GP-3105 GRAVITY & MAGNETIC · 2020. 7. 20. · Mass = Volume x Density ... melakukan perhitungan...

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 #02

GRAVITY METHOD▪ Variation of g Value

✓ Density & Rock Types✓ Noise sources

▪ What is Gravity Anomaly?▪ Gravity Data Processing (Reduction &

Corrections)✓ Tide & Drift Corrections✓ Latitude Correction (Normal Gravity)✓ Free-Air Correction✓ Bouguer Slab Correction✓ Terrain Correction

VARIATION OF g VALUE

How is the variation of g value due to the contrast density of rock types?

Density & Rock Types

In gravity data acquisition, there are two noise sources, i.e.:

▪ Temporal variation → changes of g value due to time variation;

▪ Spatial variation → changes of g value due to the spatial variation (latitude & elevation), from one point to other point, and that is not related to the geologic condition ….

Noise Sources

What is Gravity Anomaly?

Observed Gravity:

g0 = G (m0/r02) → effect of m0 (assumed as point mass)

g2 = G (m2/r22) → effect of m2 (as anomaly)

gobs = g0 + g2

Mass = Volume x Density

= V x


g


→ Gravity anomaly depends on the density contrast …..

= 2-0

g = V2(2-0) / r22 = V2 / r2

2

→ Gravity anomaly is direct proportional with density contrast …..

g

(+)

g

(+)

Distance


= 2-0

g = V2(2-0) / r22 = V2 / r2

2

(-)

g

g

(-)

Distance


→ Gravity anomaly is direct proportional with density contrast …..

= 2-0

g = V2(2-0) / r22 = V2 / r2

2


→ For the given model, could we observe the gravity anomaly on the ground? …..

= 2-0

g = V2(2-0) / r22 = V2 / r2

2g?

g

?

Distance

2

0


In Conclusions, what do Bouguer gravity anomalies mean?

▪ the difference between observed and expected gravity!

→ “expected” for average Earth (not rotating, perfect sphere, no topography, no lateral density variations) …..

▪ in calculating the Bouguer anomaly, the data have been corrected for the effects of latitude and elevation;

▪ any remaining gravity variations due to local density variations → due to subsurface geology,

negative g value = anomalously low densitypositive g value = anomalously high density

GRAVITY DATA PROCESSING(REDUCTION & CORRECTIONS)

Reduction & Corrections

Gravity Data Reduction → to transform raw gravity data into gravity anomaly through the process as follows:

▪ Temporal Correction, as time domain: ✓ Tidal correction, and✓ Instrument drift.

▪ Spatial Correction, as space domain:✓ Latitude correction, and✓ Elevation correction.


Tidal Correction

▪ To eliminate gravity effects of the sun and moon which are time function due to relative motion among earth, moon and sun → ±0.3 mGals;

▪ The size of the ocean tides is measured in terms of meters;

▪ The size of the solid earth tide is measured in terms of centimeters → as large as 0.2 mGals;

▪ In case of gravity variations in Tulsa, Oklahoma; the tides, the cyclic variation in gravity with a period of oscillation of about 12 hours, have the amplitude of the tidal variation about 0.15 mGals ….


Tidal Correction▪ To calculate vertical attractions of the tidal accelerations produced by the Moon and the Sun, at any

time and any place on the Earth’s surface, using the Longman’s formula (Longman, 1959):

Longman I. M., 1959: Formulas for Computing the Tidal Accelerations Due to the Moon and the Sun. Journal of Geophysical Research, 64, 2351–2355.

