Empty bins and inverse interpolation

WELLS NOT MATCHING THE SEISMIC MAP

Accurate knowledge comes from a well, but wells are expensive and far apart. Less accurate knowledge comes from surface seismology, but this knowledge is available densely in space and can indicate significant trends between the wells. For example, a prospective area may contain 15 wells but 600 or more seismic stations. To choose future well locations, it is helpful to match the known well data with the seismic data. Although the seismic data is delightfully dense in space, it often mismatches the wells because there are systematic differences in the nature of the measurements. These discrepancies are sometimes attributed to velocity anisotropy. To work with such measurements, we do not need to track down the physical model, we need only to merge the information somehow so we can appropriately map the trends between wells and make a proposal for the next drill site. Here we consider only a scalar value at each location. Take $\bold w$ to be a vector of 15 components, each component being the seismic travel time to some fixed depth in a well. Likewise let $\bold s$ be a 600-component vector each with the seismic travel time to that fixed depth as estimated wholly from surface seismology. Such empirical corrections are often called ``fudge factors''. An example is the Chevron oil field in Figure 3.8.

wellseis
Figure 8. Binning by data push. Left is seismic data. Right is well locations. Values in bins are divided by numbers in bins. (Toldi)

The binning of the seismic data in Figure 3.8 is not really satisfactory when we have available the techniques of missing data estimation to fill the empty bins. Using the ideas of subroutine mis1() , we can extend the seismic data into the empty part of the plane. We use the same principle that we minimize the energy in the filtered map where the map must match the data where it is known. I chose the filter $\bold A = \nabla'\nabla=-\nabla^2$ to be the Laplacian operator (actually, its negative) to obtain the result in Figure 3.9.

misseis
Figure 9. Seismic binned (left) and extended (right) by minimizing energy in $\nabla^2 \bold s$.

Figure 3.9 also involves a boundary condition calculation. Many differential equations have a solution that becomes infinite at infinite distance, and in practice this means that the largest solutions may often be found on the boundaries of the plot, exactly where there is the least information. To obtain a more pleasing result, I placed artificial ``average'' data along the outer boundary. Each boundary point was given the value of an average of the interior data values. The average was weighted, each weight being an inverse power of the separation distance of the boundary point from the interior point.

Parenthetically, we notice that all the unknown interior points could be guessed by the same method we used on the outer boundary. After some experience guessing what inverse power would be best for the weighting functions, I do not recommend this method. Like gravity, the forces of interpolation from the weighted sums are not blocked by intervening objects. But the temperature in a house is not a function of temperature in its neighbor's house. To further isolate the more remote points, I chose weights to be the inverse fourth power of distance.

The first job is to fill the gaps in the seismic data. We just finished doing a job like this in one dimension. I'll give you more computational details later. Let us call the extended seismic data $\bold s$.

Think of a map of a model space $\bold m$ of infinitely many hypothetical wells that must match the real wells, where we have real wells. We must find a map that matches the wells exactly and somehow matches the seismic information elsewhere. Let us define the vector $\bold w$ as shown in Figure 3.8 so $\bold w$ is observed values at wells and zeros elsewhere.

Where the seismic data contains sharp bumps or streaks, we want our final earth model to have those features. The wells cannot provide the rough features because the wells are too far apart to provide high spatial frequencies. The well information generally conflicts with the seismic data at low spatial frequencies because of systematic discrepancies between the two types of measurements. Thus we must accept that $\bold m$ and $\bold s$ may differ at low spatial frequencies (where gradient and Laplacian are small).

Our final map $\bold m$ would be very unconvincing if it simply jumped from a well value at one point to a seismic value at a neighboring point. The map would contain discontinuities around each well. Our philosophy of finding an earth model $\bold m$ is that our earth map should contain no obvious ``footprint'' of the data acquisition (well locations). We adopt the philosopy that the difference between the final map (extended wells) and the seismic information $\bold x=\bold m-\bold s$ should be smooth. Thus, we seek the minimum residual $\bold r$ which is the roughened difference between the seismic data $\bold s$ and the map $\bold m$ of hypothetical omnipresent wells. With roughening operator $\bold A$ we fit

$$\bold 0\quad\approx\quad \bold r \eq \bold A ( \bold m - \bold s ) \eq \bold A \bold x$$ (12)

along with the constraint that the map should match the wells at the wells. We could write this as $\bold 0 = (\bold I-\bold J) ( \bold m - \bold w )$. We honor this constraint by initializing the map $\bold m = \bold w$ to the wells (where we have wells, and zero elsewhere). After we find the gradient direction to suggest some changes to $\bold m$, we simply will not allow those changes at well locations. We do this with a mask. We apply a "missing data selector" to the gradient. It zeros out possible changes at well locations. Like with the goal (3.7), we have

$$\bold 0\quad\approx\quad \bold r \eq \bold A \bold J \bold x + \bold A \bold x_{\rm known}$$ (13)

After minimizing $\bold r$ by adjusting $\bold x$, we have our solution $\bold m = \bold x + \bold s$.

