Born and ray-theory seismograms in 2D heterogeneous isotropic models Libor ˇ Sachl Charles University, Faculty of Mathematics and Physics, Department of Geophysics, E-mail: [email protected]ff.cuni.cz Summary 3D P-wave seismograms computed using the first-order ray-based Born approximation in a 2D heterogeneous isotropic background model with a 2D heterogeneous perturbation are presented. The perturbed model contains 16 domains. We construct 16 partially perturbed models, in which the distribution of elastic parameters is the same as in the background model except for one domain where the P-wave velocity is perturbed. The Born seismograms are compared with the differences between the ray-theory seismograms computed in the partially perturbed models, or the complete perturbed model, and the background model. The seismograms are computed for a set of receivers in order to see how the seismograms change. The rays in the perturbed models are computed to be able to identify the wavegroups in the seismograms. Similarly, the travel times of the diffracted waves are computed and highlighted in the seismograms. The differences between the Born and ray-theory seismograms are studied. Key words: Born approximation, ray theory, velocity model, perturbation 1 Introduction The Born approximation is a useful tool. Moser (2012) mentions its 3 important applica- tions: 1. It can be used to test migration algorithms; 2. It can be used to study the effect of small perturbations of the existing model (scenario testing); 3. It can help in acquisition survey design. Klimeˇ s (2012a), Klimeˇ s (2012b) and Klimeˇ s (2010) deal with the aforementioned ap- plications. The first-order Born approximation can be useful for the first mentioned application, because it generates idealized synthetic data, which contain primaries only (e.g., Chauris et al., 2002). According to Jin et al. (1992), the important advantage of the ray-based Born modelling is computational efficiency. Another point in favour is that a general acquisition geometry and surface topography can be considered, as well as arbitrary locations of the scatterers in the model, with no restrictions to model topology (Moser, 2012). Besides, the Born approximation underlies most seismic migration tech- niques, but requires some assumptions to be fulfilled. We think that it is worth studying Seismic Waves in Complex 3-D Structures, Report 22, Charles University, Faculty of Mathematics and Physics, Department of Geophysics, Praha 2012, pp. 83-112 83
Transcript
Born and ray-theory seismograms in 2D heterogeneousisotropic models
Libor Sachl
Charles University, Faculty of Mathematics and Physics, Department of Geophysics,E-mail: [email protected]
Summary3D P-wave seismograms computed using the first-order ray-based Born approximation ina 2D heterogeneous isotropic background model with a 2D heterogeneous perturbationare presented. The perturbed model contains 16 domains. We construct 16 partiallyperturbed models, in which the distribution of elastic parameters is the same as in thebackground model except for one domain where the P-wave velocity is perturbed. TheBorn seismograms are compared with the differences between the ray-theory seismogramscomputed in the partially perturbed models, or the complete perturbed model, and thebackground model. The seismograms are computed for a set of receivers in order tosee how the seismograms change. The rays in the perturbed models are computed tobe able to identify the wavegroups in the seismograms. Similarly, the travel times ofthe diffracted waves are computed and highlighted in the seismograms. The differencesbetween the Born and ray-theory seismograms are studied.
Key words: Born approximation, ray theory, velocity model, perturbation
1 Introduction
The Born approximation is a useful tool. Moser (2012) mentions its 3 important applica-tions:1. It can be used to test migration algorithms;2. It can be used to study the effect of small perturbations of the existing model (scenario
testing);3. It can help in acquisition survey design.Klimes (2012a), Klimes (2012b) and Klimes (2010) deal with the aforementioned ap-plications. The first-order Born approximation can be useful for the first mentionedapplication, because it generates idealized synthetic data, which contain primaries only(e.g., Chauris et al., 2002). According to Jin et al. (1992), the important advantageof the ray-based Born modelling is computational efficiency. Another point in favour isthat a general acquisition geometry and surface topography can be considered, as well asarbitrary locations of the scatterers in the model, with no restrictions to model topology(Moser, 2012). Besides, the Born approximation underlies most seismic migration tech-niques, but requires some assumptions to be fulfilled. We think that it is worth studying
Seismic Waves in Complex 3-D Structures, Report 22, Charles University, Faculty of Mathematics andPhysics, Department of Geophysics, Praha 2012, pp. 83-112
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the Born approximation for these reasons.As already mentioned, the Born approximation can be used for inversion algorithms.