Where G is the universal gravity constant = 6.6732 x 10-11 N-m2/kg2 = 6.67408 x 10-11 m3·kg-1·s-2

m, s = the zenith angles of the Moon and Sun, respectively

Rm, Rs = the distance between the center of the Earth and Moon, the Earth and Sun, respectively

m, S = mass of the Moon and Sun, respectively

r = earth ellipsoidal, based on e = 1/298.256 (natural number)

= geographic latitude of observation point

∆g𝑟 𝑚 = −3 𝐺𝑚𝑟

𝑅𝑚3 3 𝑐𝑜𝑠2𝜓𝑚 − 1 +

3

2

𝑟

𝑅𝑚5 𝑐𝑜𝑠3𝜓𝑚 − 3 cos𝜓𝑚

∆g𝑟 𝑠 =−3 𝐺𝑆𝑟

𝑅𝑠3 3 𝑐𝑜𝑠2 𝜓𝑠 − 1 ; 𝑟 = 6,378,139.00 1 − 0.0033528 𝑠𝑖𝑛2 + 0.0000070 𝑠𝑖𝑛22

▪ To calculate g due to tides using Schureman’s formula (Schureman, 1941):

High tides are occurred twice in a day, or one-time in 14 days due to the Moon, or in 6 months due to the Sun.


Tidal Correction

r : Distance from observation point to the center of the EarthMB : mass of the MoonDB : distance between the Moon and the Earth : zenith angle from the MoonMM : mass of the SunDM : distance between the Sun and the Earth : zenith angle from the Sun

Zenith angle is a parameter that depend on the latitude position and the time of observation

▪ Observed gravity changes as a function of time at a given location;

▪ To eliminate the effect of spring fatigue of the instrument….


Instrument Drift Correction

)tt(tt

ggDrift n

n

n1

1

1 −−

−=

g1, ….gn : gravity value of 1st point and nth point, respectively

t1, ….tn : measurement time of 1st

point and nth point, respectively

gobs = ginst-reading ± gtide ± gdrift


▪ Gravimeters measure vertical gravity (g) relative to the Geoid (dashed line), which is the equipotential surface represented by mean sea level in the oceans;

▪ Normal gravity gn is defined perpendicular to the oblate ellipsoid of spheroid of revolution (full line) that is mathematical approximation of the Geoid;

▪ Geocentric latitude, which is also called co-latitude, is referenced to the center of the Earth, whereas astronomic latitude is measured relative to the true local vertical defined by the Geoid;

▪ The deflection of vertical is the angular difference between the verticals observed on the Geoid and defined by the ellipsoid, whereas height differences (N) between Geoid and ellipsoid are geoidal undulations …

Geoid, Ellipsoid and Topography

Adapted from Sheriff (2002)No scale


Geoid, Ellipsoid and Topography

No scale

Bentuk geoid yang tidak beraturan tidak memungkinkan kita untukmelakukan perhitungan matematis. Karena itu, sebagai representasimatematis dari bentuk fisik Bumi, digunakan elipsoid …

Geoid → model Bumi yang mendekati sesungguhnya, didefinisikansebagai bidang ekuipotensial Bumi yang dianggap berhimpit denganpermukaan air laut rata-rata (mean-sea level, MSL). Model Geoid global, i.e.: EGM96, EGM2008

Ellipsoid→ suatu pendekatan model Bumi (berbentuk elips ), dimanaparameternya ditentukan dari setengah sumbu panjang (a) , setengahsumbu pendek (b) dan nilai penggepengan atau flattening ( 1/f )

Contoh: Bessel 1841, GRS67, WGS72 , WGS84.

Undulasi Geoid N → jarak atau beda tinggi antara Geoid danellipsoid … Ini tidak sama di semua tempat, disebabkan adanyaketidakseragaman sebaran rapat-massa (density) Bumi

Geoid, Ellipsoid & Topography (Earth’s Surface)

▪ Ellipsoid earth gravity reference has to be applied to produce an earth gravity value at mean sea level as a function of the latitude of observation point;

▪ This reference implies an homogenous mass distribution of the earth ellipsoid model;

▪ The ellipsoid model:

✓ IGRF 1967 : :

✓ IUGG 1979 :


Latitude Correction

( ) 200000590005304018978031 22 sin.sin..)(g ++=

( ) 42 00002346200052788950185978031 sin.sin..)(g ++=

Where

g(ɸ) : theoretical (normal) gravity value (in mGal) as function of ɸ

ɸ : geographic latitude of observation point

▪ g-normal based on WGS84 (World Geodetic System) using newly constant values as described by NIMA (2000):

