Introduction

For the exploration geophysicst, velocity is the key component in creating accurate images of the earth. Velocity is in important component of almost all phases of processing. It used in implicitly or explicitly in Normal Moveout, mulitple attenuation, Dip Moveout, (DMO) and migration Velocity, and the explorationist’s final product, the earth image, also play an important part in the work of other geoscientists. On an active oil field seismic data helps target wells and monitor fluid flow (cite David). It provides rock physicists with reservoir properties (cite Isao??). And geostatisticians and reservoir engineers constraints for their simulations (cite Mao).

With the wrong velocity seismic events do not focus and reflectors are mispositioned. Without an accurate velocity, wells will never tie with the seismic data and direction drilling becomes much more difficult. Without an accurate velocity model seismic data rock property estimates is more likely to hinder rather than help the rock physicist (cite vofz), geostatistician, and reservoir engineer.

Obtaining an accurate velocity estimate is one of the most difficult problems in geophysics. Velocity estimation is a nonlinear, underdetermined problem. The first step in all velocity estimation schemes is to make an asumption(s) about the geology based on some foreknowledge of the region. The most basic assumption is the one used by NMO, a horizontally straified earth. A less stringent assumption, flat reflectors, was made by (?). When neither of these assumptions are sufficient, one is often forced to the family of methods generally refered to as ‘tomography’.

TOMOGRAPHY

Tomography starts from the idea that there is an operator (T_{nl}) that relates slowness (s) to travel times (t),

t D T_{nl}s  (1)

One way to think of this operator is in terms of rays. Figure 1 shows a simple two layered model with sources every .2 km in depth and every .5 km in x. We can think of the operator T_{nl} as the integral of slowness along the ray.

This simplistic view of the tomography has a number of inconsistancies with the reflection tomography problem. The is that we wouldn’t be doing tomography if we actually knew the slowness field before hand. What we actually have is some initial guess s_{0} at the slowness field. If we perform the same experiment using as our initial guess, a constant slowness, Figure 2, we encounter another problem, our raypaths aren’t the same. An alternate way to state this observation is that our operator is dependant on our model and therefore non-linear.

Figure 1: A simple example of tomography with the sources and receivers every .2 km in depth and every .5 km in x. Overlaid are the raypaths connecting the sources to the receivers. cor.overlay [ER]

Figure 2: The same recording geometry as Figure 1, with a constant velocity field. Note that the ray paths are significantly different. init.overlay [ER] Non-linear problems

Non-linear problems are much more difficult to solve than linear problems. A method that is often successful is to make some approximations that will turn the non-linear problem into a linear problem. We can make this conversion by doing a Taylor expansion (ignoring second and higher order terms) around our initial guess at the slowness field,

t ≈ T_{0}s_{0}C ∇T_{0}1s
t ≈ t_{0}C T_{0}1s  (2)
1t ≈ T_{0}1s

where s_{0} is our initial guess at slowness, T_{0} is the Frechet derivative, a linear approximation of T_{nl} at s_{0}, 1s is the change in slowness, t_{0} are the modeled travel times by applying T_{0} to s_{0}, 1t are the difference between the modeled travel times, t_{0}, and the measured travel times, t. Reflection seismic tomography problems are to large to do direct matrix inversion so iterative methods, such as conjugate gradient, are used instead. Once we have inverted for 1s, we update our slowness estimate,

s_{1} D s_{0}C 1s.  (3)

We can then re-linearize around this new model (s_{1}), constructing a new tomography operator T_{1}.

Unfortunately are linearization does not share many of the properties of its fully linear equivalent. First, the step length calculation that is valid in the linear problem is not valid in the non-linear problem, instead we must take into account are linearization point when calculating the step size. Second, and more importantly, our problem isn’t guaranteed to converge. To see why lets return to our simple example. Our operator is based on our initial slowness estimate, which is in error. In the case of the ray example, this means that we are back projecting along our guess at the ray position, not there true positions. As a result we will introduce changes in slowness in the incorrect position.