Thierry et al. (1999) pointed out that the ray-based Born approach allows a great sim-plification in the migration algorithm thanks to low storage space of Green’s functionscomputed by the ray theory. Cohen et al. (1986) are interested in cases described bythe acoustic wave equation in the high-frequency limit. They derive a full 3-D velocityinversion based on assumptions of the Born approximation, i.e. the data have relativetrue amplitudes and a reasonably accurate background velocity is available. The pa-per is theoretical; no sample computations in any realistic models are shown. Beydounand Mendes (1989) work with 3-D heterogeneous elastic media. They use an approximateray-theory Green function combined with the first-order Born approximation to generatea scattered wavefield. The linearized inverse scattering problem is then solved in thespace-time domain and consists of minimizing a cost function within the L2 norm with aone-step conditioned gradient procedure.
Other papers, which could be found in the literature, focus on the Born approxima-tion itself. Coates and Chapman (1990) studied the Born approximation in comparisonwith the ray perturbation theory. They restricted themselves to studying slowness per-turbations (although the Born approximation is capable of dealing with perturbations ofdensity and Lame parameters). They considered the scattered signal from that part ofthe model that is near the reference ray and is well described by the first terms of a Taylorexpansion of the model. They demonstrated the agreement, in the far field, of the twomethods to the first order in the slowness perturbation and to the leading order in theasymptotic ray series. Hudson and Heritage (1981) attempted to define the cases withinwhich the Born approximation may be regarded as reasonably accurate. The obtainedinequalities, governing the ranges of the parameters of the problem, are tested by com-parison with the exact solution for a special case. The scattering of a plane wave by ahomogeneous sphere, where the region outside the sphere is also homogeneous but withdifferent properties, is treated. Gibson and Menahem (1991) analysed Rayleigh scatteringusing the Born approximation. They assumed an anisotropic obstacle embedded withinan anisotropic matrix and showed that a perturbation of any of the 21 independent elasticconstants acts as a secondary moment tensor source which radiates energy as it is encoun-tered by the incident wave. Further on, they studied the case of an anisotropic obstaclein an isotropic background medium in more detail and examined the case of a small frac-tured volume. Ursin and Tygel (1997) used ray-theory Green functions from the sourceand receiver to the scattering point and assumed that the parameters of the medium varyslowly except along a surface, where they vary steeply. They applied these assumptionsto derive reciprocal surface scattering integrals. Beydoun and Tarantola (1988) studiedthe pressure-field response from a one-dimensional velocity slab immersed in an infiniteconstant velocity medium. The analytical expressions are used for comparison with theresults obtained by both the first Born and first Rytov approximations. They concludethat the first Born approximation is best suited for modelling and inverting the primaryreflected, or backscattered part of the wavefield.
This paper follows Sachl (2011b) and Sachl (2012). Sachl (2011b) and Sachl (2012)deals with the computation of the Born approximation in perturbed models constructedusing 2D heterogeneous isotropic models P1I and P1. The properties of model P1I,which are important for this paper, are described in Section 2. These models are also
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considered here. The paper by Sachl (2011b) is dedicated to the choice of the P-wavevelocity perturbation and the seismogram computation for one receiver in 4 perturbedmodels. Sachl (2012) deals with the effects of caustics. Now we present the seismogramsand ray diagrams computed in 16 perturbed models for 37 receivers. Some of them arealso presented in Sachl (2012). Here they appear again because (a) they constitute onecollection with the other seismograms and (b) there remain pieces of information whichhave not been mentioned yet, because they have no connection with caustics.
The theory of the first-order ray-based Born approximation in an isotropic medium,with the point-source and high-frequency approximation is described in Sachl (2012),Section 2.
2 Perturbed models and the background model
Perturbed model P1I-10% is contructed using models P1 and P1I. We give a short descrip-tion of models P1 and P1I; for detailed information we refer the reader to Bulant and Mar-takis (2011).
Model P1 is a 2D velocity model situated in a rectangle (0 km, 47.3 km)×(0 km, 6 km).The model is smooth and should be suitable for ray-theory computations. It serves asa background model for the computations using the ray-based Born approximation. P-wave velocity vp in smooth model P1 ranges from 4.64 km/s to 5.93 km/s. S-wave velocityvs = vp/
√3. Density ρ = 1000 kg/m3 everywhere.
Model P1I is quite complicated, because it is composed of 16 blocks separated bysmooth interfaces, see Figure 1. The largest absolute value of the difference between theP-wave velocity in model P1I and the P-wave velocity in smooth model P1 is |∆vp|max ≈0.21km/s. The spatial distribution of the S-wave velocity in model P1I is not shown. Wehave set vs = vp/
√3 in the models which we have constructed from model P1I, analogously
to the background model.
1 2
3 4
5
6 789
1 01 11 21 3
1 41 5
1 6
Figure 1: Blocks in model P1I. The colour of the block is determined by its index. The colour changesfrom green to red and blue. Block 1 is the leftmost green block.
