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Inverse Problems in Science and
Engineering
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Inversion of the seismic parabolic
Radon transform and the seismic
hyperbolic Radon transform
Sunghwan Moon
a
a
Department of Mathematical Sciences, Ulsan National Institute
of Science and Technology, Ulsan, Korea.
Published online: 23 Mar 2015.
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To cite this article: Sunghwan Moon (2015): Inversion of the seismic parabolic Radon transform
and the seismic hyperbolic Radon transform, Inverse Problems in Science and Engineering, DOI:
10.1080/17415977.2015.1025071
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Inverse Problems in Science and Engineering, 2015
http://dx.doi.org/10.1080/17415977.2015.1025071
Inversion of the seismic parabolic Radon transform and the seismic
hyperbolic Radon transform
Sunghwan Moon∗
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan,
Korea
(Received 5 July 2014; final version received 11 February 2015)
Reflection seismology is a method of exploration of the hidden structure of the
earth subsurface by processing received seismograms. For a long time, this processing utilizes the so-called slant stack transform, or line Radon transform. More
recently, two generalizations of the slant stack transform have been introduced
to extract new features of the seismic data and to improve their treatment. These
transforms are the parabolic and hyperbolic seismic Radon transforms. The first
transform maps a given function to its integrals over parabolas with a fixed axis
direction, whereas the second one maps a function to its integrals over hyperbolas
(more generally also over ellipses and circles) of fixed axis directions. We show
how they can be converted to a line Radon transform, and thereby obtain their
inversion formulas. Numerical simulations for each transform were performed
and commented to illustrate the suggested algorithms.
Keywords: Radon transform; tomography; parabolic; hyperbolic; seismology
AMS Subject Classifications: 44A12; 65R10; 86A22
1. Introduction
Geophysical imaging attempts to determine the structure of the earth’s interior from data
observed on its surface.[1]
Reflection seismology is the method of geophysics that estimates the properties of
the earth’s interior such as structure and composition, from reflected seismic signals. To
handle seismic data, the line Radon transform was introduced in [2–4]. Depending on the
source excitation and the inherent properties of the target signal, a special parabolic Radon
transform and a special hyperbolic Radon transform were often used.[5–12]
Reflection seismology data appear as an amplitude function f (x, t), where x and t are
the so-called offset (distance from the sound source to a detector) and the travel time of the
signal (from the source to the detector), respectively. The function f (x, t) usually peaks
on typical trajectories of the (x, t) plane, which corresponds to various processes such as
simple propagation and reflection on a plane reflector at some depth. In order to retrieve
information on reflecting layers of the earth subsurface, the slant stack (or line Radon)
transform is applied to the data yielding a more convenient form in Radon parameter space
∗ Email: shmoon@unist.ac.kr
© 2015 Taylor & Francis
2
S. Moon
for treatment, such as determination of reflecting layers, elimination of multiple reflections,
removal of noise and estimation of sound velocity. As an example, one may cite the fact
that the presence of an arc of an ellipse in Radon parameter space on which the transformed
signal peaks are the sign that a reflecting plane under the earth surface exists. Thus, the
seismic data can be reworked first in Radon parameter space and then upon applying the
inverse Radon transform a so-called synthetic data in (x, t) plane may be reconstructed and
compared to the field data.
A further refinement of the concept of the slant stack transform consists in introducing
the parabolic and hyperbolic seismic Radon transforms. These transforms have better
performances as far as some tasks are concerned: focusing on specific processes, sampling
efficiency, restoration of missing data, etc., see [5,9,12]. In this respect, these transforms
deserve to be studied and their inverse analytically derived as they are useful for full seismic
data treatment.
Let f (x, t) be a seismic wave field at travel time t and offset x. Its seismic parabolic
Raodn transform g(u, s) is defined by
g(u, s) =
R
f (x, u + s(x − c)2 )dx,
(see [6,11,13]). This is the integral of f (x, t) over the parabolas with apex (c, u) and a fixed
axis direction. We call this the ‘seismic parabolic Radon transform’.
If one assumes instead that the arrival time t of a seismic signal is given by a hyperbolic
relation of the form t 2 = u + s(x − c)2 , where u and s can be chosen at will (see [10]), our
data can be represented as:
g(u, s) =
=
f (x, t)δ(t 2 − s(x − c)2 − u)dxdt
R2
1
2
x∈R
u+s(x−c)2 ≥0
×
( f (x, u + s(x − c)2 ) + f (x, − u + s(x − c)2 ))
dx
u + s(x − c)2
.
