Program Studi Teknik Geofisika

Fakultas Teknologi Eksplorasi dan Produksi

Universitas Pertamina

Djedi S. Widarto

Dicky Ahmad Zaky

GP-3105 GRAVITY & MAGNETICMETODE GAYABERAT & MAGNETIK TA 2019/2020

LECTURE #04

GRAVITY METHOD▪ Gravity Anomalies due to Geometrical

Effect▪ Geophysical Modeling▪ Gravity Modeling

✓ 2D Forward Modeling✓ 3D Forward Modeling

▪ Gravity Interpretation

▪ Gravity acceleration at an observation point P due to a mass point m at distant r:

▪ So the vertical component of gravity acceleration at an observation point P:

▪ Thus variation of g along the line x can be estimated if we have the depth of mass point z, mass of point m and universal gravity constant G …

𝛥𝑔𝑟 =𝐺𝑚

𝑟2(1)

𝛥𝑔 =𝐺𝑚𝑧

𝑥2 + 𝑧2 3/2(2)

Gravity Anomalies due to Geometrical Effect

P

Gravity Anomaly due to a Mass Point

▪ The value of g due to the sphere with mass m is similar to the mass point with same mass m :

▪ If mass m is stated as contrast density Δ𝜌, then g due to a sphere can be estimated:

▪ Equation (3) can be manipulated to observe its relationship with Δ𝑔𝑚𝑎𝑥:

▪ Thus Eq. (5) can be used to determine the depth of anomalous source …

𝛥𝑔 =𝐺𝑚𝑧

𝑥2 + 𝑧2 3/2 (2)

𝛥𝑔 =ൗ4 3𝜋𝑅

3Δ𝜌𝐺𝑧

𝑥2 + 𝑧2 3/2(3)

Δ𝑔 =ൗ4 3𝜋𝑅

3Δ𝜌𝐺

𝑧2𝑧

1 + 𝑥2/𝑧2 3/2

Δ𝑔 = Δ𝑔𝑚𝑎𝑥

𝑧

1 + 𝑥2/𝑧2 3/2 (5)

Gravity Anomalies due to Geometrical Effect

Gravity Anomaly due to a Sphere

▪ Anomalous responses of spheres vs depth can be estimated using Eq. (5);

▪ Please pay your attention to the amplitude and width of gravity responses due to the variation of depth.

Gravity Anomalies due to Geometrical Effect

Gravity Variation due to the Depth Variations of Sphere

▪ Maximum value of gravity anomaly for a sphere model is,

Δ𝑔𝑚𝑎𝑥 =ൗ4 3𝜋𝑅

3Δ𝜌𝐺

𝑧2

▪ We can estimate the depth of the center of the sphere using Eq. (5). When Δ𝑔 = Δ𝑔𝑚𝑎𝑥/2, so

𝑧 =𝑥1/234 − 1

𝑧 = 1.3 𝑥1/2

▪ 𝑥1/2 is half of the maximum value of Δ𝑔𝑚𝑎𝑥.

Gravity Anomalies due to Geometrical Effect

Gravity Anomaly due to a Sphere

▪ Nilai anomali gayaberat maksimum untuk model tabung vertikal

𝛥𝑔𝑚𝑎𝑥 = 2𝜋𝐺Δ𝜌(𝑠1 − 𝑑) 𝐿 ⟶ 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒if

𝛥𝑔𝑚𝑎𝑥 = 2𝜋𝐺Δ𝜌𝑟 𝑑 = 0if

𝛥𝑔𝑚𝑎𝑥 = 2𝜋𝐺Δ𝜌(𝐿 + 𝑠1 − 𝑠2)