The perturbed model P1I-10% is a 2D isotropic heterogeneous velocity model situatedin rectangle (0 km, 47.3 km) × (0 km, 6 km). The density in the perturbed model is thesame as in the background model and equal to ρ = 1000 kg/m3 everywhere. The S-wavevelocity vs = vp/
√3. The P-wave velocity in the model is equal to the P-wave velocity in
the smooth background model P1 plus 10% of the difference between model P1I and thebackground model. The perturbation is reduced to 10% in order to satisfy sufficiently lowdifferences between the perturbed and the background model. This is the requirement ofthe Born approximation.
The partially perturbed models used in this paper are called P1-1-10%, P1-2-10%, . . . ,P1-16-10%. The only difference between the partially perturbed models and model P1I-10% is that the perturbation is present only in one domain. The domain corresponds to
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“Block j” of model P1I in the case of model P1-j-10%, see Figure 1. The P-wave velocityin each of these domains is again equal to the P-wave velocity in the smooth backgroundmodel P1 plus 10% of the perturbation between model P1I and the background model.
3 Numerical computations
The x1 coordinate is the horizontal coordinate, increasing from the left to the right. Thex3 coordinate is the vertical coordinate, increasing downwards. The origin of the Cartesiancoordinate system is in the upper left corner of the model.
Motivated by Bulant and Martakis (2011) we chose 37 receivers placed at the uppermodel boundary. The first receiver has the horizontal coordinate x1 = 16 km. The spacingbetween the receivers is 0.5 km, therefore, the horizontal coordinate of the last receiver isx1 = 34 km.
The explosive source is also located at the upper model boundary, x1 = 25 km, x3 =0 km. Its position is the same as the position of the 19th receiver. The source timefunction is a Gabor signal with a prevailing frequency of 10 Hz filtered by a frequencyfilter which is non-zero only for frequencies f , 1 Hz < f < 20 Hz. There is a cosinetapering for 1 Hz < f < 2 Hz and 19 Hz < f < 20 Hz while for 2 Hz < f < 19 Hz thefilter is equal to one. Only P waves are considered.
The two-point ray tracing in models P1-1-10%, . . . , P1-16-10% is performed first. Therays from the first elementary wave reflect at the interface reached first. The rays fromthe second elementary wave transmit through the first and reflect at the interface reachedsecond. The ray-theory seismograms correspond to the two-point rays.
The Born seismograms are computed in 2D rectangular grids according to equation (9)in Sachl (2012). The grid intervals are equal to 0.005 km in the directions of the x1 andx2 coordinate axes. The size of the computational grid is chosen in order to cover thedomain with non-zero perturbations in the particular model. The rays are shot into thewhole lower half-plane in computing the Born seismograms. We use the basic system ofrays containing 121 rays, which covers the straight angle into which the rays are shot.The decomposition of the model volume into ray cells on the direct wave is performed bycontrolled initial-value ray tracing. The interpolation within these ray cells is then appliedto computing the values of the required quantities at the gridpoints of the computationalgrid. The description of the algorithm is given in Bulant (1999). We use the appropriateamplitude cut-off applied to the Green function in computing the Born seismograms. Formore details see Sachl (2012).
The vertical and horizontal components of the resulting seismograms together withthe ray diagrams of the first and second elementary wave computed in models P1-1-10%,. . . , P1-16-10% are depicted in Figures 2, . . . , 5, 7, . . . , 18, respectively. The referencesolutions are the ray-theory seismograms. The scale of each set of seismograms computedin one model is given in the caption under the figure depicting the seismogram.
The abscissae in the seismograms denote the travel times of the diffracted waves. Thewaves are diffracted from the edges of the block containing the perturbation of the elasticparameters. The travel times are computed in the background model, because the Bornapproximation uses quantities from the background model. Most blocks in model P1Ihave 4 edges. The colours associated with the edges are in Table 1.
Three blocks are exceptional. Block 3 and Block 16 have only 3 edges, Block 10 has
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Edge left upper left lower right upper right lowerColour green pink yellow blue
Table 1: The edges of the block and the colours of the corresponding abscissae highlighting the traveltimes of the waves diffracted from the edges.
5 edges: In Block 3, there are no diffracted waves (computed by the ray theory) fromthe rightmost edge. The remaining two edges are taken as the left upper and left loweredges. In Block 16, there are no diffracted waves from the leftmost edge. The remainingtwo edges are taken as the right upper and right lower edges. In Block 10, there are nodiffracted waves from the two edges located at the upper model boundary. The two edgesat the right-hand side are taken as the right upper and right lower edges, the remainingedge is taken as the left lower edge.
The strange wavegroups described in Sachl (2012) and the diffractions from the caus-tics are discussed no more.