Although the integral domain is a hyperbola when s is positive and an ellipse or circle when
s is negative, we call this the ‘seismic hyperbolic Radon transform’.
There are a few works dealing with parabolic Radon transforms. Cormack considered the
Radon-type transform on parabolas with central axis rotating around the origin in [14,15].
Jollivet, Nguyen and Truong studied various types of parabolic Radon transforms in [13].
Denecker, Van Overloop and Sommen showed the relation between the Radon-type transform on isofocal parabolas, with parabolic arc length measure, and the regular Radon
transform. Also, they presented that the Radon-type transform on isofocal parabolas can be
inverted using the inversion formula for the regular Radon transform in [16].
In this article, we study the seismic parabolic and hyperbolic Radon transforms. In
Section 2, we show how to reduce the seismic parabolic Radon transform to the regular
Radon transform, and find an inversion formula. Also, numerical results are provided to
demonstrate the suggested two-dimensional algorithm. In Section 3, we do the same thing
for the seismic hyperbolic Radon transform.
Inverse Problems in Science and Engineering
3
2. Seismic parabolic Radon transform
From now on, we write the function f with the variables x = (x 1 , x2 ) ∈ R2 instead of x
and t. Let us define the seismic parabolic Radon transform.
Definition 2.1 For a continuous and compactly supported function f on R2 , we define the
seismic parabolic Radon transform by
f (x1 , s(x1 − c)2 + u)dx1 .
R P f (s, u) =
R
Then the seismic parabolic Radon transform R P f is the integral of f over parabolas
with apex (c, u) and fixed axis direction (see Figure 1(a)). Here, if f is odd with respect to
the line x1 = c, i.e. f (−x1 + c, x2 ) = − f (x1 + c, x2 ), then R P f is equal to zero. We thus
assume that f is even with respect to the line x 1 = c, i.e. f (−x1 + c, x2 ) = f (x1 + c, x2 ).
We will define a new function and obtain a relation between the regular Radon transform
of this new function and R P f .
2.1. Inversion formula
First, let us define a new function k P (y) on R2 by
⎧ √
⎨ f ( y1 + c, y2 )
√
k P (y) =
y1
⎩
0
if y1 > 0,
otherwise,
where y = (y1 , y2 ) ∈ R2 . Then we have for x1 > c,
f (x) = k P ((x1 − c)2 , x2 )(x1 − c),
so for all x1 = c,
f (x) = k P ((x1 − c)2 , x2 )|x1 − c|,
(a)
(b)
Figure 1. (a) Parabolas and (b) hyperbolas, ellipses and circles.
(1)
4
S. Moon
since f (−x1 + c, x2 ) = f (x1 + c, x2 ). Let the regular Radon transform Rk P (θ, t) be
defined as
Rk P (θ , t) =
k P (θt + τ θ⊥ )dτ,
R
where t ∈ R and θ = (cos θ, sin θ ) ∈ S 1 and θ⊥ = (− sin θ, cos θ ) ∈ S 1 . Then we have
the following relation between R P f and Rk P :
Theorem 2.2 Let f ∈ C ∞ (R2 ) satisfy f (−x1 +c, x2 ) = f (x1 +c, x2 ) and have compact
support in R/{c} × R. Then we have
1
u
s
2
1 + s R P f (s, u) = Rk P − √
,√
,√
.
1 + s2
1 + s2
1 + s2
Proof
Since f (−x1 + c, x2 ) = f (x1 + c, x2 ), R P f (s, u) can be written as:
∞
R P f (s, u) = 2
f (x1 , s(x1 − c)2 + u)dx1
c
∞
=2
k P ((x1 − c)2 , s(x1 − c)2 + u)(x1 − c)dx1 ,
c
where in the second line, we used the definition of k P . Changing the variables (x1 − c)2 →
y1 , we have
∞
k P (y1 , sy1 + u)dy1 =
k P (y1 , sy1 + u)dy1 ,
(2)
R P f (s, u) =
0
R
since k P (y) is equal to zero for y1 ≤ 0. We recognize the right-hand side as the integral
along the line perpendicular to
(−s, 1)/ 1 + s 2
√
with (signed) distance u/ 1 + s 2 from the origin. In this case, the length measure for the
line becomes
1 + s 2 dy1 .
