Summary. Reconstruction of data from non-uniformly spaced samples

Transcription

1 Is there always extra bandwidth in non-unifor spatial sapling? Ralf Ferber* and Massiiliano Vassallo, WesternGeco London Technology Center; Jon-Fredrik Hopperstad and Ali Özbek, Schluberger Cabridge Research Suary We address the question of whether or not there always is a bandwidth benefit in non-unifor spatial sapling of geophysical data. Answering this question is, for exaple, iportant in the context of rando sapling of seisic data, as it recently has been shown that there can be such a benefit under certain assuptions on the spectral structure of the data. Assue that a fixed nuber of sensors are placed either uniforly (i.e., on a regular grid) or nonuniforly (either randoly distributed or following any suitable non-unifor sapling schee). The bandwidth supported by unifor sapling is that of the yquist wavenubers corresponding to the sapling distances on the regular grid. The bandwidth supported by the nonunifor sapling we propose here refers to the axiu bandwidth of data that could be reconstructed by a linear operator at arbitrary sapling locations within the survey area without unacceptably high reconstruction error. Without aking further assuptions on the spectral structure of the data, i.e., especially without assuing sparseness of the data spectru, we will argue that we see no such bandwidth benefit in non-unifor sapling in the exaples we have investigated. Introduction Chen and Allebach (1987) set out to find a sapling location set aong a candidate group of such sets that is optial for data reconstruction over a given class of bandliited broadband signals. If the bandwidth is given, the sapling locations are usually designed to be unifor with the sapling location distances such that the corresponding yquist wavenubers are greater than the assued bandwidth. However, in any applications it is ipossible to place sensors at regular locations. The theory of copressive sapling even requires randoly distributed sensors (Candès et al., 2004; Hennenfent and Herrann, 2008; Moldoveanu, 2010). We consider here a reverse proble; that of finding the axiu bandwidth supported by a given set of non-unifor sapling locations. We consider that a set of sapling locations supports broadband signals of a certain bandwidth if the data can be reconstructed at non-sapling points within an acceptable accuracy. Hence, it akes sense to look for the axiu bandwidth supported by the set of sapling locations. To eliinate edge effects, the area of data reconstruction could be restricted to an inner part of the set of data sapling locations. To test whether a certain bandwidth is supported by the sapling locations, we suggest proceeding as follows: firstly, copute the axiu of the iniu ean-square interpolation error for a range of reconstruction locations within the selected subarea of the point set. Secondly, if this axiu is lower than the acceptable reconstruction error, flag the bandwidth as being supported by the set of sapling locations. If this test is done in an exhaustive search over bandwidth, the bandwidth doain can be split into a region that the set of sapling locations supports and a copleentary region that it fails to support. The approach developed here is valid for any diensionality of the sapling proble, while a typical exaple would be twodiensional spatial sapling of geophysical data. All exaples we give to illustrate the approach will be spatially two-diensional for ease of visualization. Reconstruction of data fro non-uniforly spaced saples We wish to reconstruct the signal fro non-uniforly spaced sapling locations by a linear convolution-like process; the signal at non-sapling locations will be estiated by a weighted suation of existing saples in the direct neighborhood of the reconstruction location. The weights, of course, ust be selected such that the reconstruction error is inial. Optial reconstruction of one-diensional signals by a weighted suation of data sapled non-uniforly is given by Yen s 4 th theore (1956). Chen and Allebach showed an extension of Yen s approach to two-diensional sapling, which can be easily extended to any nuber of diensions. The reconstruction ethod described by Yen s 4 th theore is optial for spatially broadband signals (Özbek et al., 2010); the assued signal bandwidth is a critical paraeter. In all the work entioned above, the signal bandwidth was assued known and fixed, such that, depending on the sapling locations, a reconstruction ight be possible or not. For infinitely long one-diensional data, it is well known that the axiu bandwidth supported by the sapling locations can be estiated as the yquist wavenuber corresponding to the average sapling distances (Beutler, 1974; Jerri, 1977). Moore and Ferber (2008) added a bandwidth optiization strategy to the reconstruction of n- diensional non-uniforly sapled seisic data as a way of finding the axiu bandwidth supported by the sapling locations. In this paper, we first show that the axiu bandwidth supported by non-unifor sapling locations is not uniquely defined. In fact, the wavenuber doain can be split into a region around the zero wavenuber supported by the sapling locations, and a copleentary region that is not supported. We also show that, for broadband data, there is no significant bandwidth SEG San Antonio 2011 Annual Meeting 57