4 Description of the computed seismograms
4.1 Model P1-1-10%
The vertical and horizontal components of the seismograms are depicted in Figures 2a and2b, respectively. The ray diagrams of the first elementary wave are depicted in Figure 2c.The ray reflects very close to the right lower edge of the block, see Figure 2c. Both thecomponents of the Born seismograms continue with the diffracted waves highlighted bythe blue abscissae. This phenomenon is also observed often in other seismograms.
The waves diffracted from the right upper edge are clearly visible for the receivers be-tween x1 = 16 km and x1 = 22 km in the horizontal component of the Born seismograms.These waves arrive too late for the other receivers. The waves diffracted from the leftupper and left lower edges are not visible. They arrive later. The wave first diffractedfrom these edges does not arrive earlier than in approximately 7.9 s.
See also Sachl (2012).
4.2 Model P1-2-10%
The vertical and horizontal components of the seismograms are depicted in Figures 3aand 3b, respectively. The ray diagrams of the first elementary wave are depicted inFigure 3c. The ray-theory seismograms are non-zero for the receivers between x1 =21.5 km and x1 = 33 km. As regards the diffracted waves, only the waves diffracted fromthe left upper and left lower edges should be visible, similarly as in model P1-1-10%. See,for example, the horizontal component of the seismograms: The last ray, which is incidentat the receiver at x1 = 21.5, reflects very close to the left lower edge, see the ray diagramdisplayed in Figure 3c. The ray-theory seismograms are zero for the receivers betweenx1 = 16 km and x1 = 21 km, but the Born seismograms are non-zero; they smoothlycontinue with the diffractions from the left lower edge highlighted by the pink abscissae.
We observe a triplication of rays in the ray diagram for the receivers between x1 =31.5 km and x1 = 33 km. The rays shot closer and closer to the upper model boundaryare incident more and more to the right, they then return back to the left, and are again
Figure 2: Born (red) and ray-theory (black) seismograms computed in model P1-1-10%. Vertical (a)and horizontal (b) components, scaled by 1× 106. (c) P-wave rays shot from the point source, reflectedat the interface reached first and arriving at the profile of receivers in model P1-1-10%.
Figure 3: Born (red) and ray-theory (black) seismograms computed in model P1-2-10%. Vertical (a)and horizontal (b) components, scaled by 1× 105. (c) P-wave rays shot from the point source, reflectedat the interface reached first and arriving at the profile of receivers in model P1-2-10%.
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incident more to the right. The caustic is present, and we can see the waves diffracted fromthe caustic on the reflected wave in the Born seismograms for the receivers at x1 = 30.5 km,x1 = 31 km and x1 = 33.5 km, x1 = 34 km. These diffracted waves provide a transitionto the shadow zone in the Born seismograms. The effects of caustics on the ray-Bornapproximation are studied in Sachl (2012).
See also Sachl (2012).
4.3 Model P1-3-10%
The vertical and horizontal components of the seismograms are depicted in Figure 4aand 4b, respectively. There are no two-point rays in this model, therefore, there is neithera ray diagram, nor ray-theory seismograms. There are diffracted waves in the Bornseismogram. The wavegroups visible in the seismogram for the receiver at x1 = 16 kmin approximately 1.9 s and for the receiver at x1 = 16.5 km in approximately 1.8 s arediscussed below.
See also Sachl (2012).
4.4 Model P1-4-10%
The vertical and horizontal components of the seismograms are depicted in Figure 5aand 5b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 5c and Figure 5d, respectively. There are waves penetrating the shadowin the Born seismograms, some diffracted waves (note that the reflections visible in theray-theory seismograms, e.g., between x1 = 16 km and x1 = 18 km, have virtually thesame travel times as the diffractions from the right lower edge highlighted by the blueabscissae), but the most interesting are the strong wavegroups in the Born seismogramsfor the receivers between x1 = 16 km and x1 = 19 km.
The Born seismograms computed in the background model contain not only thereflected and the diffracted waves, but also the corrections of the direct waves. Theray diagram of the direct wave computed in the background model is depicted in Fi-gure 6, together with the blocks in model P1I. The rays in Figure 6 travel throughBlocks 10, 12, 14, 15, 16. Thus, we expect the most important contribution to thecorrection of the direct wave in models P1-10-10%, P1-12-10%, P1-14-10%, P1-15-10%,P1-16-10%, because in these models the perturbation of the elastic parameters is non-zeroin some of the listed blocks. This behaviour is really observed, see the seismograms com-puted in these models. It seems that these corrections of the direct waves are generatedalso in models P1-3-10% and P1-4-10%. The corrections are rather weak in the seismo-grams computed in model P1-3-10%, but the positions of these wavegroups correspond tothe already mentioned strong wavegroups in the Born seismograms computed in modelP1-4-10%.
