THE UNB TECHNIQUE FOR
PRECISE GEOID DETERMINATION
HOW TO COMPUTE GEOID IN SEVEN (NOT SO) EASY STEPS
P. Vaníček & J. Janák
Geodesy and Geomatics Engineering
University of New Brunswick
Fredericton, New Brunswick
Presented at CGU annual meeting, Banff, May 26, 2000
.
1.
Formulation of the appropriate Boundary Value Problem (BVP)
of the third (Robin's?) kind.
*Region of Interest: Space outside the geoid;
*Boundary: The geoid (unknown);
*Unknown Function: Disturbing potential T(≅W-U); has the following
properties:
r
r
∇2 Τ(r ) = − 4πGρ (r ) (non-homogeneity is a problem),
r
lim r → ∞ T (r ) = O( r −3 ) (Hörmander condition);
*Boundary Values:
r
Τ( r )
2
r
− T(r g − Z(Ω ),Ω) = F ( T (r g)) (cannot be obtained form surface
r =rg
r
rg
observations).
2.
Transformation of BVP into a harmonic space.
Potential W has to be transformed to another potential W h , harmonic
everywhere above the geoid for which the boundary value equation can be
linked to observations in the harmonic space:
r
r
T (r ) → T h (r );
r
∇2 T (r ) = 0 (homogeneous equation);
geoid →co-geoid.
Many choices of harmonic spaces exist.
3.
Selection of an appropriate harmonic space
Helmert's space:
r
r
r
r
r
r
T h (r ) = T(r ) − V t (r ) + V ct (r ) − V a(r ) + V ca( r ) ,
where t stands for topography, a for atmosphere and c for condensation.
Choice of condensation scheme: Helmert’s second condensation. Why
Helmert’s scheme?
We use the scheme that preserves earth mass in the transformation to
Helmert's space (=> spherical model must be used).
Hörmander's condition is violated.
4.
Formulation of boundary values on the geoid (really Helmert's cogeoid)
4.1. Transformation of observations into Helmert's space
Observed gravity g( H ° ,Ω) = g(rt ,Ω) transformed to:
g h (rt ,Ω ) = g(rt ,Ω) − DTE(rt Ω) − DAE(rt ,Ω) ,
where
r
DTE(r ) =
r
DAE( r ) =
r
V ( r) − V
(
H
t
(V
H
a
ct
r
(r ))
r
r
( r ) − V ca( r ))
(these are complicated integrals for spherical models). See Figure 1.
4.2. Determination of Helmert's anomalies
(Helmert's) gravity g h (rt ,Ω ) on the earth surface is converted to
(Helmert's) anomalies:
∆gh (rt ,Ω) = g h (rt ,Ω) −
(r
o
+ H o ( Ω),Ω ) +
2 o
H ( Ω)∆gB (Ω) − SITE(rt ,Ω) − SIAE( rt ,Ω)
R
T h (r,Ω)
1
( r,Ω)
=−
+
T h (rt − Z(Ω ),Ω ).
H
r
,Ω
n
r= rt
(t )
r= rt
−
4.3. Use of simple Bouguer's anomaly
Instead of using observed gravity g( rt ,Ω) we use simple Bouguer anomaly
∆g B(Ω) :
g( rt ,Ω) −
(r + H
o
o
(Ω ),Ω) = ∆gFA (Ω) = ∆g B(Ω ) − 2
o
H o (Ω ).
Figure 1 Real and Helmert spaces.
4.4
Averaging gravity anomalies
For easier numerical integration (later on) we convert point values
∆gh (rt ,Ω) to mean values:
1
∆gh (rt ,Ω) = ∫ A ∆gh (rt ,Ω )dA.
A
This is done for 5' by 5' cells. The size of cells is dictated by available
gravity data density.
4.4.