2 gain in non-unifor sapling copared to unifor sapling. Miniu ean-square error interpolation operator As stated above, we wish to reconstruct data at nonsapling locations using a weighted suation over data in the direct vicinity of the non-sapling location. The weights w ( w1, w2,, w ) T for a reconstruction operator applied to the data d ( d, d2,, d ) T, with the data represented by a function d( x ) of an M-diensional sapling location vector. The reconstruction by eans of weighted suation can be atheatically expressed as ˆ T d w d. We select here the iniu ean-square error reconstruction operator (Chen and Allebach, 1987; Moore and Ferber, 2008), that depends on the sapling locations, the interpolation location, and the assued data bandwidth (hence used as the bandwidth paraeter of the Yen-4 operator), but not specifically on the details of the data or its wavenuber spectru. The reconstruction operator can be coputed fro the atrix/vector equation 1 w( k ) S ( k ) r( k ), with k as the bandwidth paraeter, whose coponents are the axiu wavenubers in each of the M diensions of sapling. The sapling location atrix that ust be inverted to get the reconstruction operator is M sin 2 k( xi, x j, ) S( k ) si, j, with si, j. 2 k ( x x ) 1 i, j, Siilarly, the interpolation vector, r, also a product of sinc functions, depends on the differences between sapling locations and the output location for data reconstruction, and is given by M sin 2 k( xi, y ) T r( k ) ( r1,..., r), with ri. 2 k ( x y ) 1 i, The iniu ean-square error (MMSE) can now be coputed as: MMSE Exaples 1 w r. i i i 1 As a first exaple, we show here a bandwidth support spectru for a siple unifor sapling schee, consisting of 25 sapling locations on a unifor sapling grid of 12.5 by For a range of x- and y- wavenubers, we copute the axiu value of the MMSE of the corresponding reconstruction operator, the axiu being taken over the central survey area. These axia as function of the x- and y-wavenubers are displayed colorcoded in Figure 1B. The wavenubers reside in the range of zero to 0.1 1/, for both coordinate axes of the sapling schee. Only positive wavenubers, i.e. one quadrant of Figure 1: Bandwidth support spectru (B) for 25 sensors on a unifor by grid. The lower wavenubers below / are well supported, with a transition region before the non-supported wavenuber (greater than /) appear. The left side of Figure (A) shows a non-quadratic bandwidth box and how it relates to a single entry in the bandwidth support spectru. SEG San Antonio 2011 Annual Meeting 58

3 the full bandwidth support wavenuber spectru, are displayed due to the inherent syetry. The corresponding axiu MMSE values are displayed on a logarithic color-coded scale, such that deep blue colors denote very sall reconstruction errors, while red colors denote large errors. For coparison, we indicate the yquist wavenubers of the infinite regular grid (i.e., kx = ky = 0.04 /) by a blue dot. Figure 1A clearly shows that the wavenuber doain splits into two regions: one region of low wavenubers, supported by the non-unifor sapling schee, and the other region with larger wavenubers that are not supported. There is also a transition zone between these regions. The bandwidth support spectru should be interpreted as follows: assue a bandwidth of the data characterized by two wavenubers in spatial x- and y- directions, for exaple kx = / and ky = /, i.e., one assues that the data resides in the wavenuber box highlighted in green color in Figure 1A. For this bandwidth box there is one entry in the bandwidth support spectru highlighted by the blue dot indicated by the green arrow. This particular schee of 25 sensors on a by grid does not support this bandwidth box well, as it is in the transition zone fro good to poor support. The yquist wavenubers of the by unifor grid, i.e. kx = ky = /, are also on the edge of the bandwidth support area due to the fact that only 25 sensor locations are available for data reconstruction. The size of the transition zone will shrink with a larger nuber of sensors, which would allow use of a ore accurate interpolation operator. As a second exaple we show a siple non-unifor sapling schee, also consisting of 25 sapling locations derived fro the unifor sapling grid of by used above by rando variations around the unifor sapling locations. For the sae range of x- and y- wavenubers we again copute the axiu value of the MMSE of the corresponding reconstruction operator, the axiu being taken over the central reconstruction locations (inside the red rectangle in Figure 2A). These axia as a function of the x- and y- wavenuber are again displayed color-coded in Figure 2B. For coparison we indicate again the yquist wavenubers of the underlying regular grid (i.e. kx = ky = /) by a blue dot. The diagra depicted in Figure 2B clearly shows that the wavenuber doain again splits into two regions: one region with low wavenubers, clearly supported by the non-unifor sapling schee, and the other region with larger wavenubers that are not supported. There is also a transition zone between these regions. The bandwidth support region is larger than that of the unifor sapling schee used above. Higher wavenubers in one direction, ky = 0.08 for exaple, can be supported if the wavenuber of the other direction is reduced, kx = 0.01 for exaple, see yellow dot in Figure 2B, such that non-unifor Figure 2: Bandwidth support spectru (Figure 2B) for a non-unifor sapling schee (sensor locations depicted in Figure 2A), created by rando variations around the unifor grid used in the first exaple. The axiu ean square error for the linear reconstruction operators is coputed over the inner part of the survey area inside the red rectangle in Figure 2A. SEG San Antonio 2011 Annual Meeting 59

