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 METHODโช Gravity Anomalies due to Geometrical
Effectโช Geophysical Modelingโช Gravity Modeling
โ 2D Forward Modelingโ 3D Forward Modeling
โช Gravity Interpretation
โช Gravity acceleration at an observation point P due to a mass point m at distant r:
โช So the vertical component of gravity acceleration at an observation point P:
โช 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 โฆ
๐ฅ๐๐ =๐บ๐
๐2(1)
๐ฅ๐ =๐บ๐๐ง
๐ฅ2 + ๐ง2 3/2(2)
Gravity Anomalies due to Geometrical Effect
P
Gravity Anomaly due to a Mass Point
โช The value of g due to the sphere with mass m is similar to the mass point with same mass m :
โช If mass m is stated as contrast density ฮ๐, then g due to a sphere can be estimated:
โช Equation (3) can be manipulated to observe its relationship with ฮ๐๐๐๐ฅ:
โช Thus Eq. (5) can be used to determine the depth of anomalous source โฆ
๐ฅ๐ =๐บ๐๐ง
๐ฅ2 + ๐ง2 3/2 (2)
๐ฅ๐ =เต4 3๐๐
3ฮ๐๐บ๐ง
๐ฅ2 + ๐ง2 3/2(3)
ฮ๐ =เต4 3๐๐
3ฮ๐๐บ
๐ง2๐ง
1 + ๐ฅ2/๐ง2 3/2
ฮ๐ = ฮ๐๐๐๐ฅ
๐ง
1 + ๐ฅ2/๐ง2 3/2 (5)
Gravity Anomalies due to Geometrical Effect
Gravity Anomaly due to a Sphere
โช Anomalous responses of spheres vs depth can be estimated using Eq. (5);
โช 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
โช Maximum value of gravity anomaly for a sphere model is,
ฮ๐๐๐๐ฅ =เต4 3๐๐
3ฮ๐๐บ
๐ง2
โช We can estimate the depth of the center of the sphere using Eq. (5). When ฮ๐ = ฮ๐๐๐๐ฅ/2, so
๐ง =๐ฅ1/234 โ 1
๐ง = 1.3 ๐ฅ1/2
โช ๐ฅ1/2 is half of the maximum value of ฮ๐๐๐๐ฅ.
Gravity Anomalies due to Geometrical Effect
Gravity Anomaly due to a Sphere
โช Nilai anomali gayaberat maksimum untuk model tabung vertikal
๐ฅ๐๐๐๐ฅ = 2๐๐บฮ๐(๐ 1 โ ๐) ๐ฟ โถ ๐๐๐๐๐๐๐ก๐if
๐ฅ๐๐๐๐ฅ = 2๐๐บฮ๐๐ ๐ = 0if
๐ฅ๐๐๐๐ฅ = 2๐๐บฮ๐(๐ฟ + ๐ 1 โ ๐ 2)
๐ง = ๐ฅ1/2 3
if ๐ฟ โถ ๐๐๐๐๐ก๐
Gravity Anomalies due to Geometrical Effect
Gravity Anomaly due to a Vertical Cylinder Model
โช The maximum value of gravity anomaly for 2D prism model is,
๐ฅ๐๐๐๐ฅ = 2๐บฮ๐ ๐ ln๐
๐ฟ๐ฟ โซ ๐if
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
30ยฐ & 90ยฐ
=90ยฐ
=30ยฐ
= - 30ยฐ
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:โช To increase the resolution of the subsurface features and avoid use of constraints; andโช 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 โ 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 โ using the observed data d to predict a model m
โ m = F-1(d)
F
F
(Mira Geoscience, 2014)
Geophysical Modeling
Model Types
A single physical property โhomogeneous 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 โฆ.
(Mira Geoscience, 2014)
Geophysical Modeling
Gravity Modeling
3D
Mira Geoscience Rock Property Database System โ Free ...!http://rpds.mirageoscience.com/
2D
(3D) โ presently itโs rarely done!
2.5D
x
z
x
z
y
Gravity Modeling
Forward Modeling
Inversion Modeling
2D, 2.5D & 3D Forward Modeling
Gravity Modeling
โช Cady, JW, 1980. Calculation of gravity and magnetic anomalies of finite-length right polygonal prisms, Geophysics 45 (10), 1507-1512. (PDF avaiable) โ 2.5D
โช 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.
โช Talwani, M. and M. Ewing, 1960. Rapid computation of gravitational attraction of three-dimensional bodies of arbitrary shape, Geophysics, 25, 203-225.
โช Farquharson, C.G, Mosher, C.R.W., 2009. Three-dimensional modelling of gravity data using finite differences, J. of Applied Geophysics, 68, 417-422.
โช GMsys, Gravity/Magnetic Modeling Software Userโs Guide, Version 480 4.9, Northwest Geophysical Associates, Corvallis Oregon, 2004.