Using the projection slice theorem for the regular Radon transform, we obtain an
analogue of the projection slice theorem for the parabolic Radon transform:
Theorem 2.3 Let f ∈ C(R2 ) satisfy f (−x1 + c, x2 ) = f (x1 + c, x2 ) and have compact
support in R/{c} × R. Then we have
α
k P (α, β) = R P f − , β ,
β
where k
P is the two-dimensional Fourier transform of k P and R
P f is the one-dimensional
Fourier transform of R P f with respect to u.
Proof
Theorem 2.2 is equivalent to
t) = csc θ R P f (− cot θ, t csc θ ) for θ ∈ (0, π ) and t ∈ R.
Rk P (θ,
5
t) = Rk P (θ, −t), we have
Since Rk P (−θ,
t) = | csc θ |R P f (− cot θ, t csc θ )
Rk P (θ,
for θ ∈ (0, 2π ) and θ = π.
Using the projection slice theorem, we have
=
e−iσ t Rk P (θ, t)dt =
e−iσ t | csc θ |R P f (− cot θ, t csc θ )dt.
k
P (σ θ)
R
R
Changing the variables t csc θ → t gives
=R
k
P (σ θ)
P f (− cot θ, σ sin θ ).
Corollary 2.4 If f ∈ C ∞ (R2 ) satisfies f (−x1 + c, x2 ) = f (x1 + c, x2 ) and has
compact support in R/{c} × R, then we have
∂u R P f (s, u)
|x1 − c|
duds,
P.V.
f (x) =
2
2
2π
R
R (x 2 − s(x 1 − c) ) − u
where P.V. means the Cauchy principal value.
Proof
We have
|x1 − c|
i(α,β)·((x1 −c)2 ,x2 )
f (x) = k P ((x1 − c) , x2 )|x1 − c| =
dαdβ
k
P (α, β)e
(2π )2 R2
|x1 − c|
α
i(α,β)·((x1 −c)2 ,x2 )
=
R
dαdβ,
P f − ,β e
β
(2π )2 R2
2
where in the last line, we used Theorem 2.3. Changing the variables −α/β → α gives
|x1 − c|
iβ(−α,1)·((x1 −c)2 ,x2 )
|β|R
dαdβ
f (x) =
P f (α, β)e
2
(2π ) R2
|x1 − c|
2
(−isgn(β))(∂u R P f )
(α, β)eiβ(−α,1)·((x1 −c) ,x2 ) dαdβ.
=
(2π )2 R2
f (t)
Notice that the Fourier transform of 1/π(P.V. R u−t
dt) with respect to u is −isgn(ξ ) fˆ
for any f in the Schwartz space. Hence, we have the assertion.
2.2. Numerical implementations
Here, we provide the results of two-dimensional numerical implementations.
First of all, we set c = 0. Then the phantom should be even with respect to the x 1 -axis.
In the experiments presented here, we use the phantom shown in Figure 2(a). The phantom,
supported within the rectangle [−1, 1] × [−1, 1], is the sum of eight characteristic functions
of disks. Notice that it has to be even in the x 1 -axis and there are four characteristic functions
of disks centred at (0.5, −0.5), (0.6, −0.4), (0.5, 0.4) and (0.4, 0.5) with radii 0.05, 0.05,
0.1 and 0.15, whose values are 0.5, 1, 1 and 0.5 on the left side of the x 2 -axis. Thus, we also
include their reflections on the right side of the x 2 -axis. (Actually, our phantom has support
in
{x ∈ R2 : x14 + x22 < 1}.
6
S. Moon
1.5
(b)0.81
1
0.5
−0.5
0
0.5
1
0
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
1.5
1
0.5
0
−0.5
x1
(c)0.81
x2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
x2
x2
(a)
0
0.5
1
1.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
−0.5
0
0.5
1
x1
x1
Figure 2. Reconstructions in two dimensions (Red dotted lines are the line x1 = −0.46): (a) the
phantom, (b) the reconstruction from exact data and (c) the reconstruction from noisy data.
(a) 1.5
(b) 1.6
(c)
1.4
1.2
1
1
1
0.8
0.6
0.5
0.5
0.4
0.2
0
0
0
−1
−0.5
0
x2
0.5
1
−0.2
−1
2
1.5
−0.5
0
x2
0.5
1
−0.5
−1
−0.5
0
0.5
1
x2
Figure 3. The values on the cross-section x1 = −0.46: (a) the phantom, (b) the reconstruction from
exact data and (c) the reconstruction from noisy data.