▪ g-normal using the International Terrestrial Reference Frame, as recommended by the North American Gravity Data Committee (NAGDC, 2005), with the 1980 Geodetic Reference System (GRS80; Moritz, 1980):


Latitude Correction

Where

𝑔𝑛 : theoretical (normal) gravity value (in mGal) as function of ɸ

ɸ : geographic latitude of observation point

𝑔𝑛 = 978032.533591 + 0.00193185265241 sin2 ∅

1 − 0.00669437999014 sin2 ∅𝑚𝐺𝑎𝑙

𝑔𝑛 = 978032.677151 + 0.0019131851353 sin2 ∅

1 − 0.0066943800229 sin2 ∅𝑚𝐺𝑎𝑙


Elevation CorrectionsAfter eliminating the effect of the latitude of the observation point, the gravity variations are affectedby topographic/elevation (→ free-air correction) effect and local geologic effect (→ Bouguercorrection).

▪ Free-Air Correction → variation of gravity g due to the distance between observation point at topographic surface to the center of the Earth (r):

or it can be calculated by: 𝑔 =𝐺𝑀𝐸

𝑟2→ distance r >>, g <<

where 𝐺 = 6.67408 × 10−11 m3·kg-1·s-2 and 𝑀𝐸 = (5.9722±0.0006)×1024 kg

∆𝑔 = − 0.3085672 ∆ℎ = −0.3086∆ℎ → Free-Air Correction, gFA = ∆𝑔 for the latitude ± 45°

∆𝑔 in mGal and ∆ℎ in meter ....... Minus means increasing of h will decrease of g value ….

𝑔𝐹𝐴 = −(0.3087691 − 0.0004398 𝑠𝑖𝑛2) ∆ℎ mGal


A gravity value is observed from A to B, so

The elevation increases 100 m, then the gravity value decreases,

∆𝑔 = 0.3086 x 100 = 30.86 mGal .......

This number is not a small change...!

▪ To compare the gravity values at A, B, and C, we have to apply Free-Air Correction at those observation points, and

▪ We have to define a reference level that is generally represented as mean sea level …..

▪ Free-Air Correction


▪ Free-Air Correction 𝐶𝐹𝐴= 0.3086 ∆ℎ

▪ ∆ℎ is height difference of observation point to the reference level, so:

✓ If the observation point is situated above the reference level (B), CFA is added to the observed gravity value;

✓ If the observation point beneath the reference level (C), CFA is subtracted to the observed gravity value .........

▪ The gravity value as due to the free-air correction is called Free-Air Gravity Anomaly (𝑔𝐹𝐴 )→𝑔𝐹𝐴 is gravity variation if we observed at any points at the same distant from the center of the Earth ...

▪ Free-Air Correction

gFA = gobs – (gn – CFA) = gobs – gn + CFA


▪ Bouguer Correction▪ Bouguer correction is to estimate the earth gravity

at elevation h above sea level with earth mass of density (g/cm3) fill up the space of thickness h;

▪ The gravity value observed at B is determined by the height of observation point and all masses with density beneath the observation points;

▪ The elevation of observation points are corrected by CFA;

▪ But the gravity value at B is still higher than at A due to the existence of mass of the hill beneath site B;

▪ Approaching by gravity effect due to the infinite layer/slab with homogeneous density and thickness;

▪ Then the gravity value of an infinite slab is defined as 𝑔 = 2G∆ℎ, where ∆ℎ is thickness and is density (assumed to be constant values) ....

infinite layer/slab

h


▪ Bouguer correction is given by,

𝐶𝐵 = 2G∆ℎ = 0.04193 ρ∆ℎ mGal

▪ By assuming that the average density of the Earth = 2.67 g/cm3, so

𝐶𝐵 = 0.1119∆ℎ mGal

▪ At SEA, Bouguer is negative because the water acts like a mass deficit:

▪ The density of rocks can be determined by: rock samples, well data, seismic velocity data and Nettleton method ....

