### 11.1 Spectral Ratio

Q determined from Spectral Ratio technique with VSP data is a measure of the differential attenuation of frequencies in a seismic waveform while traversing through the earth. A lower value of Q indicates the high frequencies are attenuating faster than the low frequencies.

### 11.2 Processing Steps for Computing Q from Spectral Ratios

The tasks in the VSP processing menu that are applicable to Q-Analysis are indicated below. Having accurate picked transit times is mandatory, and if not already picked, then this can be done as step-1.

The step-2 is also optional, and allows to reduce the depth extent or time extent of the VSP data set. This should be done if there is noisy cased hole data above the main data set. The multi-spectral Q also requires the entire dataset to be regular spaced in depth. Sparse level spacing will not give optimal results.

After the pre-processing steps indicated above, the next step is to compute the frequency spectrum. The data from the frequency spectrum is used for the subsequent Q-analysis computuations.

There are two Q analysis computations available.

- Spectral Ratio between 2 levels. This simply takes spectrum from 2 individual VSP waveform depths and computes the Spectral Ratio between these 2 depths.
- Multi-Spectral Q. This computes the spectral ratio between all possible combimation of depth pairs. If n is the number of VSP levels, then there will be n! (n factorial) results.

### 11.3 VSP Waveforms and Frequency Spectrum

There are no parameters required for computing the frequency spectrum. Simply select the task and run the computation, and a plot of the frequency spectrum will be produced. The axes of the plot are frequency in Hz on the lower axis, and waveform depth on the vertical axis. The vertical axis corresponds directly to the Waveform plot. A file of the spectrum can be found in the data list.

It is assumed that the data being analysed is purely compressional downgoing waves. If the data contains shear wave energy, then the results will become distorted by by the shear wave energy

### 11.4 Spectral Ratio

The processing options for Spectral Ratio are shown below. The frequency spectrum computed in the previous step is the only input required.

By default, the two level numbers presented are the top and bottom levels of the VSP dataset, which in this example are level 1 and level 76. The Q is computed between the frequency range selected, and is 7-70 Hz in this example.

After clicking "Compute" a crossplot of the results will appear. The red and blue curves are the two individual spectra, which the blue curve being the deepet curve, and can be seen to have lesser high frequencies than the shallower red curve. The green dots are the difference between the two spectra. The green line is a best fit curve to the difference, over the selected frequency range.

The Q is the reciprocal of the gradient of the green line. So a more rapid attentuation of high frequencies will give a low value of Q. Low attentaution of the high frequencies will give a high value of Q. This indicates a major short coming of the definition of Q: if the two levels have identical spectra, then the gradient will be zero, and the Q will be infinity.

### 11.5 Multi-Spectral Ratio and Q-Log

For optimal results, the VSP data used for multi-spectral Q should be at a regular spacing, and the level spacing at a typical VSP spacing of around 15 metres or similar.

The parameters for multi-spectral Q are similar to spectral ratio method above. The computations are restricted to levels that have transit time differences that are within the time range specified.

For each Q-computation done, the Q value is assigned to the midpoint depth of each of the waveform pairs, and difference in depth (or Delta Depth) is computed together with the Q value. Similarly to Delta Depth, we can compute a DeltaTime which is the difference in transit time between the 2 levels.

After performing the calculation, a crossplot is produced as shown below, and a new dataset with the results is made.

- SPDEP - midint depth between the 2 levels.
- SPTIM - mid point transit time of the 2 levels
- SPRAT - spectral ratio between the two levels
- SPRAT-R - reciprocal of the spectral ratio
- DeltaTime - difference in transit time between the 2 levels
- DeltaDepth - difference in deth between the 2 levels

The data is cross-plotted with SPDEP against SPRAT (mid-point depth vs spectral ratio) and is color coded by DeltaTime. Therefore the red points are from levels that are far apart and give an average measure of the Q over the entire dataset, while the blue points are from closely spaced levels, and give an indication of the variation of Q over the VSP interval.

Right-clicking on the cross-plot will display a menu with an option to "Compute a Q-Log". This computation bins all the computed Q values into a series a depth-based bins, and then computes the average Q for each bin. The parameters required are:

- Depth width of the bin. In the example above, 30 metres is used.
- Upper and Lower Delta-Time. The aim is to produce a "log-like" results, so we need to remove the widely spaced lower resolution data (the red points).
- Restrict the out-liers. By default, out-liers are considered as points that are off the plot, or have a Q <=0. The table shown in the top part of the Q-Log menu is used for excluding the out-liers (see above).
- Averaging method. Default is a variance based weighted mean average. Since Q will go to infinity with low attentuation, then mean averaging can go crazy, and in this case a harmonic mean may work better.

In the example, the data is restricted to Q values that have a DeltaTime of between 0.05 an 0.2 sec. Q values from very finely spaced levels, although potentially higher resolution are too erratic and should be removed. Q values from widely spaced levels may give a good average Q, but have poor resolutuon and should be removed. The out-liers are specified by SPDEP and SPRAT are left to default values.

After clicking "Compute Q-Log" the black curve is added to the crossplot. Each step in the Q-Log curve represents a new bin in which the data has been averaged. The QLOG channel is written to datalist, and can be later saved to a LAS file if required.

### 11.6 Multi-Spectral Ratio - Restricting the input data for the Q-Log

The Q-Log computed above was done over a restricted range of Delta Time but all the Q data is still displayed, so it is difficult to see the correlation between the Q-Log and the Q data.

The Q data that is shown on the crossplot can be restricted by using the "Exclude Data Points" task. This can be found in the drop-down menu when right clcking on the crossplot. Use this task to restrict the Delta Time to the same range as the Q-Log Computation, as shown below.

Then repeat the Q-Log computation. Since the binning is only done from the data that is shown on the cross-plot, then the Upper and Lower DT parameters for Q-Log are now superfluous, but they should be kept to the same values as the Exclude task.

### 11.7 Multi-Spectral Ratio with Reciprocal Q

An issue that arises when computing Q from the spectral ratio technique, is that zero attenuation gives a Q value equal to infinity. With VSP data and when computing Q from closely spaced levels, zero attenuatuion and negative Q is often encountered.

When the Q value trends from positive Q to negative Q, it does so by asymptoting to positive infinity and comes back through negative infinity. Curve fitting doesn't work very well in this situation.

Negative Q indicates that the lower frequencies are attentuating faster than the higher frequencies. Whether this is caused by genuine formation attenuation or if it results from attenuation of multiple interfering downgoing waves, may not be known. But the Q measurement is real and valid, to within the accuracy of the method.

It turns out the 1/Q is a much better measurement to use for spectral ratio from VSP data. The 1/Q will simply go through zero and back again for small excursions to negative Q. Infinity or extremely large values of Q will no longer be encountered, so computations will be much more stable.

This is also the reason why Harmonic Mean is offered as a choice for averaging the binned Q data - a harmonic mean is the reciprocal of the average of the reciprocal's. (Harmonic Mean should only be used with Q data, it should not be used with 1/Q data.)

To compute a QR-Log (a reciprocal Q-Log), change the y-axis curve to SPRAT-R, instead of SPRAT. The figures below are repeats of the earlier examples, but using SPRAT-R instead.

When computing a QR-Log from 1/Q, values less than zero should no longer be automatically considered as out-liers. Therefore, the minimum value of SPRAT-R in the exclusion table should not be left at the default value of zero, but expanded to include more of the data.