There are no rays crossing Block 3 or Block 4 in Figure 6. All two-point rays areincident at the receivers situated on the right-hand side of the model. However, therewould probably also be rays incident at the receivers situated on the left-hand side ofthe model, if the model was defined for x3 < 0 km. Note that the amplitudes of thewavegroups grow from x1 = 19.5 km to x1 = 16 km in both Figure 4a and Figure 5a. Thisbehaviour is expected as the length of the ray affected by the perturbation of the elasticparameters grows. Our conclusion is that the strong wavegroups observed in the Born
Figure 5: Born (red) and ray-theory (black) seismograms computed in model P1-4-10%. Vertical (a)and horizontal (b) components, scaled by 4× 104. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-4-10%.
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Figure 6: The ray diagram of the direct wave computed in the background model, depicted togetherwith the blocks in model P1I.
seismograms in model P1-4-10% for the receivers between x1 = 16 km and x1 = 19 kmand the wavegroups observed in model P1-3-10% are the corrections of the direct waves.
See also Sachl (2012).
4.5 Model P1-5-10%
The vertical and horizontal components of the seismograms are depicted in Figure 7aand 7b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 7c and Figure 7d, respectively. There are 4 arrivals for the receiversbetween x1 = 16 km and x1 = 20.5 km, 2 arrivals for the receivers between x1 = 21 kmand x1 = 23 km, and 1 arrival for the receivers at x1 = 29 km and x1 = 29.5 km, see theray diagrams. Four waves visible in the Born seismograms x1 = 20.5 km are:
The first two waves are from the first elementary wave. The stronger wave is thereflection from the upper part of the block. The weaker wave is reflected close to the rightupper edge.
The next two waves are from the second elementary wave. The wave which arrivesearlier is the reflection from the right lower interface. This wave continues for the receiversat x1 ≥ 21 km with very weak diffractions from the right lower edge highlighted by theblue abscissae. The wave which arrives later is reflected from the left lower interface. Thiswave continues for the receivers at x1 ≥ 21 km with very weak diffractions from the leftlower edge highlighted by the pink abscissae.
A similar continuation to the shadow zone is present in case of the waves visible atx1 = 29 km and x1 = 29.5 km.
This model is discussed in Sachl (2012).
4.6 Model P1-6-10%
The vertical and horizontal components of the seismograms are depicted in Figure 8aand 8b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 8c and Figure 8d, respectively. The ray-theory seismograms containmainly the rays from the first elementary wave. Only one ray from the second elementarywave is incident at the receiver at x1 = 21 km. There is a stronger wavegroup in theray-theory seismogram for the receiver at x1 = 28.5 km, which is probably a consequenceof the caustic. We also observe diffracted waves in the seismograms, see, e.g., the waveshighlighted by the yellow abscissae in the vertical component of the seismograms forthe receivers between x1 = 21 km and x1 = 28 km. The remaining unexplained wavesdistinguishable between the yellow and blue abscissae for the receivers between x1 = 16 kmand x1 = 23 km could be the reflections from the lower boundary of the computationalgrid. This problem is discussed in Sachl (2011a).
Figure 7: Born (red) and ray-theory (black) seismograms computed in model P1-5-10%. Vertical (a)and horizontal (b) components, scaled by 1× 105. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-5-10%.
Figure 8: Born (red) and ray-theory (black) seismograms computed in model P1-6-10%. Vertical (a)and horizontal (b) components, scaled by 1× 105. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-6-10%.
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4.7 Model P1-7-10%
The vertical and horizontal components of the seismograms are depicted in Figure 9aand 9b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 9c and Figure 9d, respectively. We observe the diffracted waves whichprovide a smooth transition to the shadow zone in the Born seismograms. One example:The single wave visible in the ray-theory seismogram for the receiver at x1 = 21 km isreflected very close to the left upper edge of Block 7, see Figure 9c. The receivers betweenx1 = 16 km and x1 = 20.5 km record the waves diffracted from the left upper edgehighlighted by the green abscissae.
See also Sachl (2012).