▪ Bouguer Correction

infinite layer/slab

h

= 1.03 − 2.67 g/cm3

𝐶𝐵. 𝑠𝑒𝑎 = − 0.0688 ℎ𝑤 mGal


▪ Bouguer correction at B,

mGal

▪ Site B is at above of the reference level →subtracted the value of CB (→ to eliminate gravity effect of the hill);

▪ Site C is below the reference level → added the value of CB (→mass of the hill is vanish) ....

▪ After this Bouguer correction, the gravity value is then so-called Bouguer Gravity Anomaly 𝑔𝐵 ....

infinite layer/slab

▪ Bouguer Correction

hCB= 0.1119 h = 0.1119 x 100 = 11.19


▪ Select for a reference level,

𝐶𝐹𝐴 = 0.3086 ∆ℎ𝐶𝐵 = 0.1119∆ℎ

Elevation of Observe Point

CFA CB

Above reference level ADD SUBTRACT

Beneath reference level SUBTRACT ADD

Simple Bouguer Gravity Anomaly, gSBA = gobs - (gn - CFA + CB)

infinite layer/slab

h

PLEASE NOTE that this Simple Bouguer Anomaly is evaluated based on simple slab model, the effect of mass excess from the hill and mass deficit from the valley have not considered yet ….

✓ Nettleton (or graphical) method;

✓ Bouguer density is determined by creating topographic profile and Bouguer anomalysimultaneously for various density;

✓ Corrected g value has no correlation to the topographic profile;

✓ From this figure (right), it can be determined that the appropriate Bouguer density that has no correlation to topographic profile is 𝜌=2.3 g/cc ….


▪ Bouguer Correction: Nettleton’s Density Profile

• Metode Nettleton (Analitik): Merujuk pada dasar metode Nettleton secara grafik.

• Mencari nilai koefisien korelasi antara anomali Bouger sederhana dan profil topografi denganvariasi nilai 𝜌

• Penentuan rapat-massa Bouguer (Bouguer density) adalah yang memberikan nilai 𝑘 = 0


▪ Bouguer Correction: Bouguer Density Determination


▪ Terrain Correction

If our observation site is situated in or close to rugged terrain area, there is topographic effect to the gravity values at the observation sites;

At Sites A and C: the existence of a hill with 100 m height will affect the pendulum of gravimeter to move upward(see Figure) → this will cause the gravity value becomes lower.

At Site B: the gravity value becomes slightly lower due to missing mass at surrounding valleys.

For those cases, we have to run Terrain Correction in order to eliminate the topographic effect near the observation sites .....


▪ Terrain Correction

✓ To compensate the topography: the existence of mass excess of the hill (M1) and missing mass at valley (M2);

✓ M1 will decrease gravity value at A (negative vertical attraction), Bouguer correction considers that there is M2, so mass attraction equivalent to M2 must be extracted from values at A;

✓ So terrain correction CT will give negative gravity value, or the other word by adding the value of Simple Bouguer Anomaly gSBA;

✓ CT value can be obtained from a table of CT or calculation …


▪ Terrain Correction: Hammer Chart Correction


▪ Terrain Correction: Hammer (Cylindrical Ring/Segment) Correction

𝐶𝑇−𝑠𝑒𝑔 =2πρ𝐺

𝑁𝑟0 − 𝑟𝑖 + 𝑟𝑖

2 + 𝑧21/2

− 𝑟𝑜2 + 𝑧2 1/2

N : number of segments/sectors in the ring;

z : the different in elevation between the gravity station and mean elevation of the segment/sector;

𝜌 : the Bouguer correction density (g/cm3)Segment/sector


▪ Terrain Correction: Table


▪ Complete Bouguer Anomaly

✓ Complete Bouguer Anomaly is then determined by:

gCBA = gobs - (gn - CFA + CB - CT)= gobs - gn + CFA - CB + CT)

