Time preserving tomographyLior Liram, Yoav Naveh*, Gali Dekel, Zvi Koren, Paradigm

Summary

Time Preserving Tomography is an accurate, efficient and flex-ible tool for constructing kinematically equivalent subsurfacemodels given a background model and a set of model param-eter perturbations. In this method, parameters of the back-ground model are allowed to change while preserving travel-times of all ray pairs. Perturbations can be applied for all typesof model parameters. In the case of tilted transverse isotropy(TTI) the model parameters are the axial compressional ve-locity, Thomsen anisotropic interval parameters epsilon anddelta, and the depth values of the model horizons. The travel-times of all ray pairs traced during the tomographic inversionare kinematic invariants. In the migrated domain, all the kine-matically equivalent models should provide more or less flatreflection events along common image gathers (CIG). In fact,time-preserving tomography does not require CIG at all. Anexample of the application of this type of method is the useof misties between well markers and seismic depth horizonsto obtain Thomsen delta parameters (Mancini, 2013). Time-preserving tomography is a very useful tool for depth interpre-tation, uncertainty analysis and risk management.

Introduction

Subsurface geological models derived from seismic imagingare not unique. Many combinations of subsurface geologicalmodel representations and model parameters can be used tosatisfy the imaging conditions and flatten common image gath-ers. This is mainly due to the limitations of seismic data acqui-sitions, in particular the limited offset between sources and re-ceivers and the lack of azimuth coverage. Other factors for thenon-uniqueness are finite frequency band, noise, poorly illumi-nated zones and limitations in the asymptotic high-frequencyray method and theory of wave propagation used (e.g. not ac-counting for factors like the “real” anisotropy, dispersion fromattenuation, scattering, etc.). The process of deriving a sub-surface anisotropic velocity model from seismic data is verydemanding, normally requiring massive computational powerand intensive human involvement. This non-uniqueness prob-lem demonstrates the motivation and need for the followingprocesses:

1. Integration of additional external information into the modelbuilding procedures in order to reduce the ambiguity / non-uniqueness. The most prominent example of this is wellmarkers that are used to constrain the depths of the seismichorizons.

2. Analysis of equivalent subsurface models that differ fromthe current model by small perturbations in the model pa-rameters. Starting with a given background model, we wantto define small changes in one or more of the model param-

eters, and construct a new model that respects these smallchanges, while keeping the rays traveltimes unchanged (i.e.fully in agreement with the recorded seismic data). Such aconstruction is natural in the interpretation process (askinginterpretative queries like, “What are the location changesof the interface horizons due to given perturbations of theaxial compression velocity and Thomsen parameters?”).

Figure 1: Background subsurface velocity model and modelperturbation maps. The layer above the deepest horizon wasassigned a laterally variant depth perturbation map (mistiemap). Second formation from the top was assigned a constantvelocity perturbation of −200m/s.

We present the concept of Time Preserving Tomography - anaccurate and efficient tool for constructing a kinematically equiv-alent subsurface model given an initial background model anda set of model perturbations (which we call constraints), whilekeeping the total traveltime along specular incident and re-flected rays traveling through the subsurface model unchanged.Under the assumptions of tilted transverse isotropy (TTI), bymodel perturbations we mean small changes in the axial com-pressional velocity, anisotropic interval parameters epsilon anddelta, and the depth values of the model horizons. In TimePreserving Tomography we first calculate a set of kinematiccharacteristics inherent to the background model. These kine-matic characteristics are computed on a relatively coarse grid(the tomographic inversion grid) by calculating the discretetraveltime integrals along specular rays shot in wide openingangles and all azimuths through our background anisotropicmodel. This results in a set of linear homogeneous equations(the right hand side of the system describes traveltime errorswhich are set to zero in this case). The kinematic characteris-tics are the coefficients of the above-mentioned homogeneous

Time preserving tomography

system. These coefficients are stored in a matrix called the in-fluence matrix. In the second stage we add the imposed modelconstraints (perturbations) to the influence matrix and solvethe system using the Conjugate Gradient algorithm.