4.8 Model P1-8-10%
The vertical and horizontal components of the seismograms are depicted in Figure 10aand 10b, respectively. Quite complicated ray diagrams of the first and second elementarywave are depicted in Figure 10c and Figure 10d, respectively. Only waves from the firstelementary wave occur in the seismograms for the receivers between x1 = 16 km andx1 = 20.5 km. Further, we see that the ray from the second elementary wave is reflectedfrom the lower interface of Block 8 close to the left lower edge and is incident at thereceiver at x1 = 21 km at approximately 2 s. The receivers placed at x1 ≤ 21 km detectvery weak waves diffracted from the left lower edge. The waves are highlighted by thepink abscissae in the Born seismograms. Two waves from the second elementary wave,which reflect from the left interface of Block 8, arrive at the receivers at x1 = 29 km andx1 = 29.5 km at approximately 2.6 s, but they are also weak. The stronger wavegroups arehighlighted by the yellow abscissae in the vertical component of the Born seismograms forthe receivers between x1 = 30 km and x1 = 31 km. These wavegroups are the transitionof the waves reflected close to the right upper edge to the shadow zone.
This model is discussed in Sachl (2012).
4.9 Model P1-9-10%
The vertical and horizontal components of the seismograms are depicted in Figure 11aand 11b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 11c and Figure 11d, respectively. The ray diagrams are quite compli-cated, similarly to the ray diagrams in model P1-8-10%. We observe a triplication on bothelementary waves. The waves diffracted from each of the four edges of Block 9 shouldbe present, but they are not clearly visible. We only observe diffracted waves highlightedby the yellow abscissae in the Born seismograms for the receivers at x1 ≤ 31.5. Thesediffracted waves follow the stronger wave visible in the seismograms at x1 = 31.5 km thatis reflected very close to the right upper edge of the model, see Figure 11b.
See also Sachl (2012).
4.10 Model P1-10-10%
The vertical and horizontal components of the seismograms are depicted in Figure 12aand 12b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 12c and Figure 12d, respectively. The waves from the first elementary
Figure 9: Born (red) and ray-theory (black) seismograms computed in model P1-7-10%. Vertical (a)and horizontal (b) components, scaled by 1× 105. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-7-10%.
Figure 10: Born (red) and ray-theory (black) seismograms computed in model P1-8-10%. Vertical (a)and horizontal (b) components, scaled by 4× 104. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-8-10%.
Figure 11: Born (red) and ray-theory (black) seismograms computed in model P1-9-10%. Vertical (a)and horizontal (b) components, scaled by 1× 104. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-9-10%.
Figure 12: Born (red) and ray-theory (black) seismograms computed in model P1-10-10%. Vertical (a)and horizontal (b) components, scaled by 1×104, except the horizontal components between x1 = 28 kmand x1 = 34 km scaled by 3× 103. P-wave rays shot from the point source: (c) reflected at the interfacereached first; (d) transmitted through the interface reached first, reflected at the interface reached secondand arriving at the profile of receivers in model P1-10-10%.
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wave are visible for the receivers between x1 = 18.5 km and x1 = 27 km; they arrivefirst. There is a triplication on the second elementary wave apparent in the ray diagramin Figure 12d. The triplication is visible in the seismograms for the receivers betweenx1 = 25 km and x1 = 26.5 km at approximately 1.3 s. The rays from the second elementarywave reflected from the right interface of Block 10 arrive at the receivers at x1 = 32 km andx1 = 32.5 km at approximately 2.3 s. The remaining waves in the ray-theory seismogramsare the waves from the second elementary wave.
The Born seismograms coincide well with the ray-theory seismogram in the wholeset of seismograms. However, the Born seismograms are non-zero even in the shadowzone. See, e.g., the wavegroups in the vertical component of the Born seismograms forthe receivers at x1 = 31 km and x1 = 31.5 km highlighted by the blue abscissae. Thesewavegroups are the diffractions from the right lower edge of Block 10. The amplitudes ofthese diffracted waves decrease for the receivers with coordinates x1 ≤ 30.5 km.
The wavegroups arriving first in the Born seismograms for the receivers between x1 =27.5 km and x1 = 34 km are the corrections of the direct waves. These wavegroupsare present in the ray-theory seismograms for the receivers between x1 = 32 km andx1 = 34 km, where the signal is computed by subtracting the direct wave computed inthe background model from the direct wave computed in the perturbed model. The ray-theory and Born corrections coincide well. The travel times coincide perfectly, but somedifferences in the amplitudes can be observed.
See also Sachl (2012).
4.11 Model P1-11-10%
The vertical and horizontal components of the seismograms are depicted in Figure 13aand 13b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 13c and Figure 13d, respectively. The waves from the first elementarywave arrive first in the ray-theory seismograms for the receivers between x1 = 16 kmand x1 = 27 km. There are no waves from the first elementary wave in the ray-theoryseismograms for the receivers at x1 ≥ 27.5 km, but the diffractions from the right upperedge of Block 11 highlighted by the yellow abscissae continue to the shadow zone inthe Born seismograms. The waves from the second elementary wave reflected from theleft interface of Block 11 are detected by the receivers between x1 = 24 km and x1 =27.5 km. These waves arrive last in the ray-theory seismograms and continue in theBorn seismograms with the waves diffracted from the left lower and left upper edges ofBlock 11, see, e.g., the wavegroups highlighted by the green abscissae for the receiversbetween x1 = 22 km and x1 = 23.5 km. The remaining wavegroups in the ray-theoryseismograms are the reflections from the right and lower interface.