โช 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
โ ๐
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
โ ๐
ai
โ๐๐ง = 2๐บฯ โฎ ๐ง ๐๐
โ๐๐ฅ = 2๐บฯ โฎ ๐ฅ ๐๐
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, ๐ is the volume density of the body, and โฎ 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 = ๐ฅ tan๐ (1)
z = ๐ฅ โ ๐๐ tanโ ๐ (2)
Gravity Modeling
2D Modeling (Talwani, Worzel, and Landisman, 1959)
z =๐๐ tan ๐ tan โ ๐
tan โ ๐ โ tan ๐
From (1) and (2),
or
เถฑ๐ต๐ถ
๐ง ๐๐ = เถฑ๐ต
๐ถ ๐๐ tan ๐ tanโ ๐tan โ ๐ โ tan๐
๐๐ = ๐๐
เถฑ๐ต๐ถ
๐ฅ ๐๐ = เถฑ๐ต
๐ถ ๐๐ tanโ ๐tan โ ๐ โ tan๐
๐๐ = ๐ณ๐
The vertical (โ๐๐ง) and horizontal (โ๐๐ฅ) components of gravitational attraction due to the whole polygon,
โ๐๐ง = 2๐บฯ
๐=1
๐
๐๐ โ๐๐ณ = 2๐บฯ
๐=1
๐
๐ณ๐
The summations being made over the n sides of the polygon. To solve the integrals in the expressions for ๐๐and ๐ณ๐;
๐๐ = ๐๐ sin โ ๐ cos โ ๐ ๐๐ โ ๐๐+1 + tanโ ๐ ๐๐๐๐cos ๐๐ tan ๐๐ โ tanโ ๐
cos ๐๐+1 tan ๐๐+1 โ tanโ ๐
๐ณ๐ = ๐๐ sin โ ๐ cos โ ๐ tan ๐๐ (๐๐+1 โ ๐๐) + ๐๐๐๐cos ๐๐ tan ๐๐ โ tanโ ๐
cos ๐๐+1 tan ๐๐+1 โ tanโ ๐
Gravity Modeling
2D Modeling (Talwani, Worzel, and Landisman, 1959)
where
๐๐ = tanโ1๐ง๐๐ฅ๐
โ ๐ = tanโ1๐ง๐+1 โ ๐ง๐๐ฅ๐+1 โ ๐ฅ๐
๐๐+1 = tanโ1๐ง๐+1๐ฅ๐+1
๐๐ = ๐ฅ๐+1 + ๐ง๐+1๐ฅ๐+1 โ ๐ฅ๐๐ง๐ โ ๐ง๐+1
๐ถ๐๐ ๐ ๐ด โ ๐๐ ๐ฅ๐ = 0
๐๐ = โ๐๐ sin โ ๐ cos โ ๐ ๐๐+1 โฯ
2+ tanโ ๐๐๐๐๐ cos ๐๐+1 tan๐๐+1 โ tanโ ๐
๐ณ๐ = ๐๐ sin โ ๐ cosโ ๐ tan ๐๐ ๐๐+1 โฯ
2โ ๐๐๐๐ cos ๐๐+1 tan๐๐+1 โ tanโ ๐
๐ถ๐๐ ๐ ๐ต โ ๐๐ ๐ฅ๐+1 = 0
๐๐ = ๐๐ sin โ ๐ cos โ ๐ ๐๐ โฯ
2+ tanโ ๐๐๐๐๐ cos ๐๐ tan ๐๐ โ tanโ ๐
๐ณ๐ = โ๐๐ sin โ ๐ cosโ ๐ tanโ ๐ ๐๐ โฯ
2โ ๐๐๐๐ cos ๐๐ tan ๐๐ โ tanโ ๐
Gravity Modeling
2D Modeling (Talwani, Worzel, and Landisman, 1959)
๐ถ๐๐ ๐ ๐ถ โ ๐๐ ๐ง๐ = ๐ง๐+1
๐๐ = ๐ง๐ ๐๐+1 โ ๐๐
๐ณ๐ = ๐ง๐๐๐๐๐sin ๐๐+1sin ๐๐
๐ถ๐๐ ๐ ๐ท โ ๐๐ ๐ฅ๐ = ๐ฅ๐+1
๐๐ = ๐ฅ๐๐๐๐๐cos ๐๐cos ๐๐+1
๐ณ๐ = ๐ฅ๐ ๐๐+1 โ ๐๐
๐ถ๐๐ ๐ ๐ธ โ ๐๐ ๐๐ = ๐๐+1
๐๐ = 0
๐ณ๐ = 0
๐ถ๐๐ ๐ ๐น โ ๐๐ ๐ณ๐ = ๐๐ = 0
๐๐ = 0
๐ณ๐ = 0
๐ถ๐๐ ๐ ๐บ โ ๐๐ ๐ณ๐+1 = ๐๐+1 = 0
๐๐ = 0
๐ณ๐ = 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
โช 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
โช Direct Interpretation โ The information of anomalous bodies at the subsurface are obtained directly from gravity anomaly profile, i.e. depth and density of anomalous bodies;
โช Indirect Interpretation โ 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;
โช 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 โฆ. โ the modeling needs additional geologic information, well data, etc. as a constraint โฆ
Gravity Anomaly Interpretation
โช Direct interpretation is conducted to the gravity anomaly of the Salt Dome;
โช Drawing the gravity anomaly profile. From the profile, estimate ฮ๐๐๐๐ฅ and ๐ฅ1/2;
โช Salt Dome is assumed to be a sphere model. The depth to the center of the sphere is ๐ง, radius of the sphere is ๐ , and the depth from the top of Salt Dome hcan be estimated;
โช By assuming the density of Salt Dome, so mass of the Salt Dome can be calculated โฆ
Gravity Anomaly Interpretation
Direct Interpretation
Ambiguity in Gravity Interpretation
Thank you,See you for the next lecture ....