𝑧 = 𝑥1/2 3

if 𝐿 ⟶ 𝑓𝑖𝑛𝑖𝑡𝑒

Gravity Anomalies due to Geometrical Effect

Gravity Anomaly due to a Vertical Cylinder Model

▪ The maximum value of gravity anomaly for 2D prism model is,

𝛥𝑔𝑚𝑎𝑥 = 2𝐺Δ𝜌 𝑏 ln𝑑

𝐿𝐿 ≫ 𝑏if

Gravity Anomalies due to Geometrical Effect

Gravity Anomaly due to a 2D Prism Model

Gravity Anomalies due to Geometrical Effect

Gravity Anomaly due to a Semi-infinite Horizontal Sheet Model

Gravity Anomalies due to Geometrical Effect

Gravity Anomaly due to a Faulted Horizontal Sheet Model

h1=750 mh2=1350 m

t=300 m

30° & 90°

=90°

=30°

= - 30°

Gravity Anomalies due to Geometrical Effect

Gravity Anomaly due to a Faulted Horizontal Bed Model

Geophysical Modeling

In general, geophysical data modeling is carried out in order to construct a model that involves several physical or multiple properties from a set of data in a single model ...

Objective:To obtain a model as a complete figure representing the real Earth.

Advantages:▪ To increase the resolution of the subsurface features and avoid use of constraints; and▪ To increase the accuracy of exploration target by using 3D Geographic Information

System (GIS) analysis.

Modeling

Forward ModelingFor a given model m and the data d that will be

predicted or calculated → d = F(m)F is an operator in the form of equations related to model and data ...

Inversion ModelingA technique that uses mathematics and statistic to obtain the information of subsurface physical properties (i.e. Magnetic susceptibiliy, density, electrical conductivity, velocity, etc) from the observed data → using the observed data d to predict a model m

→ m = F-1(d)

F

F

(Mira Geoscience, 2014)

Geophysical Modeling

Model Types

A single physical property →homogeneous half-space model

Object parameterization(i.e. density, susceptibility, length,depth, orientation)

Physical properties varies with depth

Plate within free-spacemodel in vacuum condition

Plate within a half-space model

Plate within a layered-Earth model

(Mira Geoscience, 2014)

Geophysical Modeling

Model Types

A fix model and perpendicular to the profile

Model with a finite strike length

A combination of 1Dmodels

Physical properties changes in three directions (x, y, z)

Boundaries of geologic units will help in locating & constructing 3D bodies shape ….

(Mira Geoscience, 2014)

Geophysical Modeling

Gravity Modeling

3D

Mira Geoscience Rock Property Database System – Free ...!http://rpds.mirageoscience.com/

2D

(3D) – presently it’s rarely done!

2.5D

x

z

x

z

y

Gravity Modeling

Forward Modeling

Inversion Modeling

2D, 2.5D & 3D Forward Modeling

Gravity Modeling

▪ Cady, JW, 1980. Calculation of gravity and magnetic anomalies of finite-length right polygonal prisms, Geophysics 45 (10), 1507-1512. (PDF avaiable) → 2.5D

▪ Talwani, M, Worzel, JL, and Landisman, M, 1959. Rapid Gravity Computations for Two-Dimensional Bodies with Application to the Mendocino Submarine Fracture Zone, Journal of Geophysical Research, 64:49-61.

▪ Talwani, M. and M. Ewing, 1960. Rapid computation of gravitational attraction of three-dimensional bodies of arbitrary shape, Geophysics, 25, 203-225.

▪ Farquharson, C.G, Mosher, C.R.W., 2009. Three-dimensional modelling of gravity data using finite differences, J. of Applied Geophysics, 68, 417-422.

▪ GMsys, Gravity/Magnetic Modeling Software User’s Guide, Version 480 4.9, Northwest Geophysical Associates, Corvallis Oregon, 2004.

▪ Zhang, J., Wang, C.-Y., Shi, Y., Cai, Y., Chi, W.-C., Dreger, D., Cheng, W.-B., Yuan, Y.-H., 2004. Three-dimensional crustal structure in 535 central Taiwan from gravity inversion with a parallel genetic algorithm, Geophysics, 69, 917-924.