4.12 Model P1-12-10%
The vertical and horizontal components of the seismograms are depicted in Figure 14aand 14b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 14c and Figure 14d, respectively. The waves from the first elementarywave are incident at the receivers between x1 = 16 km and x1 = 25.5 km. The wavesfrom the second elementary wave are incident at the receivers between x1 = 20.5 km and
Figure 13: Born (red) and ray-theory (black) seismograms computed in model P1-11-10%. Vertical (a)and horizontal (b) components, scaled by 1× 104. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-11-10%.
Figure 14: Born (red) and ray-theory (black) seismograms computed in model P1-12-10%. Vertical (a)and horizontal (b) components, scaled by 6×103, except the horizontal components between x1 = 26 kmand x1 = 34 km scaled by 1× 103. P-wave rays shot from the point source: (c) reflected at the interfacereached first; (d) transmitted through the interface reached first, reflected at the interface reached secondand arriving at the profile of receivers in model P1-12-10%.
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x1 = 27 km. The receivers between x1 = 20.5 km and x1 = 25.5 km detect the wavesfrom both elementary waves. The waves from the first elementary wave arrive earlier.
The strong wavegroups in the Born seismograms for the receivers between x1 = 26 kmand x1 = 34 km are the corrections of the direct waves similarly as in model P1-10-10%.
The waves diffracted from the left lower edge of Block 12 are observed in the Bornseismograms. The waves are highlighted by the pink abscissae for the receivers betweenx1 = 16 km and x1 = 20 km. The waves diffracted from the other edges of Block 12 arenot visible.
4.13 Model P1-13-10%
The vertical and horizontal components of the seismograms are depicted in Figure 15aand 15b, respectively. The ray diagrams of the first and second elementary wave aredepicted in Figure 15c and Figure 15d, respectively. We recognize the waves from thesecond elementary wave reflected from the left interface. These waves are visible in theseismograms for the receivers between x1 = 22.5 km and x1 = 23.5 km; they arrive afterabout 1.5 s.
It is interesting to note the significant discrepancy between the ray-theory and Bornseismograms for the receivers between x1 = 19.5 km and x1 = 21 km, although the ray-theory and Born seismograms coincide well for the receivers between x1 = 22.5 km andx1 = 24.5 km. The probable explanation is the focusing of rays which is observed for thereceivers between x1 = 19.5 km and x1 = 21 km.
The waves from the second elementary wave continue with the waves diffracted fromthe left lower edge of Block 13 for the receivers at x1 ≤ 22 km, see the green abscissae,and from the left upper edge of Block 13 for the receivers at x1 ≥ 24 km, see the pinkabscissae. The waves diffracted from the right upper and right lower edge of Block 13 arevirtually not observed; they are probably weak. The exceptions are the diffracted waveshighlighted by the yellow abscissae in the horizontal component of the seismograms forthe receivers between x1 = 28 km and x1 = 29 km.
4.14 Model P1-14-10%
The vertical and horizontal components of the seismograms are depicted in Figure 16aand 16b, respectively. The ray diagram of the first elementary wave is depicted in Fi-gure 16c. There are no two-point rays from the second elementary wave. Although theray diagram in Figure 16c is not complex, we observe some discrepancies between theray-theory and Born seismograms. The strong waves in the Born seismograms for thereceivers between x1 = 25.5 km and 34 km are probably the corrections of the directwaves. The ray-theory correction does not appear in too many seismograms, but if itdoes, it is similar to the correction predicted by the Born approximation.
4.15 Model P1-15-10%
The vertical and horizontal components of the seismograms are depicted in Figure 17aand 17b, respectively. The ray diagram of the second elementary wave is depicted inFigure 16c. There are no two-point rays from the first elementary wave. The ray diagramis very simple. There is just one arrival for the receiver at x1 = 33 km.
Figure 15: Born (red) and ray-theory (black) seismograms computed in model P1-13-10%. Vertical (a)and horizontal (b) components, scaled by 4× 103. P-wave rays shot from the point source: (c) reflectedat the interface reached first; (d) transmitted through the interface reached first, reflected at the interfacereached second and arriving at the profile of receivers in model P1-13-10%.
Figure 16: Born (red) and ray-theory (black) seismograms computed in model P1-14-10%. Vertical (a)and horizontal (b) components, scaled by 1× 103. (c) P-wave rays shot from the point source, reflectedat the interface reached first and arriving at the profile of receivers in model P1-14-10%.
