Geoscience Reference
In-Depth Information
As for the mathematical formulation, let
r
be the position of an arbitrary station
where the surface deformation shall be determined. The station displacements (ver-
tical, east and north directions) evoked by surface pressure loads
P
r
,
(
t
)
over the
entire surface of the Earth
S
are written as:
r
,
r
)
]
ϑ
d
ϑ
d
λ
,
U
r
(
r
,
t
)
=
[
P
(
t
)
−
P
ref
(
G
r
(ψ)
cos
(1)
S
r
,
r
)
]
ϑ
d
ϑ
d
λ
,
U
e
(
r
,
t
)
=
[
P
(
t
)
−
P
ref
(
G
h
(ψ)
sin
α
rr
cos
(2)
S
r
,
r
)
]
ϑ
ϑ
d
λ
.
U
n
(
r
,
t
)
=
[
P
(
t
)
−
P
ref
(
G
h
(ψ)
cos
α
rr
cos
d
(3)
S
r
)
denotes the reference pressure, which represents the pressure of an unper-
turbed atmosphere. Various methods for determination of the reference pressure are
summarized by Schuh et al. (
2009
).
P
ref
(
λ
is the longi-
tude. The Green's functions are computed from Load Love Numbers (LLNs)
h
n
and
l
n
according to
ϑ
is the geocentric latitude and
∞
GR
g
2
h
n
P
n
(
G
r
(ψ)
=
cos
ψ),
(4)
n
=
0
∞
GR
g
2
l
n
∂
P
n
(
cos
ψ)
G
h
(ψ)
=
,
(5)
∂ψ
n
=
0
where
G
is the universal gravitational constant,
is the angular distance between the
station with the position
r
and the pressure source with the position
r
g
is the mean
gravitational acceleration at the surface of the Earth,
R
is the mean Earth radius and
P
n
is the Legendre polynomial of degree
n
.
Cosine and sine of the azimuth angle
ψ
α
rr
between the station and the pressure
load can be calculated using the formalism described by Hofmann-Wellenhof and
Moritz (
2005
):
ϑ
−
ϑ
(λ
−
λ)
cos
ϑ
sin
sin
ϑ
cos
cos
cos
α
rr
=
(6)
sin
ψ
ϑ
(λ
−
λ)
cos
sin
sin
α
rr
=
(7)
sin
ψ
In the calculation, a complex IB model describing the oceanic response to
atmospheric pressure and wind forcing should be introduced (Geng et al.
2012
).
Instead of using such a complex model, van Dam and Wahr (
1987
) proposed a