Now we prepare some roughening operators $\bold A$. We have already coded a 2-D gradient operator igrad2 . Let us combine it with its adjoint to get the 2-D laplacian operator. (You might notice that the laplacian operator is ``self-adjoint'' meaning that the operator does the same calculation that its adjoint does. Any operator of the form $\bold A'\bold A$ is self-adjoint because $(\bold A'\bold A)'=\bold A'\bold A''=\bold A'\bold A$. )

user/fomels/laplac2.c

void laplac2_lop(bool adj, bool add, 
		 int np, int nr, float *p, float *r)
/*< linear operator >*/
{
    int i1,i2,j;

    sf_adjnull(adj,add,np,nr,p,r);

    for (i2=0; i2 < n2; i2++) {
	for (i1=0; i1 < n1; i1++) {
	    j = i1+i2*n1;
	    if (i1 > 0) {
		if (adj) {
		    p[j-1] -= r[j];
		    p[j]   += r[j];
		} else {
		    r[j] += p[j] - p[j-1];
		}
	    }
	    if (i1 < n1-1) {
		if (adj) {
		    p[j+1] -= r[j];
		    p[j]   += r[j];
		} else {
		    r[j] += p[j] - p[j+1];
		}
	    }
	    if (i2 > 0) {
		if (adj) {
		    p[j-n1] -= r[j];
		    p[j]    += r[j];
		} else {
		    r[j] += p[j] - p[j-n1];
		}
	    }
	    if (i2 < n2-1) {
		if (adj) {
		    p[j+n1] -= r[j];
		    p[j]    += r[j];
		} else {
		    r[j] += p[j] - p[j+n1];
		}
	    }
	}
    }

Subroutine lapfill() is the same idea as mis1() except that the filter $\bold A$ has been specialized to the laplacian implemented by module laplac2 .

user/fomels/lapfill.c

void lapfill(int niter   /* number of iterations */, 
	     float* mm   /* model [m1*m2] */, 
	     bool *known /* mask for known data [m1*m2] */)
/*< interpolate >*/
{
    if (grad) {
	sf_solver (igrad2_lop, sf_cgstep, n12, 2*n12, mm, zero, 
		   niter, "x0", mm, "known", known, "end");
    } else {
	sf_solver (laplac2_lop, sf_cgstep, n12, n12, mm, zero, 
		   niter, "x0", mm, "known", known, "end");
    }
    sf_cgstep_close ();
}

Subroutine lapfill() can be used for each of our two problems, (1) extending the seismic data to fill space, and (2) fitting the map exactly to the wells and approximately to the seismic data. When extending the seismic data, the initially non-zero components $\bold s \ne \bold 0$ are fixed and cannot be changed.

The final map is shown in Figure 3.10.

finalmap
Figure 10. Final map based on Laplacian roughening.

Results can be computed with various filters. I tried both $\nabla^2$ and $\nabla$. There are disadvantages of each, $\nabla$ being too cautious and $\nabla^2$ perhaps being too aggressive. Figure 3.11 shows the difference $\bold x$ between the extended seismic data and the extended wells. Notice that for $\nabla$ the difference shows a localized ``tent pole'' disturbance about each well. For $\nabla^2$ there could be large overshoot between wells, especially if two nearby wells have significantly different values. I don't see that problem here.

My overall opinion is that the Laplacian does the better job in this case. I have that opinion because in viewing the extended gradient I can clearly see where the wells are. The wells are where we have acquired data. We'd like our map of the world to not show where we acquired data. Perhaps our estimated map of the world cannot help but show where we have and have not acquired data, but we'd like to minimize that aspect.

A good image of the earth hides our data acquisition footprint.

diffdiff
Figure 11. Difference between wells (the final map) and the extended seismic data. Left is plotted at the wells (with gray background for zero). Center is based on gradient roughening and shows tent-pole-like residuals at wells. Right is based on Laplacian roughening.

To understand the behavior theoretically, recall that in one dimension the filter $\nabla$ interpolates with straight lines and $\nabla^2$ interpolates with cubics. This is because the fitting goal $\bold 0 \approx \nabla \bold m$, leads to ${\partial \over\partial \bold m'} \bold m'\nabla'\nabla \bold m = \bold 0$ or $\nabla'\nabla \bold m = \bold 0$, whereas the fitting goal $\bold 0 \approx \nabla^2 \bold m$ leads to $\nabla^4 \bold m = \bold 0$ which is satisfied by cubics. In two dimensions, minimizing the output of $\nabla$ gives us solutions of Laplace's equation with sources at the known data. It is as if $\nabla$ stretches a rubber sheet over poles at each well, whereas $\nabla^2$ bends a stiff plate.

Just because $\nabla^2$ gives smoother maps than $\nabla$ does not mean those maps are closer to reality. This is a deeper topic, addressed in Chapter . It is the same issue we noticed when comparing figures 3.3-3.7.

Empty bins and inverse interpolation