Figure 17: Born (red) and ray-theory (black) seismograms computed in model P1-15-10%. Vertical(a) and horizontal (b) components, scaled by 2 × 105, the seismograms between x1 = 31.5 km andx1 = 34 km scaled by 2× 103 km. (c) P-wave rays shot from the point source, transmitted through theinterface reached first, reflected at the interface reached second and arriving at the profile of receivers inmodel P1-15-10%.
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The seismograms for the receivers between x1 = 31.5 km and x1 = 34 km again depictthe corrections of the direct waves. The corrections seem to be weak, but note that thescale is much smaller for the receivers between x1 = 31.5 km and x1 = 34 km than forthe other receivers. The ray-theory seismogram for the receiver at x1 = 31.5 is zero, butthe Born approximation predicts some correction even for this receiver.
We observe the diffracted waves from the left lower, left upper, right lower and rightupper edges highlighted by pink, green, blue and yellow abscissae, respectively.
There are other, not yet discussed, wavegroups in the seismograms. Namely we speakabout some wavegroups visible for the receivers between x1 = 16 km and x1 = 17.5 km.What these wavegroups are is still an opened question.
See also Sachl (2012).
4.16 Model P1-16-10%
The vertical and horizontal components of the seismograms are depicted in Figure 18aand 18b, respectively. The seismograms computed in this model are similar to the seis-mograms computed in the previous model. There are no two-point rays in this model,therefore, there is neither a ray diagram, nor ray-theory seismograms. The Born seis-mograms are non-zero; they contain diffracted waves and corrections of the direct waves.The seismogram for the receiver at x1 = 31.5 km is not shown. The ray-theory seismo-gram contains a huge correction of the direct wave. The reason is that a new ray of thedirect wave, which is incident at this receiver in the perturbed model, is generated. Thissudden change is not correct. The amplitude should change smoothly as in the Bornapproximation.
4.17 Model P1I-10%
At the end of this section we present the seismograms computed in the complete per-turbed model. The vertical and horizontal components of the seismograms are depictedin Figure 18a and 18b, respectively.
5 Discussion
As you see, we understand the main features of the presented seismograms. However,there still remain some unrecognized wavegroups or differences between the ray-theoryand Born seismograms.
One possibility is that the unrecognized wavegroups are the diffracted waves arisendue to the imperfect coverage of the block which contains non-zero perturbations. Fi-gure 20 depicts how the ray shot close to the upper model boundary can bend down-wards. Blocks 2, 10, 12, 14, 15, 16 are not covered perfectly. New incorrect edges andwaves diffracted from these edges are created.
6 Concluding remarks
The Born seismograms are compared with the differences between the ray-theory seismo-grams in heterogeneous 2D models and the background model. We try to identify and
Figure 18: Born (red) and ray-theory (black) seismograms computed in model P1-16-10%. Vertical (a)and horizontal (b) components, scaled by 1 × 106, at x1 = 28.5 km scaled by 1 × 105, at x1 = 29 kmscaled by 1× 104, between x1 = 29.5 km and x1 = 34 km scaled by 1× 103.
Figure 19: Complete P-P scattered wave Born (red) and ray-theory (black) seismograms computed inmodel P1I-10%. Vertical (a) and horizontal (b) components, scaled by 1× 103. Scattering corrections tothe direct P waves are much stronger than the reflected and diffracted waves. Since the amplitude scaleis chosen according to the corrections to the direct waves, most of the reflected and diffracted waves areinvisible.
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Figure 20: Coverage of smooth model P1 with the rays in shooting from the position of the 2nd receiver.The rays are depicted together with the blocks in model P1I.
discuss the individual wavegroups. We claim that the Born seismograms contain, apartfrom the reflected waves, the diffracted waves and the corrections of the direct waves.The ray-theory seismograms computed in the perturbed model are zero in the shadowzone. The Born seismograms are non-zero. The reflected waves often continue with thediffracted waves.
Acknowledgements
First of all I would like to thank Ludek Klimes, who helped me greatly with the workwhich led to this paper. I would also like to thank Petr Bulant for providing models P1and P1I and the history files, which he used for seismogram computation and visualizationof the medium parameters. Hence, if I wanted to compute the reference seismogram orto visualize some quantities, I used parts of these history files (or all the history files) andadjusted them as required.
The research has been supported by Grant SVV-2012-265308, by the Grant Agency ofthe Czech Republic under Contract P210/10/0736, by the Ministry of Education of theCzech Republic within Research Project MSM0021620860, and by the members of theconsortium “Seismic Waves in Complex 3-D Structures” (see “http://sw3d.cz”).
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