Figure 3 show the velocity estimate after applying linearized tomography. Generally we have moved towards the correct velocity model, Figure 1, but we can also see the effects of linear approximation. In the upper,right portion of Figure 3 the velocity has been incorrectly increased. If we look at the ray connecting (z D 1.6,x D 0) and (z D 0,x D 9.5) in Figures 2 and 1 we can begin to see why. Our guess at the raypath indicates that we spend approximately 30% of the length in the lower layer while in fact 50% of the rays length is spent in the lower layer. The inversion has left over travel time errors which it accounts for by increasing the velocity in the upper portion of the model.

The above is an example of transmission tomography, reflection tomography has another major source of non-linearity, the reflection point is unknown. Figure ?? shows a synthetic anticline model. Overlaid in solid lines are the correct reflector position and the correct raypaths to these reflector position. The dashed lines are the reflector positions and raypaths that would be estimated using a s(z) initial slowness model. Note that our guess at the reflector position is in significant error, making our guess at the raypaths, and the corresponding tomography operator in greater error. In addition in traveltime error can be accounted for by either shifting Figure 3: The tomography result using the slowness model and raypaths of Figure 2 as our initial estimates. first.tomo [ER]

the reflector, changing the slowness field, or some combination of the two. The result is that our reflection tomography is much more likely to get stuck in local minima and maxima than its transmission counterpart.

The possibility of converging to a local minima or even maxima is well known. The general solution is to try to get to as accurate an initial model as possible (to move our initial point as close the global minima as possible). This is usually done through conventional velocity analysis methods such as semblance analysis and interval velocity estimation by Dix or some derivation on Toldi’s (?) approach. In addition, smaller step sizes are used to avoid going outside the valid range of our linear approximation. Both of these approaches have potential negative consequences. A smaller step size means more expensive non-linear iterations. And conventual velocity estimation often does not provide a good initial estimate and might in fact lead us to a local maxima.

Tomography and null space

If our model space is cartersian mesh or somthing similar we face another problem when doing reflection tomography, its large null space. There are several reasons for this null space:

shadow zones -Certain parts of the velocity model simply might not be illuminated, or illuminated poorly by our acquisition geometry. A classical example is under the edge of a salt body (?).

limited angle coverage Velocity information is contained in higher reflection angle data. At larger depths, given our limited surface recording geometry our angle coverage will decrease (Figure 5). As a result larger depth model components after often under or unilluminated.

Figure 4: The solid lines show raypath through the correct velocity and the correct reflector position. The dashed lines are raypaths through our initial model and our initial reflector positions. Note that our estimated raypaths have significant error, therefore our tomography operator will have significant error. rays [ER]

not enough equations In most reflection tomography methods we invert a limited set of positioning errors. When doing tomography, especially 3-D tomography, it is possible to have more model points than moveout errors.

To deal with the null space in of the reflection tomography operator we have two alternative paths we could follow, reparameterization or regularization.

Reparamterization involves using something other than regular Cartesian mesh to represent our model space. It usually takes the form of either a layered model with simple velocity function (such as velocity gradient) within the layers or reparamterizing the model in terms of a number of splines nodes. The layered model approach is attractive because it allows a direct link to be kept with a interpeter’s geologic model. It also provides a fairly accurate description of velocity structure in regions such as the North Sea. The spline node approach can also be beneficial. It can guarantee the velocity model will be smooth (necessary for ray based methods) and with selective placement of nodes can allow less freedom in areas where the data provides information about the svelocity field. The downside of both of these approaches is that the parameterization most be chosen a priori. At early iterations our guess at the velocity model can be in significant error. If we choose a poor job of parameterizing the model we could actually hurt or stop the problem from converging (?).

The second option is regularization. In the regularization approach we add a second term to our objective function,

Q(1s) D ∥1t − T_{0}1s∥^{2}C ϵ^{2}∥A1s∥^{2},  (4)

Figure 5: Given a fixed recording geometry (s being the source and g_{m}ax being the farthest receiver) as we go deeper in depth the angle range that we at each reflector decreases. ray-limit [NR]

where A penalize some definition of roughness in our model. The regularization approach has its drawbacks. The largest is that our model space no longer has any direct connection with our concept of geology. In addition choosing A is problematic and can introduce unrealistic elements into our slowness estimate.