This implies that k P has support in the unit ball and this makes it sufficient to consider the range [−1, 1] in t.) The 256 × 256 images are used in Figure 2. To reconstruct
the image in Figure 2(b), we have 256 × 256 projections for θ and t in Rk P (θ, t) =
| csc θ |R P f (− cot θ, t csc θ ) obtained by Theorem 2.2. After finding the function k P using the inversion code for the regular Radon transform, we obtain the function f using
Equation (1). (When using the inversion code for the regular Radon transform, the builtin function ‘iradon’ in MATLAB was used. The function ‘iradon’ is the inversion of the
built-in function ‘radon’ in MATLAB which considers the number of the pixels where
the line pass through. Thus, when computing R P f , we also considers the number of the
pixels where the parabola passes through. We used the default version of the function
‘iradon’, in which the filter, whose aim is to de-emphasize high frequencies, is set to the
Ram-Lak filter and the interpolation is set to be linear.) While Figure 2(b) demonstrates
the image reconstructed from the exact data, Figure 2(c) shows the result of the reconstruction from noisy data. Actually, we show absolute values of the reconstruction from
noisy data to compare Figures 2(a) and (b) easily. The noise is modelled by normally
distributed random numbers and this is scaled so that its L 2 norm was equal to 5% of
the L 2 norm of the exact data. In the case of Figure 2(c), the noisy data are modelled
by adding the noise values scaled to 5% of the L 2 norm of the exact data to the exact
data.
In Figure 3, we show the cross-section x1 = −0.46 of all images in Figure 2. (Red
dotted lines in Figure 2 are the line x1 = −0.46.)
7
3. Seismic hyperbolic Radon transform
It would appear natural to define the seismic hyperbolic Radon transform as follows: for a
continuous and compactly supported function f on R2 such that R H f (s, u) is equal to
1
dx1
( f (x1 , s(x1 − c)2 + u) + f (x1 , − s(x1 − c)2 + u)) .
x
∈R
1
2
s(x1 − c)2 + u
s(x −c)2 +u≥0
1
(3)
From now on, we assume that f is smooth and compactly supported on R2 and satisfy
f (x1 , 0) = 0. Then f (x1 , x2 )/x2 is continuous at x2 = 0 and the integral (3) exists. If f is
odd with respect to the line x1 = c or odd with respect to the x2 -axis, then R H f is equal to
zero. We thus assume that f is even with respect to the line x 1 = c and even with respect
to the x2 -axis, i.e. f (−x1 + c, x2 ) = f (x1 + c, x2 ) and f (x1 , x2 ) = f (x1 , −x2 ). Then we
have
dx1
f
(x
,
s(x1 − c)2 + u) 1
x1 ∈R
s(x1 − c)2 + u
s(x1 −c)2 +u≥0
dx1
f
(x
,
−
s(x1 − c)2 + u) .
=
1
x1 ∈R
s(x1 − c)2 + u
s(x −c)2 +u≥0
1
Definition 3.1 For a smooth and compactly supported function f on R2 satisfying
f (x1 , 0) = 0, f (−x1 + c, x2 ) = f (x1 + c, x2 ) and f (x1 , x2 ) = f (x1 , −x2 ), we define the
seismic hyperbolic Radon transform as:
dx1
f
(x
,
s(x1 − c)2 + u) R H f (s, u) =
.
1
x1 ∈R
s(x1 − c)2 + u
s(x −c)2 +u≥0
1
As mentioned before, although we call R H f the hyperbolic Radon transform, the
integration domain is an ellipse or circle (s = −1) when s < 0, and a hyperbola when
s > 0 (see Figure 1(b)).
As in Section 2.1, we define a new function and reduce the hyperbolic Radon transform
to the regular Radon transform of this new function.
3.1. Inversion formula
As in Section 2.1, let us define a function k H (y) on R2 by
⎧ √
√
⎨ f ( y1 + c, y2 )
if y1 > 0 & y2 > 0,
√ √
k H (y) =
y1 y2
⎩
0
otherwise,
y = (y1 , y2 ) ∈ R2 .