Evaluation of 'fundamental gravimetric equation'
Τ h (r,Ω)
h
∆g (rt ,Ω) = −
− Τh ( rt − Z( Ω) ,Ω) + n (rt ,Ω)
r
r =rt
−
g
(rt ,Ω) = F[Τh (rt ,Ω)] + EC (rt ,Ω ).
This equation links the observed values g( rt ,Ω) to boundary values
[
]
F T h (rg ,Ω) , it is valid for all r≥rg!
4.5.
Downward continuation of F( T h) to the geoid
Τh
Since Τ h is harmonic above geoid, so is r
and∇2 rF[Τ h (r,Ω)] = 0 ,
r
∀r≥rg. Direchlet's BVP can be formulated and solved ⇒Poisson's solution
can be used. (Actually, EC(r,Ω ) is also harmonic for ∀r≥rg, but we have
(
)
elected to work with rF h[Τ ] only. We get:
F Τh ( rg ,Ω) = ∆gh (rg ,Ω) − EC ( rg ,Ω) .
[
]
We have elected to continue 5' by 5' mean values rather than point values.
(Poisson's solution appears to be a linear operation.) Mean values
F Th (rg ,Ω) are the discrete boundary values for the BVP of the third
[
]
kind.
5.
Solution of the BVP of the third kind.
5.1.
Reformulation (generalization) of the BVP
To take advantage of the availability of global field (Τ )L from satellite
orbit analysis we construct (T h ) L through transformation to Helmert's
space (Hemertization). We take L=20; for L>20 global field consists of
mostly noise.
20
T h = (T h )20 + (T h )
Note: W h = U + T h = U + ( T h )20 + (T h )
reference potential, [U + (T h )20 ]/
5.2.
20
and
U + (T h )20 is a new
is a new reference spheroid.
Stokes’s solution to the BVP
This is the classical solution of Green's kind:
T h ( rg ,Ω) = 4 R ∫ geoid F T h (rg ' ,Ω' ) S(rg' ,Ω,Ω' ) dΩ'
[
]
 R  j+1 i 2 j + 1
S (rg ' ,Ω,Ω' ) = ∑   ∑ .
Pj ( cos Ψ).
j = 2  r' g 
m =− j j −1
∞
For the generalized BVP:
Τ h (rg ,Ω) = Τ h ( rg,Ω ) + 4 R ∫ spheroid F T h ( rg' ,Ω ) S20 (rg ' ,Ω,Ω' ) dΩ'
(
)
20
[
 R  j +1 j 2 j + 1
S (rg ', Ω,Ω' ) = ∑   ∑.
Pj ( cos Ψ).
j = 21  rg ' 
m =− j j − 1
∞
20
So far, we have taken only rg'=R.
]
5.3.
Splitting the integration domain
We evaluate the Stokes integral in its above form only over a spherical cap
C0 of radius Ψ0 . For the rest of the domain (spheroid - C0) we use the
spectral form. We choose Ψ0 = 6˚ (somewhat arbitrarily).
Using a simplified notation we write:
+ M20 ) =
(S 2042
∫ sph FS 20 = ∫ sph F20 S 20 = ∫ sph F 20 S = ∫ sph F20 1
43
(
)
S*20
=
∫
C0
F20 S*20 + ∫ sph − C0 F20 S*20
.
S *20 is a modified spheroid Stokes kernel. We have chosen M20 such as
to minimize the upper bound of the far-zone contribution (second term).
We evaluate this FZC in a spectral form using a global field model. For
computational reasons we construct the F20 already on the topography and
only F20 [Th(r, Ω)] is continued down to the geoid.
6.
Solution of the geoid
Still in the Helmert space, we construct the (Helmert co-) geoid:
20 
1  h
h
Ν h (Ω ) =
Τ
r
,Ω
+
Τ
r
,Ω
(
)
(
)

g
g


20
0 ( Ω)
(
= ( Νh ( Ω)) 20 + ( Ν h( Ω))
) (
20
)
.
The first term is the reference spheroid, the second term is the residual
geoid form the solution to the BVP of the third kind.