Gravity Modeling

2D Modeling (Talwani, Worzel, and Landisman, 1959)

Z

D

B (xi, zi)

A

QP

X

R (x, z)

C (xi+1, zi+1)

F

E

∅𝜃

ai

Gravity Modeling

2D Modeling (Talwani, Worzel, and Landisman, 1959)

Z

D

B (xi, zi)

A

QP

X

R (x, z)

C (xi+1, zi+1)

F

E

∅𝜃

ai

∆𝑔𝑧 = 2𝐺ρ ∮ 𝑧 𝑑𝜃

∆𝑔𝑥 = 2𝐺ρ ∮ 𝑥 𝑑𝜃

Based on the method similar to Hubbert 1948), the vertical and horizontal components of gravitational attraction are given by:

Where G is the universal constant, 𝜌 is the volume density of the body, and ∮ is the line integral.As an example, we first compute BC that meets the x axis at Q at an angle i . Let PQ = ai, then

for any arbitrary point R on BC. Also

z = 𝑥 tan𝜃 (1)

z = 𝑥 − 𝑎𝑖 tan∅𝑖 (2)

Gravity Modeling

2D Modeling (Talwani, Worzel, and Landisman, 1959)

z =𝑎𝑖 tan 𝜃 tan ∅𝑖

tan ∅𝑖 − tan 𝜃

From (1) and (2),

or

න𝐵𝐶

𝑧 𝑑𝜃 = න𝐵

𝐶 𝑎𝑖 tan 𝜃 tan∅𝑖tan ∅𝑖 − tan𝜃

𝑑𝜃 = 𝑍𝑖

න𝐵𝐶

𝑥 𝑑𝜃 = න𝐵

𝐶 𝑎𝑖 tan∅𝑖tan ∅𝑖 − tan𝜃

𝑑𝜃 = 𝒳𝑖

The vertical (∆𝑔𝑧) and horizontal (∆𝑔𝑥) components of gravitational attraction due to the whole polygon,

∆𝑔𝑧 = 2𝐺ρ

𝑖=1

𝑛

𝑍𝑖 ∆𝑔𝒳 = 2𝐺ρ

𝑖=1

𝑛

𝒳𝑖

The summations being made over the n sides of the polygon. To solve the integrals in the expressions for 𝑍𝑖and 𝒳𝑖;

𝑍𝑖 = 𝑎𝑖 sin ∅𝑖 cos ∅𝑖 𝜃𝑖 − 𝜃𝑖+1 + tan∅𝑖 𝑙𝑜𝑔𝑒cos 𝜃𝑖 tan 𝜃𝑖 − tan∅𝑖

cos 𝜃𝑖+1 tan 𝜃𝑖+1 − tan∅𝑖

𝒳𝑖 = 𝑎𝑖 sin ∅𝑖 cos ∅𝑖 tan 𝜃𝑖 (𝜃𝑖+1 − 𝜃𝑖) + 𝑙𝑜𝑔𝑒cos 𝜃𝑖 tan 𝜃𝑖 − tan∅𝑖

cos 𝜃𝑖+1 tan 𝜃𝑖+1 − tan∅𝑖

Gravity Modeling

2D Modeling (Talwani, Worzel, and Landisman, 1959)

where

𝜃𝑖 = tan−1𝑧𝑖𝑥𝑖

∅𝑖 = tan−1𝑧𝑖+1 − 𝑧𝑖𝑥𝑖+1 − 𝑥𝑖

𝜃𝑖+1 = tan−1𝑧𝑖+1𝑥𝑖+1

𝑎𝑖 = 𝑥𝑖+1 + 𝑧𝑖+1𝑥𝑖+1 − 𝑥𝑖𝑧𝑖 − 𝑧𝑖+1

𝐶𝑎𝑠𝑒 𝐴 → 𝑖𝑓 𝑥𝑖 = 0

𝑍𝑖 = −𝑎𝑖 sin ∅𝑖 cos ∅𝑖 𝜃𝑖+1 −π

2+ tan∅𝑖𝑙𝑜𝑔𝑒 cos 𝜃𝑖+1 tan𝜃𝑖+1 − tan∅𝑖

𝒳𝑖 = 𝑎𝑖 sin ∅𝑖 cos∅𝑖 tan 𝜃𝑖 𝜃𝑖+1 −π

2− 𝑙𝑜𝑔𝑒 cos 𝜃𝑖+1 tan𝜃𝑖+1 − tan∅𝑖

𝐶𝑎𝑠𝑒 𝐵 → 𝑖𝑓 𝑥𝑖+1 = 0

𝑍𝑖 = 𝑎𝑖 sin ∅𝑖 cos ∅𝑖 𝜃𝑖 −π

2+ tan∅𝑖𝑙𝑜𝑔𝑒 cos 𝜃𝑖 tan 𝜃𝑖 − tan∅𝑖

𝒳𝑖 = −𝑎𝑖 sin ∅𝑖 cos∅𝑖 tan∅𝑖 𝜃𝑖 −π

2− 𝑙𝑜𝑔𝑒 cos 𝜃𝑖 tan 𝜃𝑖 − tan∅𝑖

Gravity Modeling

2D Modeling (Talwani, Worzel, and Landisman, 1959)

