Monte Carlo Simulations of Gravimetric Terrain Corrections Gravimetric Terrain Corrections Using LIDAR Data J. A. Rod Blais Dept. of Geomatics Engineering Pacific Institute for the Mathematical Sciences University of Calgary, Calgary, AB www.ucalgary.ca/~blais
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Monte Carlo Simulations of Gravimetric Terrain CorrectionsGravimetric Terrain Corrections
Using LIDAR Data
J. A. Rod BlaisDept. of Geomatics Engineering
Pacific Institute for the Mathematical SciencesUniversity of Calgary, Calgary, ABy g y, g y,
www.ucalgary.ca/~blais
Outline
G i i i C i• Overview of Gravimetric Terrain Corrections
• Example of Current Application with Airborne Gravimetry
C t ti A h f G i t i T i C ti• Computation Approaches for Gravimetric Terrain Corrections
• Airborne LIDAR Dense Grids of Accurate Terrain Data
• Simulations for Gravimetric Terrain Corrections• Simulations for Gravimetric Terrain Corrections
• Accuracy of Simulated Terrain Corrections
• Concluding Remarks• Concluding Remarks
Gravimetric Terrain Correction
Newtonian Potential U(P) at some point P = (x, y, z):( )( ) G d ( )
| |
QU P Q
P QE
Vertical gradient assuming z ~ height:( )z( )( ) G d ( )
Q QU P Q
| |P QE
in which ρ(Q) denotes the density of the Earth (E) at location Q and G is Newtonian’s gravitational constant, i.e. G = 6.672x10-11 m3 s-2 kg-1
3
( ) ( )( ) G d ( )z | |
Q QU P QP QE
g g
Note: ρ(crust) ≈ 2.67 g cm-3 and 1 mgal = 10-5 m s-2 = 10-8 km s-2
Source: EOS, Vol.91, No.12, 23 March 2010
Source: EOS, Vol.91, No.12, 23 March 2010
T l f M l i id Q dTemplates for Multigrid Quadratures
Integral Approach
Direct IntegrationgL L H(x,y)
o o o 2 2 2 3/2L L 0o o o
zdzdydxg(x , y ,z ) G((x x ) (y y ) (z z ) )
orR 2 H(r, )
o o o 2 2 3/20 0 0
r h dh d drg(r , ,h ) G(( ) (h h ) )
2 2 3/20 0 0o o
R H(r)
2 2 3/20 0o o
((r r ) (h h ) )r h dh dr2 G
((r r ) (h h ) )
Cartesian Prism Approach
Direct IntegrationDirect Integration2
22
1
zyx
x y
zrg G x log(y r) ylog(x r) zarctanxy
or simplifying to a known cross-section s
11
1y z
h
2 2 3/2 2 20
zdz 1 1g G s G s(d z ) d d h
which is usually called the line mass formula.
Airborne LIDAR Light Detection and Ranging
Airborne laser, GPS & INS
DEM id d t ll tiDEM rapid data collection
Grid with sub-metre resolution
Height accuracy: 15-25 cm
Ideal for special projects
Airborne LIDAR System (author unknown)
Ideal for special projects(e.g., www.ambercore.com )
LIDAR Data Coverage ExampleLIDAR Data Coverage Example