Part II: A step by step guide to learning and mastering depth conversion in geophysics.
In Part I of this guide to the important subject of depth conversion we looked at the parameters involved. We now look at the steps in the process.
What Are The Main Steps in Depth Conversion
The key steps involved in depth conversion are:
Determine layer scheme
QC input data
Compare methods within layers
Depth convert
Residual tie to wells
1. Determine layer scheme
The choice of layering will be determined by local geological knowledge. Layers generally are comprised of rocks of similar ages, lithologies or burial history. The boundaries for the layers are often chosen because they are important geological markers or major reflection events on the seismic. More layers are not necessarily better; sometimes simple single layer models work well. It depends on whether lateral velocity variations are present and represented by horizons that can be picked on seismic.
2. QC input data
Figure 1a: A plot of well depth and seismic time shows one point (highlighted red) as an outlier. This could be an unusual velocity.
Quality control of the input data is important for depth conversion. Horizons should be picked on a consistent seismic event as far as possible: any mis-tie at wells is then representative of the true uncertainty in the depth conversion. Note that it is a poor strategy to force seismic horizon picks to match well markers in time, as then the QC information is hidden.
When using wells, the first step is to compare the calibrated velocity log times from logs to the seismic times picked as interpreted horizons. By cross-plotting the two, any mis-ties can be identified and corrected, with outliers possibly indicating potential issues (Figure 1).
3. Compare methods within layers
Figure 1b: Plotting well and seismic times shows the outlier is still present and so must be a mis-pick.
Having checked the input data and layering then the usual approach is to test different methods in the layers, working from the top down. By fitting functions of different types (or using seismic velocities), the methods are compared by examining the depth residuals both within the layer and at the base of several layers as the depth conversion proceeds. The residuals are the difference between the depth predicted by the velocity model using the seismic time at the well marker and the actual depth observed in the well. A small mean residual difference (indicating an unbiased depth conversion) and a low standard deviation (meaning the method explains the depth quite well) are indications of a good depth conversion model. The spatial distribution of residuals should not show a strong trend, which would indicate a velocity variation unexplained by the model. The residuals are also a direct measure of the depth conversion uncertainty at any horizon.
4. Depth convert
The depth conversion itself is simply a matter of multiplying out the various functions and stacking up the layers to get the depth at each horizon. At this stage a depth converted surface is arrived at, which is optimal but may not tie the wells exactly.
5. Residual tie to wells
Finally, the depth residuals can be gridded up in order to arrive at depth converted surfaces that tie wells – useful for many purposes, including drilling prognosis and reservoir modelling. However, because the depth residuals are only known at wells and these are very sparse, the process of depth residual gridding is very subjective and results in spreading local well errors over an area in an arbitrary way (Figure 2).
Uncertainty in Depth Conversion
Figure 2: Typical residual map example. The choice of radius of influence in the gridding is ad hoc but significantly affects the area over which well misties are spread.
In a depth conversion context there are two main contributions to uncertainty. Firstly, uncertainty rises at a single point through there being multiple valid models and parameters by which we can depth convert, each model having a residual uncertainty; and secondly, there is the spatial uncertainty resulting from lateral prediction between data points and the spatial correlation/dependency model.
The impact of uncertainty due to choosing different valid models and parameters is assessed by doing just that – choosing different methods and building up a range of uncertainty from a variety of depth conversion models.
The second uncertainty is best handled by geostatistical analysis of the residuals and the residual mapping.
Equation 1.
The uncertainty in depth for a single layer obtained from multiplying together time and velocity uncertainty in the layer is given by the following equation (see Equation 1).
Where σ is the standard deviation and the letters z, t and v denote depth, time and velocity respectively. Note that the last term also includes ρ, the correlation between time and velocity. This clearly exists in models such as velocity functions and is expected to be a positive correlation.
The propagation of uncertainty through summing together a stack of depth layers in a multi-layer depth conversion is given by (see Equation 2).
Equation 2.
Again, note that the last term includes ρ, the correlation between layer 1 and layer 2 thickness. In general, this correlation is negative because the boundary between two adjacent layers is shared in the form of a time surface. For this reason, an increase in time thickness in one layer must necessarily be a decrease in thickness of the other layer, hence they are anti-correlated. However, if residuals through a depth conversion are allowed to float (i.e. not be force-tied at each horizon) then these anti-correlations (as well as noise) will partially cancel. An alternative way of thinking about this is that simply summing the uncertainty for each layer together through the multi-layered depth conversion will hugely inflate the uncertainty at the final depth converted horizon. This is the strongest argument for not tying at intermediate horizons in a depth conversion to a deeper horizon.
Simple Guide to Depth Converstion
In summary:
1. Don’t force horizon picks to tie wells during interpretation.
2. Always QC seismic times against well times by cross-plotting.
3. Seismic velocities may be useful but they are:
a. Noisy.
b. Subject to bias up to +/– 20%.
c. Become less useful with depth and in high velocity areas.
d. Variogram analysis and kriging are very good for analysis of stacking velocities.
4. Test multiple methods of depth conversion and consider different layer schemes and functions within layers.
5. Examine residuals and look for mean close to zero (no bias) and low SD.
6. V0 regression methods are usually just error residual corrections in disguise and should not be applied.
7. Velocity functions can become non-physical very quickly so QC the instantaneous velocities.
8. Look for unwarranted velocity inversions between layers.
9. Residuals should float between layers, not be tied. Only tie at the target horizon; intermediate horizons should be left untied.
10. Uncertainty propagation through depth conversion models is not trivial; floating residuals is a simple and robust way to deal with this.
a. Uncertainties exhibit anti-correlation between layers.
b. Correlation coefficients strongly influence uncertainty.
11. Depth prognosis uncertainty can be estimated from depth residuals and kriging.
12. GRV estimates and uncertainty can only be generate through geostatistical simulations.
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