Then we have for x1 > c and x2 > 0,
f (x) = k H ((x1 − c)2 , x22 )|x1 − c|x2 ,
(4)
so for all x = (x1 , x2 ) ∈ R/{c} × R/{0},
f (x) = k H ((x1 − c)2 , x22 )|x1 − c||x2 |,
(5)
8
S. Moon
since f (−x1 + c, x2 ) = f (x1 + c, x2 ) and f (x1 , x2 ) = f (x1 , −x2 ). We have the following
relation between R H f and Rk H :
Theorem 3.2 Let f ∈ C ∞ (R2 ) satisfy f (−x1 + c, x2 ) = f (x1 + c, x2 ) and f (x1 , x2 ) =
f (x1 , −x2 ) and have compact support in R/{c} × R/{0}. Then we have
s
1
u
.
1 + s 2 R H f (s, u) = Rk H − √
,√
,√
1 + s2
1 + s2
1 + s2
By definition, R H f (s, u) can be written as:
R H f (s, u) =
k H ((x1 − c)2 , s(x1 − c)2 + u)|x1 − c|dx1
x ∈R
1
2
s(x1 −c) +u≥0
=
k H ((x1 − c)2 , s(x1 − c)2 + u)|x1 − c|dx1 ,
R
where in the second line, we used k H (y) = 0 for y2 ≤ 0.
The remaining part of the proof is the same as that of Theorem 2.2.
As Theorem 2.3, we obtain an analogue of the projection slice theorem:
Theorem 3.3 Let f ∈ C ∞ (R2 ) satisfy f (−x1 + c, x2 ) = f (x1 + c, x2 ) and f (x1 , x2 ) =
f (x1 , −x2 ) and have compact support in R/{c} × R/{0}. Then we have
α
,
β
,
(α,
β)
=
R
f
−
k
H
H
β
where k
H is the two-dimensional Fourier transform of k H and R
H f is the one-dimensional
Fourier transform of R H f with respect to u.
The proof is identical to that of Theorem 2.3, except for replacing k P and R P by k H
and R H , respectively.
Corollary 3.4 If f ∈ C ∞ (R2 ) satisfies f (−x1 +c, x2 ) = f (x1 +c, x2 ) and f (x1 , x2 ) =
f (x1 , −x2 ) and has compact support in R/{c} × R/{0}, then we have
|x1 − c||x2 |
∂u R H f (s, u)
f (x) =
duds.
P.V.
2
2
2
2π
R
R (x 2 − s(x 1 − c) ) − u
Using Theorem 3.3 as in the proof of Corollary 2.4, we have for x1 > c and x2 > 0,
|x1 − c||x2 |
i(α,β)·((x1 −c)2 ,x22 )
dαdβ
k
f (x) = k H ((x1 − c)2 , x22 )|x1 − c||x2 | =
H (α, β)e
2
(2π )
R2
|x1 − c||x2 |
α
i(α,β)·((x1 −c)2 ,x22 )
=
R
dαdβ.
H f − ,β e
2
β
(2π )2
R
Proof
Changing the variables −α/β → α, we have
|x1 − c||x2 |
iβ(−α,1)·((x1 −c)2 ,x22 )
|β|R
dαdβ
f (x) =
H f (α, β)e
2
2
(2π )
R
|x1 − c||x2 |
iβ(−α,1)·((x1 −c)2 ,x22 )
(−isgn(β))∂
dαdβ.
=
u R H f (α, β)e
2
(2π )2
R
9
3.2. Numerical implementations
Here, we provide the results of two-dimensional numerical implementations as in the
Section 2.2.
Again, we set c = 0. Then the phantom should be even with respect to the x 1 -axis and
even with respect to the x2 -axis. In the experiments presented here, we use the phantom
shown in Figure 4(a) and the phantom, supported within the rectangle [−1, 1] × [−1, 1],
is the sum of eight characteristic functions of disks. We notice that it has to be even with
respect to the x1 -axis and there are two characteristic functions of disks centred at (0.5, 0.4)
and (0.4, 0.5) with radii 0.1 and 0.15, whose values are 1 and 0.5 in the first quadrant.
Hence, it includes their reflections on the second quadrant. Additionally, since it has to
be even with respect to the x 2 -axis and there are four characteristic functions above the
x2 -axis, it also includes their reflection below the x 2 -axis with the same value. (Again,
k H has support in the unit ball and this makes it sufficient to consider the range [−1, 1]
in t.) The 256 × 256 images are used in Figure 4. To reconstruct the image in Figure 4(b),
we have 256 × 256 projections for θ and t in Rk H (θ, t) = | csc θ |R H f (− cot θ, t csc θ )
obtained by Theorem 3.2. After finding the function k H using the built-in function ‘iradon’
in MATLAB for the regular Radon transform, we obtain the function f using Equation (5).