4 sapling could be regarded as being ore versatile than the unifor sapling schees. With this type of nonunifor sapling the wavenuber transition zone sees to follow a curve in which the product of the corresponding wavenubers is the product of the wavenubers of the 2 2 underlying unifor grid ( k k 0.04 ). The bandwidth corresponding to the yquist wavenubers of the underlying unifor grid however is not better supported by this non-unifor sapling schee, as it still sits on the edge of the transition zone fro good to poor bandwidth support. As a third exaple, we look at another non-unifor sapling schee for 25 sensors placed in the sae survey area as before, but this tie using Haersley points (Tien-Tsin et al., 1997), depicted in Figure 3A, to define the sensor locations. These points for a well-known lowdiscrepancy sequence and have been used, for exaple, for quasi-monte Carlo integration. The bandwidth support spectru, Figure 3B, now is ore anisotropic and soewhat larger than that of the rando sapling schee above, but fundaentally siilar, and again not showing a bandwidth benefit as copared to unifor sapling (blue dot in Figure 3B). x y Conclusions For spatially broadband signals, i.e., without assuing anything else about the signals other than bandliitation, we presented a technique to calculate the axiu bandwidths that a non-unifor sapling schee supports. We introduced the so-called bandwidth support spectru as a function that contains the axiu iniu ean square reconstruction error of the corresponding optiu linear reconstruction operator, the Yen-4 operator, as a function of those liiting wavenubers. The bandwidth support spectru splits the wavenuber plane into three distinct areas; those low-wavenubers well supported by the sapling schee and those higher wavenubers clearly not supported, with a transition zone between. We showed that non-unifor sapling schees can be ore versatile than unifor schees, as they can support wavenubers that a particular unifor schee with the sae nuber of sensor locations cannot support. However, we also showed, that under the restriction of an identical nuber of sensor locations in the survey area, the nonunifor sapling schees show no bandwidth gain beyond that of the yquist wavenubers associated with the unifor sapling schee. Figure 3: Bandwidth support spectru (Figure 3B) for a non-unifor sapling schee using Haersley points (sensor locations depicted in Figure 3A), created by rando variations around the unifor grid used in the first exaple. The axiu ean square error for the linear reconstruction operators is coputed over the inner part of the survey area inside the red rectangle in Fig. 3A. SEG San Antonio 2011 Annual Meeting 60

5 EDITED REFERECES ote: This reference list is a copy-edited version of the reference list subitted by the author. Reference lists for the 2011 SEG Technical Progra Expanded Abstracts have been copy edited so that references provided with the online etadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERECES Beutler, F. J., 1974, Recovery of randoly sapled signals by siple interpolators: Inforation and Control, 26, , doi: /s (74) Candès, E. J., J. Roberg, and T. Tao, 2006, Robust uncertainty principles: exact signal reconstruction fro highly incoplete frequency inforation: IEEE Transactions on Inforation Theory, 52, , doi: /tit Chen, D. S., and J. P. Allebach, 1987, Analysis of error in reconstruction of two-diensional signals fro irregularly spaced saples: IEEE Transactions on Acoustics, Speech and Signal Processing, 35, Hennenfent, G., and F. J. Herrann, 2008, Siply denoise: wavefield reconstruction via jittered undersapling: Geophysics, 75, no. 3, V19 V28, doi: / Jerri, A. J., 1977, The Shannon sapling theore its various extensions and applications: a tutorial review: Proceedings of the IEEE, 65, , doi: /proc Moldoveanu,., 2010, Rando sapling: a new strategy for arine acquisition: 80th Annual International Meeting, SEG, Expanded Abstracts, Moore, I., and R.-G. Ferber, 2008, Bandwidth optiization for copact Fourier interpolation: 70th Conference and Exhibition, EAGE, Extended Abstracts. Özbek, A., M. Vassallo, A. K. Özdeir, D. Molteni, and Y. K. Alp, 2010, Anti-alias optial interpolation with priors: 80th Annual International Meeting, SEG, Expanded Abstracts, Tien-Tsin, W., W.-S. Luk, and P.-A. Heng, 1997, Sapling with Haersley and Halton points: Journal of Graphics Tools, 2, Yen, J. L., 1956, On non unifor sapling of bandliited signals: IRE Transactions on Circuit Theory, 3, , doi: /tct SEG San Antonio 2011 Annual Meeting 61