Although seismic processing utilises some of the most sophisticated numerical algorithms known, the processing geophysicist can probably get by with simply understanding what the processes do; not how they do it!
It is useful, however, to understand some of the basic principles involved. A basic
knowledge of arithmetic, geometry and trigonometry is almost essential. Although we rely on the computer to do most of the work, we need to supply parameters for each processing stage that are mathematically (as well as geologically) sensible, and to be able to check the computer's results.
The single most common mathematical thread that runs through seismic processing is that of "Least-Squares", the process of minimising the errors involved in making some
approximation to our data. This technique is used again and again within the seismic processing sequence and is worth spending a few moments on. If you are quite happy with this technique, or simply bored or terrified at the prospect then please feel free to click the
"Next Page" button above.
In order to explain the principle of Least-Squares, here's a set of numbers representing the time of my journey to work over several days:-
Day 1 - 25 minutes Day 2 - 37 minutes Day 3 - 28 minutes Day 4 - 35 minutes
I would like to establish the "best-estimate" for my journey time, without having any concept of the meaning of "average", but with a knowledge of some basic maths.
I'll start by making a guess at the best time - say 30 minutes. In order to see how good a guess this is, I'll simply subtract it from each actual value:-
Day 1 - 25: 25 - 30 = -5 Day 2 - 37: 37 - 30 = 7 Day 3 - 28: 28 - 30 = -2 Day 4 - 35: 35 - 30 = 5
The total error could be described as -5 + 7 + -2 + 5 = 5, but this does not take into account the fact that some of the numbers are positive, and others negative. I'm really only
interested in the magnitude of the error, not it's sign, so I'll do the simplest thing
(mathematically) I can do and square the numbers before adding them. This gives me (for a guess of 30) a total error squared of 25 + 49 + 4 + 25 = 103. My best guess will be where this number is a minimum.
Here's a whole series of guesses, with the associated "error-squared":
Guess 20 - error squared = 603 Guess 30 - error squared = 103 Guess 22 - error squared = 439 Guess 32 - error squared = 99 Guess 24 - error squared = 307 Guess 34 - error squared = 127
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Guess 26 - error squared = 207 Guess 36 - error squared = 187 and so on ...
Here's a plot of the error squared against guess for a whole range of numbers:-
It's fairly obvious that the Total Error-Squared reaches a minimum value at the base of the curve, and this corresponds to the position of the "best-fit". In order to calculate its value, we need to express our calculations in general terms.
If we call our initial data values X1, X2 etc., then any given value's error-squared is:-
where G is the guess. The total error squared (E) is:-
where N is the number of values. We can expand this to:-
or (by simplifying):-
Now, the only (slightly) heavy bit. In order to minimise this total error-squared, we need to differentiate the right-hand side of the above with respect to G (the thing we're trying to find). This will give us the slope of the resultant curve, and the "turn-over" point where
31 this equals zero.
The differential of E is then:-
which we set equal to zero to find the minimum error position for G:-
and rearrange (and divide by 2):-
and solve for G:-
This appears to tell us that the best guess for G is the sum of all of the individual values (25+37+28+35=125), divided by the number of values (125/4=31.25). A very long-winded way of proving that the best-fit of a constant value to a set of numbers is the AVERAGE!
Of course, this process can be extended to "best-fit" any mathematical function to any set of numbers of any dimension; the solution in each case having the minimum
error-squared. Here's our data points once again with a whole set of higher-order curves fitted:-
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The red line is the average we just computed, the green is a straight line, the blue quadratic and the brown cubic. The brown (cubic) curve fits four points exactly, but is hardly the most meaningful "fit" in this case. (It implies that, next week, it'll take me a month to get to work, I suppose the traffic's going to be really bad!).
We will come back to the concept of least-squares much later in this course!
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Chapter 2 - Seismic Acquisition
This Chapter presents a very brief look at Seismic Acquisition, with particular emphasis on those topics relevant to the seismic processor. If you've spent the last 15 years on a land crew in the Amazon, or an equivalent length of time in the middle of the Arctic Ocean collecting marine data (or are doing either of these as you read this), you might want to skip this bit!
Page 02.02 - The ideal seismic source
What goes into the ground comes back out again! The characteristics of the ideal seismic source are discussed.
Page 02.03 - Onshore seismic sources
The use of explosives onshore - still the most common source for "Land" recording.
Page 02.04 - Other Land sources
Other land sources are mentioned, and vibratory sources (the second most common source) are examined in some detail.
Page 02.05 - Recording the data - Onshore
The logistics and mechanics of recording data onshore can be enormous. A brief explanation of the standard recording device is followed by a wider review of recording techniques.
Page 02.06 - Offshore seismic sources
From land to sea! The differences in the marine environment, and details of Air-Guns, the most common offshore seismic source.
Page 02.07 - Recording the data - Offshore
Highlights the differences between onshore and offshore recording and addresses the general problems associated with Marine Acquisition.
Page 02.08 - Questions
A set of questions to test your knowledge of this very brief introduction to Seismic Acquisition.
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