Key features of our system include:

Accuracy - the perturbed models preserve the total traveltimeof all rays travelling through the model. In particular, normalincident traveltimes are preserved, while vertical time is gen-erally not.Flexibility - the system can receive perturbations for differentmodel parameters for each subsurface geological layer. Theperturbations are defined as layer parameter maps which canvary laterally for each subsurface layer.User control - the user has full control over the inversion pro-cess. The constraints are added as soft constraints; that is, wedon’t set any variables to a specific value, but rather introducethe constraints as additional equations in our system, and findthe best solution to the entire system in a least squares sense.The user can control the weight he/she wishes to assign to theconstraints relative to the influence matrix. The user can definethe reflection angle range for which traveltimes will be pre-served. Accurate depth-to-depth conversion can be achievedby using only small angles around the Normal Incidence rays.A wider range (up to 30 degrees half-opening angles) may beused to convert isotropic models into anisotropic models usingadditional information about well-based depth misties.Efficiency - the most time-consuming step is the first one, thatis run only once for a given model. The specific constraints aredefined and used only in the second part, which is fast (orderof minutes for medium-size surveys). Therefore, the user canrun several scenarios quickly to generate a series of equivalentmodels.

Theory

Ray-based residual traveltime tomography in general and time-preserving tomography in particular, is based on solving a verylarge, over-determined system of linear equations. These lin-ear equations can be viewed as a set of linear constraints. Intime-preserving tomography we distinguish between two typesof linear constraints: (a) imposing zero traveltime errors alongthe traced rays and (b) explicitly setting model parameter per-turbations for either (or both) horizons’ depths and anisotropicvelocity parameters. Solving this system of linear equationswill result in an updated model that simultaneously satisfiesboth types of constraints.

Preserving TraveltimesIn ray-based reflection tomography (Kosloff et al., 1996; Wood-ward et al., 1998; Billette and Lambare, 1998) an influence ma-trix is constructed, relating variations in medium parameters toresidual traveltimes computed from residual moveouts (RMO)measured along CIG. In the case of time-preserving tomogra-phy, the assumption is that the starting point is a model whichis consistent with the short offset data at least, and possiblywith longer offset events. The data consistency can be viewedand defined with respect to the range of flattened events withinthe CIGs.

We use a model-based approach in which the subsurface modelis defined by a set of geological layers separated by curvedinterfaces and faults. Our 3D grids for the model update pa-rameters and for the shooting points for fans of ray pairs areregular in the horizontal directions, and irregular in the verti-cal direction - the grid points coincide with the structural in-terfaces. The updated model parameters are represented on acoarse grid. Rays are shot from a finer horizontal grid in whichpairs of incident and reflected rays are shot with respect to theNormal direction. The traveltime variation with respect to themodel parameters of each ray pair is given by

0 = δ t =∑

i

(AiV ∆V i +Ai

ε ∆εi +Ai

δ∆δ

i +Aiz∆zi) (1)

with i an index running on all model nodes. ∆V,∆ε,∆δ ,∆zare the variations (updates) of the axial compressional veloc-ity, Thomsen parameters and the horizon depth values respec-tively. Ai

V , Aiε , Ai

δ, Ai

z are the components of the influence ma-trix and are given by

Aim =

∂ t∂mi (2)

with t the total ray pair traveltime and m = V,ε,δ ,z. Each raypair produces such an equation; together they define an over-determined system of linear equations (the number of rays isconsiderably larger than the number of model nodes). Thiscould be written in a matrix-vector notation,

AV ∆V +Aε ∆ε +Aδ ∆δ +Az∆z = 0 (3)

This equation represents our first type of traveltime preserva-tion constraints. It can be brought to an even simpler form ifwe define

∆m =

∆V∆ε

∆δ

∆z

, A =(

AV Aε Aδ Az)

; A∆m = δ t

(4)If the total number of ray pairs shot was NR and the number ofmodel parameters is Nm then A is a NR×Nm matrix.

Introduction of Model PerturbationsIn order to obtain a desired value mi of a model parameter atnode i, we define a perturbation to the background value mi

Cim = mi−mi (5)

Our second type of constraint is simply the demand that

∆mi = Cim (6)

We may introduce such an equation for each model parame-ter that we wish to modify to a desired value. Our completesystem of equations is (4) combined with all the constraints oftype (6). However, if the number of such constraints is Nc, it isnecessary (but generally not sufficient) that Nc < Nm to solvethe combined system of equations.

One may notice that the model constraints are of different unitsthan (4). We therefore scale equations (6) to the same units bymultiplying them by a scale factor Si

m,

Sim ·∆mi = Si

m ·Cim (7)

Time preserving tomography

For reasons to be shown, we choose this factor to be

Sim =

√∑i

(A ji

m

)2(8)

where A jim is the j-th row element of A that correspond to node

i and model parameter of type m.We also define

Dim = Si

m ·Cim (9)

Now the second type of constraints can also be written in amatrix-vector notation

S∆m = D (10)

For convenience, we define matrix S as a Nm×Nm diagonalmatrix, with zero entries for parameters for which no con-straint was defined. Vector D is defined in a similar manner.

