Geoscience Reference
In-Depth Information
∆∬
∑
∆
(
,
)
(
)
(,,ℎ)=−
exp
(
−ℎ
)
exp[
(
+
)
]
(32)
!
Where ∆Z
n
(u, v) is define from the following relation
∆
(,)=
∬
∆
(,)exp[(+)]
When ∆z(x
0
,y
0
)<<h
(,,ℎ)=−
exp
(
−ℎ
)
exp[
(
+
)
]
(33)
∆∬ ∆(,)
The magnetic field due to a random interface may be obtained either by repeating the
mathematical procedure for a magnetic field or by making use of the relationship between
the magnetic and gravity potential (Poisson's relation). The magnetic potentials and fields
can be estimated from gravitational potential using Poisson's relation expressed as
=
, where φ is the magnetic potential, ψ is the gravitational potential, I, the magnetic
polarization, ρ, the volume density of the medium and G is the gravitational constant. The
horizontal and vertical components of the magnetic field are respectively,
=−
and
=−
.
By using the latter approach, the magnetic potential, f
T
(x, y, h) is expressed as (Naidu &
Mathew, 1998)
∑
dΔZ
(u,v)
(
)
!
(,,ℎ)=−
∆∬
(
,
)
exp
(
−ℎ
)
exp[
(
+
)
]
(34)
Where Γ(u, v) = (im
x
u + im
y
v - m
z
s)(iαu + iβv - γs) and (m
x
, m
y
, m
z
) are components of the
magnetization. Note that m
x
= I
x
∆k, m
y
= I
y
∆k and m
z
= I
z
∆k, where ∆k = (k
medII
- k
medI
) is
the magnetization contrast between the medium II and medium I and (α, β, γ) are the
direction cosines of the gradient direction and (I
x
, I
y
, I
z
) are the three components of the
inducing magnetic field. When
|
∆(
,
)
|
≪ℎ
∬
(
,
)
(,,ℎ)=−
exp
(
−ℎ
)
d∆Z(u,v)exp[
(
+
)
]
(35)
6.4 Discrete sampling of potential fields
Potential fields are continuous functions of space. They have to be sampled for processing
on a digital computer. How do we sample them?
A homogeneous random field, f(x) is sampled at x = n∆x, n = 0, 1, 2, ..., where ∆x is the
sampling interval. The accuracy of the sampling is judged by whether by using the samples,
we can recover the original function with as small an error as possible (i.e. with the mean
square error identically zero). This can be achieved as we have earlier noted that (i) the
spectrum of the field be band-limited and (ii) sampling rate (= 1/∆x) is at least twice the
highest frequency present. Band-limitedness of the spectrum of the field means
()=
0||≥
, where u
0
= 2π/λ
0
; λ
0
being the smallest wavelength corresponding to the
highest frequency and condition (ii) implies that ∆x = λ
0
/2.
A random field in two dimensions can be sampled in more than one way: we may use a
square or rectangular grid, polar grid or hexagonal grid. Such a choice of sampling patterns
is not available for one dimensional field. The Fourier transform of discrete data turns out to
Search WWH ::
Custom Search