www.bgu.ac.ilgeolclassesgeophysicsFrontgeophy by yurtgc548

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									       Gravity anomaly due to a simple-shape buried body


The general equation for gravity anomaly is:
                                               1
                             gZ            2
                                                   cosdV ,
                                           V   r
where:
 is the gravitational constant
 is the density contrast
                    
r is the distance to the observation point
 is the angle from vertical
V is the volume
Example: a sphere
                            4 a 3      1              z
                      gZ                                                 .
                                3     x 2  z 2    x   2
                                                              z   2
                                                                       

               
         Gravity anomaly due to a simple-shape buried body


A horizontal wire of infinite length
    Starting with:
           dm
    dgZ  2 cosb sin  .
            r                                                         b
    Substituting:
    dm  dl,
    cosb  R /r,
    sin   Z /R
    and :                                                                  is mass per length
                                                                           R is the distance to the wire
    r 2  R 2  l 2,                                                       r is the distance to an element dl
    we get:
                                                                        
                                dl                           l
    g Z  Z                              Z                         |
                     R   2
                                l    
                                     2 3/2
                                                    R R  l
                                                     2   2
                                                                  
                                                                 2 1/ 2
                                                                          


    2Z         Z
          2 2     .
     R2       x  Z2
        Gravity anomaly due to a simple-shape buried body


An infinitely long horizontal cylinder
To obtain an expression for a horizontal cylinder of a radius a and
density , we replace  with a2 to get:
                                      Z
                       gZ  2a  2
                                   2
                                          .
                                   x Z 2


It is interesting to compare the solution for cylinder with that of a
sphere.
           
      cylinder                             sphere
           Gravity anomaly due to a simple-shape buried body


A horizontal thin sheet of finite width

                                                 we
Starting from the expression for an infinite wire, write:
         Z
dgZ  2    dx,
         r2
where  is mass per area.
Replacing sin with Z/r:
          sin 
dgZ  2        dx 
             r
                          2
          x2
             sin 
gZ  2         dx 2  d = 2  .
          x1   r           1



Remarkably, the gravitational effect of a thin sheet is independent
of its depth.
        Gravity anomaly due to a simple-shape buried body


A thick horizontal sheet of finite width
                                             we
 Starting from the expression for a thin sheet, write:
 dgZ  2dh ,
 with  being in units of mass per volume.
                                 :
 Integration with respect to depth
            h2
 gZ  2  dh  2h .                                                 Station of two
                                                              station   Dimensional structure
            h1
                                           surface



Actually, you have seen this
expression before

                                                     z
                                                         
        Gravity anomaly due to a simple-shape buried body


A thick horizontal sheet of infinite width


To compute the gravitational effect of an infinite plate we need to
replace  with :
                           gZ  2h .



              
                          Geoid anomaly


Geoid is the observed equipotential surface that defines the sea
level.

Reference geoid is a
mathematical formula
describing a theoretical
equipotential surface of a
rotating (i.e., centrifugal
effect is accounted for)
symmetric spheroidal earth
model having realistic radial
density distribution.
                           Geoid anomaly




The international gravity formula gives the gravitational
acceleration, g, on the reference geoid:

                 g( )  gE   sin 2   b sin 4 
                            1
                 where:
                 gE is the g at the equator
                  is the latitude
                   5.278895103
                 b  2.3462 105
                         Geoid anomaly



The geoid height anomaly is the difference in elevation between
the measured geoid and the reference geoid.




Note that the geoid height anomaly is measured in meters.
                         Geoid anomaly

Map of geoid height anomaly:




                     Figure from: www.colorado.edu/geography


Note that the differences between observed geoid and reference
geoid are as large as 100 meters.

Question: what gives rise to geoid anomaly?
                         Geoid anomaly



Differences between geoid and reference geoid are due to:

• Topography

• Density anomalies at depth




                                                 Figure from Fowler
                              Geoid anomaly
What is the effect of mantle convection on the geoid anomaly?
                                           downwelling   upwelling


                                                                     Flow
Two competing effects:
                                                                     Temp.
1. Upwelling brings hotter and
   less dense material, the
   effect of which is to reduce
   gravity.

2. Upwelling causes
   topographic bulge, the effect
   of which is to increase
   gravity.


       Figure from McKenzie et al., 1980
                         Geoid anomaly


SEASAT provides water topography




Note that the largest features are associated with the trenches.
This is because 10km deep and filled with water rather than rock.
                 Geoid anomaly and corrections


Geoid anomaly contains information regarding the 3-D mass
distribution. But first, a few corrections should be applied:
• Free-air
• Bouguer
• Terrain
                  Geoid anomaly and corrections


Free-air correction, gFA:




This correction accounts for the fact that the point of
measurement is at elevation H, rather than at the sea level on the
reference spheroid.
                  Geoid anomaly and corrections

Since:                                  2
                             R()               2h 
             g(,h)  g(,0)           g( ,0)
                                                    1       ,
                            R()  h            R( ) 
where:
•  is the latitude
• h is the topographic height
     
• g() is gravity at sea level
• R() is the radius of the reference spheroid at 

The free-air correction is thus:
                                              2h
                 gFA  g( ,0)  g( ,h)  g( ,0).
                                             R( )
This correction amounts to 3.1x10-6 ms-2 per meter elevation.

       
Question: should this correction be added or subtracted?
                 Geoid anomaly and corrections




The free-air anomaly is the geoid anomaly, with the free-air
correction applied:


     gFA  reference gravity - measured gravity  gFA .
                  Geoid anomaly and corrections


Bouguer correction, gB:




This correction accounts for the gravitational attraction of the
rocks between the point of measurement and the sea level.
                  Geoid anomaly and corrections



The Bouguer correction is:

                           gB  2h ,
where:
 is the universal gravitational constant
 is the rock density
               
h is the topographic height

For rock density of 2.7x103 kgm-3, this correction amounts to
1.1x10-6 ms-2 per meter elevation.


Question: should this correction be added or subtracted?
                Geoid anomaly and corrections




The Bouguer anomaly is the geoid anomaly, with the free-air and
Bouguer corrections applied:


  gB  reference gravity - measured gravity gFA  gB .
                  Geoid anomaly and corrections


Terrain correction, gT:




This correction accounts for the deviation of the surface from an
infinite horizontal plane. The terrain correction is small, and except
for area of mountainous terrain, can often be ignored.
                   Geoid anomaly and corrections




The Bouguer anomaly including terrain correction is:
 gB  reference gravity - measured gravity gFA  gB  gT .


Bouguer anomaly for offshore gravity survey:
• Replace water with rock
• Apply terrain correction for seabed topography

After correcting for these effects, the ''corrected'' signal contains
information regarding the 3-D distribution of mass in the earth
interior.

								
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