Solution of the SystemNow both matrix S and A multiply the same vector in equations(4, 10) and have the same units. Note that since ST = S, thecombined least squares equation reads(

AT A+βS2)∆m = β ·S ·D (11)

where we also introduce an additional (positive) constant β

that can control the relative weight of the model constraints vs.the traveltime constraints. Our choice of S (8) ensures us thatall non-zero diagonal elements of S2 are equal to those of AT A.Thus when setting β = 1, the solution of (11) will equally re-spect the two types of constraints. For any other choice of β ,the solution will be biased in favor of one type of constraints.The system is over-determined and requires additional regular-ization terms. The final equation reads(

AT A+βS2 +CM +LT L)

∆m = β ·S ·D (12)

with CM a damping term and LT L a structural smoothing op-erator. This system of linear equations is solved using theConjugate-Gradient algorithm.

Example

As an example we used a synthetic dataset of a VTI layeredmodel (Figure 1). It consists of 6 layers, most of which arerelatively thin (∼ 200m). Our initial model in this case is thecorrect one (i.e. the model used for generating the syntheticseismic data). The velocity and anisotropy interval parametersare constant per layer (apart from some smoothing). Pertur-bations were specified for two of the layers. The layer abovethe deepest horizon was assigned a laterally variant depth per-turbation map with peak values of −180m, while the secondlayer from the top was assigned a constant velocity pertur-bation of −200m/s. Time-Preserving Tomography was ap-plied, allowing model parameters of all types to vary (veloc-ity, depth, ε and δ )1. In Figure 2 we display the velocityupdated by time-preserving tomography (2(a)) compared tothe initial velocity (2(b)). It is evident that the deepest hori-zon was pushed up and that the velocity in the second layerfrom the top has decreased. The velocity update in the second

1More accurately, the difference ε−δ was kept fixed.

layer exactly matches the input perturbation (−200m/s) whilethe bottom horizon depth updates match the perturbations by80%−100%. Since perturbations were specified only for twolayers, all other layers received compensating updates so thattraveltimes will be preserved. Also shown are seismic imagesobtained from Kirchhoff PSDM VTI migration using the up-dated velocity and anisotropy volumes (2(c)), and from mi-grating with the true initial model (2(d)). The updated horizonmaps are displayed on top of the image and colored accordingto depth updates. The updated maps are the result of adding thedepth updates to the initial horizon maps. They are clearly con-sistent with the seismic image. In Figure 3 we show migratedgathers that were created using the initial model (3(b)) and theupdated model (3(a)). The gathers’ lateral position is the cen-ter of the dome-shaped structure where the largest model per-turbations were given (and consequently, largest updates). Allevents remained flat with their vertical position shifted. Thisindicates that the model is consistent not only with short offsetdata (as the stack image suggests) but also with far offset data(i.e. all rays’ traveltimes are preserved). Therefore, we maycall the updated model a true equivalent of the initial model.

Conclusions

We present in this paper a practical method for incorporat-ing diverse external information into the velocity model up-dating process, and for generating alternative anisotropic ve-locity models that explain the measured seismic data equallywell. Our results show that the tomographic inversion is sta-ble, accurate and remains loyal to the seismic data. Its speed ofcomputation and flexibility make it a good candidate to be usednot only in the process of velocity model updating, but also inthe interpretation phase as a tool for analyzing the correctnessof the current interpretation and for risk management.

Time preserving tomography

(a) (b)

(c) (d)

Figure 2: Comparison of the background and updated velocity model and their resultant seismic images. The background velocitymodel is shown in (b) on top of which are the model horizon maps. Shown in (a) is the updated model which is a result of time-preserving tomography. The updated horizon maps are also a direct result of tomography (obtained from {∆zi}). A seismic imagemigrated using the background velocity model is shown in (d). An image migrated using the updated model is shown in (c). Themodel horizon maps are displayed on top of the image and colored according to depth updates. Clearly the updated model isconsistent with the seismic data.

(a) (b)

Figure 3: Depth migrated gathers using original velocity model (b) and updated velocity model from time-preserving tomography(a). The gathers correspond to the center of the dome-shaped structure (see Figure 2) where the largest model perturbations weregiven. All events remained flat with their vertical position shifted.

Time preserving tomography

REFERENCES

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