𝐶𝑎𝑠𝑒 𝐶 → 𝑖𝑓 𝑧𝑖 = 𝑧𝑖+1

𝑍𝑖 = 𝑧𝑖 𝜃𝑖+1 − 𝜃𝑖

𝒳𝑖 = 𝑧𝑖𝑙𝑜𝑔𝑒sin 𝜃𝑖+1sin 𝜃𝑖

𝐶𝑎𝑠𝑒 𝐷 → 𝑖𝑓 𝑥𝑖 = 𝑥𝑖+1

𝑍𝑖 = 𝑥𝑖𝑙𝑜𝑔𝑒cos 𝜃𝑖cos 𝜃𝑖+1

𝒳𝑖 = 𝑥𝑖 𝜃𝑖+1 − 𝜃𝑖

𝐶𝑎𝑠𝑒 𝐸 → 𝑖𝑓 𝜃𝑖 = 𝜃𝑖+1

𝑍𝑖 = 0

𝒳𝑖 = 0

𝐶𝑎𝑠𝑒 𝐹 → 𝑖𝑓 𝒳𝑖 = 𝑍𝑖 = 0

𝑍𝑖 = 0

𝒳𝑖 = 0

𝐶𝑎𝑠𝑒 𝐺 → 𝑖𝑓 𝒳𝑖+1 = 𝑍𝑖+1 = 0

𝑍𝑖 = 0

𝒳𝑖 = 0

Gravity Modeling

2D Modeling for Simple Bodies

Adopted from Hinze et al. (2013)

Gravity Modeling

2D Modeling for Simple Bodies: Half-Strike Length

Adopted from Hinze et al. (2013)

Gravity Modeling

Gravity Anomalies of 1D, 2D and 3D Bodies

Gravity Anomalies derived from depths of 1,000, 2,000, and 4,000 feets for (a) long horizontal line, (b) long, wide tabular, and (c) concentrated spherical sources

Adapted from Romberg (1958)

Gravity Modeling

2D Modeling for Polygonal Body

▪ Talwani, M, Worzel, JL, and Landisman, M, 1959. Rapid Gravity Computations for Two-Dimensional Bodies with Application to the Mendocino Submarine Fracture Zone, Journal of Geophysical Research, 64:49-61.

Gravity anomaly simulationAlan Levine (ASU Geology, 1987)

Gravity Modeling

▪ Direct Interpretation → The information of anomalous bodies at the subsurface are obtained directly from gravity anomaly profile, i.e. depth and density of anomalous bodies;

▪ Indirect Interpretation → The anomalous bodies are modeled to obtain theoretical gravity values. The model parameters can be changed trial and error to obtain the theoretical values that well-agree with observed gravity values. This method is then well-known as forward modeling;

▪ Gravity (and magnetic) modeling is always a non-unique problem, this leaves the actual extent of the ambiguity domain, i.e. the range of variability of the solutions to a potential field problem …. → the modeling needs additional geologic information, well data, etc. as a constraint …

Gravity Anomaly Interpretation

▪ Direct interpretation is conducted to the gravity anomaly of the Salt Dome;

▪ Drawing the gravity anomaly profile. From the profile, estimate Δ𝑔𝑚𝑎𝑥 and 𝑥1/2;

▪ Salt Dome is assumed to be a sphere model. The depth to the center of the sphere is 𝑧, radius of the sphere is 𝑅, and the depth from the top of Salt Dome hcan be estimated;

▪ By assuming the density of Salt Dome, so mass of the Salt Dome can be calculated …

Gravity Anomaly Interpretation

Direct Interpretation

Ambiguity in Gravity Interpretation

Thank you,See you for the next lecture ....