ꟷꟷꟷ∂x ∂y ∂z

Gxx Gxy Gxz

= Gyx Gyy Gyz

Gzx Gzy Gzz15

Simulation of FTG

Gz

Gx

Gy

Gzz

Gzy

Gzx

Gyz

Gyy

GyxGxz

Gxy

Gxx

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• Fourier Transform pairs (Mickus & Hinojosa, 2001)

∂Gx ∂Gzꟷꟷꟷ -i kx G'x ; G'x = F(Gx) ꟷꟷꟷ -i kx G'z ; G'z = F(Gz)∂x ∂x

∂Gx ∂Gzꟷꟷꟷ -i ky G'x ꟷꟷꟷ -i ky G'z∂y ∂y

∂Gx ∂Gzꟷꟷꟷ |k| G'x ꟷꟷꟷ |k| G'z∂z ∂z

|k| G'x = -i kx G'z ; G'x = -i kx |k|-1G'z

∂Gx ∂Gx ∂Gxꟷꟷꟷ -kx2 |k|-1G'z ; ꟷꟷꟷ -ky kx |k|-1G'z ; ꟷꟷꟷ -i kx G'z∂x ∂y ∂z

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Simulation of FTG

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• Tensor of gravity gradient

-kx2 -kx kyꟷꟷꟷ ꟷꟷꟷꟷꟷ -i kx |k| = (kx2 + ky2)1/2|k| |k|

-ky2 = -i kzG = [Gij ] = F -1 G'z ꟷꟷꟷ -i ky|k|

|k|

→ Tensor components of FTG can be calculated from observed gravity data Gz

→ Simulated FTG for preliminary analysis

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Simulation of FTG

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• 3D forward modelling of a simple box model with density contrast 0.5 gr/cm3

→ Gravity anomaly Gz

→ Gravity gradient tensor G

18

Simulation of FTG

mGal

mGal/km

(Grandis & Dahrin, 2014)

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• 3D forward modelling of a simple box model with density contrast 0.5 gr/cm3

→ Gravity anomaly Gz

→ Gravity gradient tensor G

• Gravity gradient tensor G from Gz by using FFT

→ Without and with 5% Gaussian noise

→ Comparison with analytically calculated G

19

Simulation of FTG

mGal

mGal/km

(Grandis & Dahrin, 2014)

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• 3D forward modelling of a simple box model with density contrast 0.5 gr/cm3

→ Gravity anomaly Gz

→ Gravity gradient tensor G

• Gravity gradient tensor G from Gz by using FFT

→ Without and with 5% Gaussian noise

→ Comparison with analytically calculated G

20

Simulation of FTG

mGal

mGal/km

(Grandis & Dahrin, 2014)

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• 3D forward modelling of a simple box model with density contrast 0.5 gr/cm3

→ Gravity anomaly Gz

→ Gravity gradient tensor G

• Gravity gradient tensor G from Gz by using FFT

→ Without and with 5% Gaussian noise

→ Comparison with analytically calculated G

21

Simulation of FTG

mGal

mGal/km

(Grandis & Dahrin, 2014)

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• Spectral analysis based filtering similar to conventional gravity data Gz

• Parameters extracted from gravity gradient tensor that can be related to structures (Pedersen & Rasmussen, 1990; FitzGerald & Holstein, 2005; 2006)

→ Invariants

→ Determinant

→ Eigen values

→ Strike angle

→ Curvature magnitude

→ Azimuth of maximum curvature

→ etc.

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Advanced Processing

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• FTG is not a stand-alone technology

→ FTG surveys complement airborne gravity surveys

→ To resolve ambiguity in subsurface structural and stratigraphic 2D/3D seismic interpretation

→ 3D forward modelling with integration with seismic and log data to improve interpretation accuracy and risk reduction

• Subsurface imaging

→ Complex salt geometry, top and base salt

→ Basalt geometry and characterisation

→ Sub basalt, sub-basalt sediments and basement geometry

Applications

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• High frequency ~ high resolution

• Amplitude decays

Gravity Gravity gradient

Gz ~ r-2 ~ z -2 Gzz ~ r-3 ~ z-3

→ Gradiometer is more detail than gravity for shallow < 2 km to 3 km depths, usually with more complex geology

→ Gravity has better resolution for structures > 2 to 3 km in depth

Applications

(Stadtler et al, 2014)

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Applications

(Stadtler et al, 2014)

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(Carlos et al, 2013)

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(Carlos et al, 2013)

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a) Gzx, b) Gzy, c) Gzz, d) Adaptive tilt angle,

e) Depth colour map of the Vinton Salt Dome (USA)

Applications

(Salem et al, 2013)

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Applications

(Salem et al, 2013)

a) Gzz, b) Adaptive tilt angle,

c) Depth colour map of Faeroe-Shetland Basin (UK)

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(a) (b)

a) Gzz from FTG

b) Gzz derived from airborne gravity

c) Gzz from FTG filtered with 5 km low-pass filter

(c)

(Barnes, 2013)

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• Full Tensor Gradiometry measures gravity gradient along all cartesian coordinate axes

→ High frequency ~ high resolution ~ shallow

→ Prospect scale, salt / sub-salt imaging

→ Detailed structures of a basin, basal / sub-basalt

• Preliminary modelling / processing with existing data or synthetic data

→ Target confirmation and survey design

• Advanced data processing of FTG data

→ Based on directional and "matrix" properties of the tensor

→ Extract more information, i.e. structural and depth delineation

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Summary

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