(Again, we used the default version of this function in which the filter is set to the Ram-Lak
filter and the interpolation is set to be linear.) As in Section 2.2, Figure 4(b) demonstrates
1.5
(b)
1
x2
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
0.5
−0.5
0
0.5
1
0
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
1.5
(c) 0.81
1
x2
(a) 0.81
x2
f (t)
dt) with respect to u is
Again, the fact that the Fourier transform of 1/π(P.V. R u−t
ˆ
−isgn(ξ ) f for any f in the Schwartz space completes the proof.
0.5
0
−0.5
0
0.5
1
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
−0.5
x1
x1
0
0.5
1
x1
Figure 4. Reconstructions in two dimensions (Red dotted lines are the line x1 = −0.49): (a) the
phantom, (b) the reconstruction from exact data and (c) the reconstruction from noisy data.
(a)1.5
(b) 1.6
(c) 2
1.4
1.5
1.2
1
1
1
0.8
0.6
0.5
0.5
0.4
0.2
0
0
0
−1
−0.5
0
x2
0.5
1
−0.2
−1
−0.5
0
x2
0.5
1
−0.5
−1
−0.5
0
0.5
1
x2
Figure 5. The values on the cross-section x1 = −0.49: (a) the phantom, (b) the reconstruction from
exact data and (c) the reconstruction from noisy data.
10
S. Moon
the image reconstructed from the exact data and Figure 4(c) shows the absolute value of
the reconstruction from noisy data. Here, the noisy data are again modelled by adding the
noise values scaled to 5% of the L 2 norm of the exact data to the exact data.
In Figure 5, the cross-sections x1 = −0.49 of all images in Figure 4 is shown. (Red
dotted lines in Figure 4 are the line x1 = −0.49.)
4. Conclusion
This paper is devoted to the study of the seismic parabolic and hyperbolic Radon transforms
arising in reflection seismology. We suggest the reduction in these transforms to the regular
Radon transform and provide inversion formulas for both transforms. Also, numerical
implementations were performed to demonstrate our algorithms. In [9], Nurul Kbir and
Vershuur also found the approximate inversion of the discrete version of the parabolic
transform and showed the applications of their inversion. We think that our inversion also
has similar applications.
Acknowledgements
The author would like to thank the referees for many helpful suggestions.
Disclosure statement
No potential conflict of interest was reported by the author.
Funding
This work has been supported in part by the National Research Foundation of Korea (NRF), grant
funded by the Korea government (MSIP) [grant number NRF-2012R1A1B3001167] and by the Basic
Science Research Program through the National Research Foundation of Korea (NRF) funded by the
Ministry of Education, Science and Technology [grant number NRF-2012R1A1A1015116].
References
[1] Robinson EA. Spectral approach to geophysical inversion by Lorentz, Fourier, and Radon
transforms. Proc. IEEE. Sept 1982;70:1039–1054.
[2] Harding AJ. Inversion methods for τ -ρ maps of near offset data-linear inversion. Geophys.
Prospect. 1985;33:674–695.
[3] Thorson JR, Claerbout JF. Velocity-stack and slant-stack stochastic inversion. Geophysics.
1985;50:2727–2741.
[4] Turner G. Aliasing in the tau-p transform and the removal of spatially aliased coherent noise.
Geophysics. 1990;55:1496–1503.
[5] Bickel SH. Focusing aspects of the hyperbolic Radon transform. Geophysics. 2000;65:652–655.
[6] Foster DJ, Mosher CC. Suppression of multiple reflections using the Radon transform.
Geophysics. 1992;57:386–395.
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[9] Nurul Kabir MM, Verschuur DJ. Restoration of missing offsets by parabolic Radon transform.
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[11] Maeland E. An overlooked aspect of the parabolic Radon transform. Geophysics. 2000;65:
1326–1329.
[12] Schonewille M, Duijndam A. Parabolic Radon transform, sampling and efficiency. Geophysics.
2001;66:667–678.
[13] Jollivet A, Nguyen MK, Truong TT. Properties and inversion of a new Radon transform on
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[14] Cormack AM. The Radon transform on a family of curves in the plane. Proc. Am. Math. Soc.
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[15] Cormack AM. The Radon transform on a family of curves in the plane. II. Proc. Am. Math. Soc.
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