This article was downloaded by: [Ulsan National Institute of Science and Technology (UNIST)] On: 09 April 2015, At: 19:36 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Inverse Problems in Science and Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gipe20 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. Click for updates 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 To link to this article: http://dx.doi.org/10.1080/17415977.2015.1025071 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. 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Terms & Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Inverse Problems in Science and Engineering, 2015 http://dx.doi.org/10.1080/17415977.2015.1025071 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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: [email protected] © 2015 Taylor & Francis Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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, Inverse Problems in Science and Engineering 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) Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 (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.) Inverse Problems in Science and Engineering 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 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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 Inverse Problems in Science and Engineering 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 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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 Downloaded by [Ulsan National Institute of Science and Technology (UNIST)] at 19:36 09 April 2015 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. 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