Seismic Inversion and Synthetic Sonic Logs
Calibration of seismic attributes to ground truth is still an emerging science. No single attribute (eg. amplitude, frequency content, attenuation, compressional or shear velocity, or acoustic impedance) can be related directly to a specific rock property (eg. porosity, lithology, hydrocarbons). However, geoscientists have found that one or more attributes may predict some reservoir property in a particular project. Trial and error is the only way to find out what works.

One of the most successful is the use of Poisson's Ratio to indicate the presence of porosity or gas. This requires close calibration to log data - many early studies obtained impossible numbers for Poisson's ratio, indicating poor quality inversion of the compressional or shear data into velocity. A successful example is shown later in this Chapter.

When attributes are used to locate hydrocarbons, they are often called direct hydrocarbon indicators or (DHI). Bright spots and dim spots (amplitude anomalies on conventional seismic displays) were the earliest form of DHI. Many bright spot studies failed because many different factors create the amplitude variation. Log modeling will show what kind of amplitude to expect for different reservoir conditions.

DHI is no longer a popular term because hydrocarbon indicators are really porosity or lithology indicators, with a major contribution from gas if it is present. The difference in acoustic and density properties between oil and water is very small and well below the noise level of even the best logs, let alone seismic.

The same story is true of amplitude versus offset (AVO) anomalies. Models are the only way to see what a particular AVO output might mean. Examples are shown below.

Wavelet processing of modern seismic field data yields results containing much more information than is found on conventionally processed data. These sections are usually called wide band or broad band sections. Yet the results may not bring joy to the average interpreter due to the noisy appearance of the data. 

Normal and broadband seismic data

Instead of simplifying the interpretation, the additional detail appears to mask the more obvious features on the conventional section and make the horizons more difficult to map. In fact, some of the principal markers on the conventional section practically disappear on the broad band section, while others appear to be displaced in time.

The broad band data approaches the response of the reflection coefficients and more accurately represents the acoustic impedance changes in the rock sequence. However, if the broad band data is to be used, some other means, other than the seismic wiggly trace, must be found to display it in a manner which can be adapted to routine interpretation.

One way to do this is to rearrange the reflection coefficient equation to solve for velocity, and display these velocities versus time or depth just like a sonic log. This requires the first velocity to be known, but thereafter all others can be derived by applying the formula in succession to each reflection coefficient.

The acoustic impedance from inversion of seismic data is:
1: Zp2 = Zp1 * (1 + Refl1) / (1 - Refl1)

If density is assumed based on lithology, the inversion can produce velocity instead of acoustic impedance. Inversion can be applied to both compression
al and shear seismic data.

This equation suffers from progressive errors as successive layers are computed. I wrote a program to do this calculation on a TIAC in 1966 but it failed miserably - the data was too low in bandwidth and I hadn't thought of finding the low frequency component from nearby sonic logs. Roy Lindseth produced the first commercial synthetic sonic logs around 1969 in Calgary.

The problem is reduced by filtering the results and stretching or squeezing to fit real, filtered sonic logs.

If this procedure is used to create an approximation of reflection coefficients from seismic data, and is expected to correlate to a real sonic log, some compensation must be made for the effects of density. Acoustic impedance is the product of velocity and density, so an inverted seismic trace is an acoustic impedance log rather than a sonic log. Fortunately, velocity is, to some degree, a linear function of acoustic impedance. The inverted data can be corrected accordingly.

Filtered sonic log

A serious constraint to inversion is the limited bandwidth caused by filtering which may occur through the system. Both the earth (subsurface) and electronic filters reduce frequency content. A sonic log has a very broad frequency bandwidth, extending from DC to approximately 1000 Hz. Current field practice and equipment limits the low end of the seismic spectrum to about 8 to 10 Hz while the natural filter of the earth eliminates frequencies much over 100 Hz, depending upon the depth. Careful stacking and de-convolution will recover a good portion of the spectrum, often almost doubling the bandwidth of about 50 Hz on conventional data.

A sonic log can be filtered to demonstrate the loss of resolution caused by high cut filtering (Figures 24.03 and 24.04). The effect is roughly analogous to logging with a very long tool spacing, which decreases the resolution of the log by smoothing out high frequency information. A seismic trace of the same frequency will have resolution no better than the log.

Low frequency content of a sonic log

Separating low and high frequency components on a sonic log

Of greater concern are low frequencies, which are usually lost through geophone response or band-limiting by the recording instruments.

Frequencies lost from the spectrum cannot be restored by de-convolution. Depending upon the geophones used and the seismic system response, frequencies below 5 to 10 Hz will be irrevocably lost from the spectrum. The absence of these frequencies is very serious, since they carry the basic velocity structure of the log.

In fact, a sonic log can be considered in terms of a low frequency carrier function modulated by higher frequencies. The effect is illustrated above, where 6 Hz has been chosen as a crossover frequency to separate a time integrated sonic log into its low frequency and high frequency components. The sum of the two components yields the original sonic log.

The first step in generating the low frequency data is to extract reliable vertical velocity information from stacking velocities.

A computer can pick a great number of sample points, which then can be statistically evaluated. The example at the left illustrates the results of a constant velocity analysis machine picked each 8 ms. The results are extremely erratic and apparently of little use, but application of a 7 Hz low pass filter yields a smooth continuous low frequency velocity curve. A single curve of this type probably contains residual errors, but several curves, closely spaced, can be averaged to produce more reliable results. The average velocities are then converted to interval velocities by ray path modeling.

Low frequency component of synthetic sonic log derived from a velocity analysis of seismic data

With the low frequency velocity information developed, the density corrected, inverted seismic data above the crossover frequency can be summed with the velocity data below the crossover to yield the synthetic sonic, scaled in time and velocity. This log can be easily converted to scales of depth and interval transit time and then compared to real sonic logs.

This is the procedure used to obtain the synthetic sonic log, generally termed Seislog, shown at the right. It has been plotted together with a borehole compensated sonic log for comparison. The vertical scale is depth, and the horizontal scale is microseconds per foot, both normal parameters for sonic logs. The seismic data has been converted into the geological domain. It is expressed in terms familiar to a geologist and is directly correlative to conventional geological data.

Comparison of filtered sonic log and seismic inversion trace

It would be natural to calibrate the synthetic sonic logs at each well along the seismic lines, and then extend the work to all seismic traces along the lines. This creates synthetic sonic log cross sections. The process is called seismic inversion, and is covered in more detail on the next Chapter. A sample inverted seismic section is shown below

This example shows a cross section of synthetic sonic log traces calibrtaed at 3 well bores along the seismic line. This product is called a seismic inversion.

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