OVERVIEW OF THESIS

The goal of this thesis is to present two different techniques that can help overcome reflection tomography’s problems with nonlinearity and null space.

Tau tomography

In Chapter ?? I introduce the concept of tau (vertical travel-time) tomography and derive the operator relating changes in slowness to moveout errors in tau space. I show how tau tomography is less sensitive to our initial guess at reflector position and slowness estimate. Figure 6 shows the model of Figure ?? with the correct and initial reflector positions in depth (left) and tau (right).

On a synthetic I show how performing tomography in tau rather than depth better constrains our slowness changes. I show that the tau tomography problem converges significantly quicker and to a more reasonable result than its depth counterpart. The resulting migration are better focused and the reflectors are better positioned by doing tau rather than depth tomography.

Figure 6: Reflector movement in depth (left) and tau (right). The solid lines are where there reflectors migrating using the initial v(z) field. The dashed lines are the correct reflector positions in tau and depth. Note that correct and initial positions are much close in the tau case. ref-movement [ER]

Steering filters

In Chapter ?? I introduce a new way to regularize our tomography estimate that takes into account our a priori estimate of geology. I start from a simple missing data problem. I show the geophysicist estimate and use and the model covariance matrix to solve the missing data problem. I show how nonstationarity makes both solutions suboptimal. I then introduce another way to approximate model covariance, steering filters. I show how to build steering filters with as little information as early migrated reflector position. For example if we know the dip of the second layer in the model of Figurer̃effig:init.overlay we can do a better job constraining the slowness field using steering filter regularization, Figure 7.

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Using steering filters I develop a new tomography objective function. I then compare and contrast the results of using steering filters to regularize our tomography problem versus an isotropic regularizer in both depth and tau. I show that the most reasonable velocity model, and the best migrated image comes from using steering filters in the tau domain.

Non-stationary filter regularization

When dealing with discontinuties in the velocity field steering filters aren’t ideal representation of the model covariance. In these cases a better representation of the model covariance can be obtained by using non-stationery Predict Error Filters (PEF). In Chapter ?? I show how Figure 7: The same tomography shown in Figure 3 now using steering filters to regularize the velocity estimate. reg.tomo [ER]

to construct and estimate a non-stationary PEF. I show it does an excellent job dealting with discontinuties and is as effective as steering filters when applied to smooth models.

Field tests

In Chapter ?? I apply the method to a 2-D marine enviornment.

Appendices

In the appendices I try to address several issues that I have mentioned briefly in this chapter, but not adequately dealt with in the succeeding chapters. In Appendix ?? I show that doing a full line search rather than using conjugate gradient criteria isn’t necessary because of regularization fitting goal. In Appendix ?? I show how parameterizing our model in terms of splines can be effective, but relies on choosing the correct parameterization. And finally in Appendix ?? I show how we can produce error bars in our velocity estimate.

ASSUMPTIONS AND LIMITATIONS

The largest weakness of the method is what tries to overcome, the non-linearity of tomography. When we are far from the correct solution and/or we have inappropriate view of the model covariance matrix the method is susceptible to non-convergence, or convergence to an unreasonable model.

The ray based tomography used in this thesis assumes that the ray tracing high frequency approximation, that the model varies smoothly over a wavelength, is valid for the dataset.

When the macro model is well defined this is a fairly valid solution, when we are far from the correct velocity it can easily break down.

The steering filters of Chapter ?? assume that we can adequately describe the model covariance with steering filter. Often velocity does not follow dip, and the velocity gradient for the region would need to be used instead of early migrated reflector position. In addition steering filters can not describe a complex media’s covariance if the slow varying function like velocity this is not a concern.

This thesis assumes that no anisotropy exists in the data. In theory it is not hard to incorporate anisotropy into the velocity estimation process in this thesis, in fact the method presented is great potential for anisotropic estimation. The reflector stability of the tau domain method would lead to much better ability to constrain our η or equivalent anisotropic parameter (?). In addition steering filters, with there ability spread information along layers, are ideal for defining anisotropic layers.