Gravity Anomaly Map, an example

Isostasy & Isostatic Gravity Anomaly

▪ Isostatic anomalies represent one of the most useful geological reduction of the gravity field;

▪ With the isostatic correction it is possible to remove a significant part of the effect of deep density heterogeneity, which dominates in the Bouguer gravity anomalies;

▪ This correction is based on the fact that a major part of the near-surface load is compensated by variations of the lithosphere boundaries (chiefly the Moho and Lithosphere-Astenosphere Boundary) and by density variations within the crust and upper mantle;

▪ Isostatic (residual) gravity anomaly maps are produced by subtracting long-wavelength anomalies produced by masses deep within the crust or mantle from the Bouguer anomaly map. The long-wavelength anomalies are assumed to result from isostatic compensation of topographic loads …..


▪ 18th and 19th century surveys set out to measure the shape of the Earth;

▪ They used plumb bobs and expected them to be attracted toward adjacent mountain chains, eg. the Andes and Himalayas;

▪ But the plumb bob was not attracted as much as expected…..

They calculated that the observed deflection could be explained if the excess mountain mass was matched by an equal mass deficiency beneath

→ The mountains were in isostatic equilibriumThe Earth’s lithosphere is “floating” on the denser asthenosphere

Excess Mass of Mountains

Excess Mass of Mountains

Mass Deficiency Beneath Mountains

Angle of deflection

Plumb bobs

Expected deflectionsActual deflections


▪ Airy’s Hypothesis:

To account for the mass deficiency beneath mountains

▪ Two densities, that of the rigid upper layer𝜌𝑢, and that of the substratum 𝜌𝑠

▪ Mountains therefore have deep roots …..

𝜌𝑠

𝜌𝑢

The compensation depth is the depth below which all pressures are hydrostatic


▪ Airy’s Hypothesis:

𝑟1 =𝜌𝑢

𝜌𝑠 − 𝜌𝑢ℎ1 𝑟3 =

𝜌𝑢 − 𝜌𝑤𝜌𝑠 − 𝜌𝑢

𝑑

𝜌𝑢 : crustal density

𝜌𝑠 : mantle density

𝜌𝑤 : water densityt : crustal thicknessh1, h2 : elevationsd : water depth

𝜌𝑠

𝜌𝑢

Equating the masses in vertical columns above the compensation depth:

The compensation depth …

𝑡𝜌𝑢+𝑟1𝜌𝑠 = ℎ1 + 𝑡 + 𝑟1 𝜌𝑢 t𝜌𝑢+𝑟1𝜌𝑠 = 𝑑𝜌𝑤 + 𝑡 − 𝑑 − 𝑟3 𝜌𝑢 + 𝑟1 + 𝑟3 𝜌𝑠

31 12

A mountain height ℎ1 is underlain by a root of thickness:

Ocean basin with depth d is underlain by an anti-root of thickness:


▪ Pratt’s Hypothesis:

To account for the mass deficiency beneath mountains

▪ The depth of the base of the upper layer is constant;

▪ Mountains therefore have low density roots …..

𝜌𝑢

The compensation depth is the depth below which all pressures are hydrostatic

𝜌𝑑𝜌1 𝜌2𝜌𝑢

𝜌1 = 𝜌𝑢𝐷

ℎ1 + 𝐷𝜌𝑑 =

𝜌𝑢𝐷 − 𝜌𝑤𝑑

𝐷 − 𝑑

Equating the masses in vertical columns above the compensation depth:

𝐷𝜌𝑢 = ℎ1 + 𝐷 𝜌1 𝐷𝜌𝑢 = 𝑑𝜌𝑤 + 𝜌𝑑 𝐷 − 𝑑31 12

A mountain height ℎ1 is underlain by a low density material, density 𝜌1 :

Ocean basin with depth d is underlain by an anti-root of thickness:


▪ Vening Meinesz ‘s Hypothesis:In this type of isostasy, short-wavelength topography is supported by the elastic strength of the crustal rocks. The load is instead distributed by the bent plate over a broad area. This distributed load is compensated.

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

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