Chapter 1 Introduction
1.1 Understanding equations
Geophysicists are often turned off by equations. This is unfortunate because equations are simply compact, quantitative expressions of relationships, and one should make an effort to understand the information that they convey. They tell us what factors are important in a relationship and their relative importance. They also suggest what factors are not relevant, except perhaps through indirect effects on the relevant factors. Graphs often help us visualize equations more clearly. We may think of derivatives as simply measures of the slopes of curves, maxima and minima being merely the places where the slopes are zero, and integration as simply summing up the area under a curve. An imaginary exponential indicates a periodic function. Limitations imposed by initial assumptions or by approximations in their derivations apply to most equations, and these should be appreciated in order to avoid drawing erroneous conclusions from the equations.
1.2 Solving equations
The problems in this volume are of various types. Some involve proofs or derivations, others involve calculating magnitudes, still others are designed merely to provoke thought. Several are short and may even appear trivial, whereas others are long and somewhat tedious. Some require that values of a specified quantity be found or that the relation between certain variables and/or parameters be deduced; others are hypotheses or statements to be proven. We hope that all emphasize important concepts.
Students (and others) try to solve problems in different ways. Some adopt a “road” that they hope will lead directly to the solution, others “mill around” with algebraic manipulations in the hope that they will stumble onto the solution, while still others start with what is to be proven and work backwards, manipulating quantities until they arrive at the proper starting point. In the process of solving a problem, some students mix up units or give answers without specifying the units, while many give more significant figures than the data warrant. This chapter attempts to help the student avoid such pitfalls.
The first step in a solution is to try to understand the problem; we should have a clear picture of what we wish to achieve, that is, what is the unknown? Secondly, we need to decide what quantities or factors must be considered in arriving at the correct answer. Thirdly, we must consider the available information from which the answer is to be deduced. Some of the available data may not be relevant. Finally, we must determine how the relations between the various factors can be used to get from the starting point to the final goal, that is, the path that must be followed.
We should plan how to obtain the solution. Often a diagram will help to understand interrelationships. Usually diagrams should be approximately to scale so that the relative magnitudes of different elements are preserved; nonessential and irrelevant elements should be omitted. Sometimes exaggerated or detailed diagrams are desirable. If a variable has a wide range of values, more than one diagram may be necessary.
Often we must introduce new symbols or notations; these should be defined or stated explicitly, not only to help avoid errors but also for the benefit of others trying to follow the derivations. Care must be taken to avoid using the same symbol to represent different quantities, for example, using to denote both location and offset. Wherever possible, a notation should suggest the nature of the quantity it represents, for example, using for amplitude, for velocity, for frequency, for time. Symbols should be kept as simple as possible. A single subscript should be related to the subgroup that it represents, for example, for apparent velocity. Double subscripts should generally be avoided if a feasible alternative exists; however, double subscripts are at times necessary or at least desirable, for example, in a double summation over both and or denoting the up-dip apparent velocity at the second interface. Symbols, especially subscripts, superscripts, and Greek letters, must be written clearly to avoid mistaking them for other symbols.
The chances of making an error in copying mathematical equations increases with the number and complexity of terms to be copied, so it may be advisable to use a symbol to represent a single complicated term or group of terms; for example, in problem 2.5, in problem 2.6, in problem 7.5b. However, this practice tends to hide individual symbols and terms so that their significance is obscured, hence should be used with caution.
Problems to be solved are of different kinds and require different approaches. “To solve” problems generally start with known or assumed relationships (hypotheses) and ask that we establish some other relationships (the conclusions). “Inductive” problems seek to discover new relationships based on a set of observations; solutions of this type are often probabilistic rather than rigorous, that is, the conclusion may be merely strongly suggested rather than proven exactly.
Some problems cannot be solved exactly by algebraic means but are amenable to numerical or graphical solution. The “accuracy” of a numerical solution usually can be as high as we wish, depending upon how much time we wish to spend; however, nothing is gained by attempting to achieve greater accuracy than that of the given data. The accuracy of a graphical solution depends on the scale of the graph and the care with which it is drawn. Graphical solutions often illustrate relationships and principles better than algebraic solutions so that we sometimes use them even where we may ultimately resort to an algebraic or numerical solution to achieve the desired accuracy.
In numerical and graphical solutions attention must be paid to the number of significant figures. The basic principle is that significant figures represent values to the accuracy of the measurement. The following examples illustrate the concept, the number of significant figures in the answers being four in all cases (note that is a significant figure except when it is used solely to indicate the magnitude).
Thus, in the first example, the digits to the right of the decimal point must be discarded (the second and third values being rounded to 21 and 30) because the first quantity is known only to the nearest unit. The following examples apply the same principle to subtraction, multiplication, and division. When rounding off a value which has the digit 5 at the right hand end, we can round upwards or downwards; however, people often use a rule such as rounding up or down so that the quantity ends in an even digit, for example, rounding 1.265 and 4.735 to become 1.26 and 4.74. To avoid accumulating error, we often carry an extra figure while working out a numerical solution and only round off to significant figures when we arrive at the answer.
Once we have solved a problem, we are not finished; we need to understand it in detail. What is of significance? What does it tell us that is of interest? What is the function of each component part? How will the relationships change as the parameters vary? Which input parameters have the greatest effects on the results and which are almost irrelevant? Under what circumstances will our conclusions not hold? What happens when parameters assume extreme values; does the solution “blow up”? Did we omit any important aspects? By examining our results critically we may convert a mere exercise into a worthwhile learning experience and develop an in-depth understanding of the subject. Sometimes an analysis will show that we did not actually solve the problem at all.
1.3 Checking solutions
Incorrect solutions can often be identified using simple checks, such as verifying that the dimensions are the same on both sides of an equation. For example, suppose that we remember the basic form of the one-dimensional wave equation but are not sure on which side of the equation the factor belongs—or even that it is instead of . Thus we may want to decide which form of the four equations below is correct:
Both sides of all equations have so we ignore this factor. Denoting dimensions of length and time by and , has dimensions , so the first equation equates to , hence cannot be correct. The second equation equates to , while the fourth equates to , hence both are incorrect. The third equation has dimensions on both sides and so is at least dimensionally correct. Note that dimensional analysis cannot prove that an equation is correct even though it can prove that one is not correct. As another example: which of the following equations for the traveltime of a head wave from a horizontal refractor are incorrect?
Since all terms in a sum must have the same dimensions, we examine the dimensions of each term and readily find that (a) and (b) are incorrect while (c) is dimensionally correct. As an additional example, consider the following equation for the angle of approach:
Recalling that the arguments of trigonometric, exponential, logarithmic, and similar functions must be dimensionless (because they can be expanded in infinite series), we see that the equation must be incorrect because the argument has the dimensions .
Another check is to see if varying the parameters produces reasonable changes in the calculated quantity. For example, which of the following equations for the reflection from a horizontal bed must be incorrect:
In the first equation the time becomes smaller as the distance increases, which is not reasonable, hence the equation must be incorrect. In the second equation, increases as increases, which is reasonable (but not a proof of correctness). A somewhat different example is to determine which of the following equations relating the critical distance to the depth of a refractor is incorrect:
Because for a head wave to exist, is imaginary, so the first equation equates an imaginary quantity to a real quantity and therefore must be incorrect.
At times, equations exhibit varying degrees of symmetry, and this may be useful, not only in remembering them, but also in detecting errors. The following equations illustrate the value of symmetry:
(the last pair might be termed “antisymmetric” because of the minus sign). As the complexity of the equation increases, the value of symmetry generally decreases rapidly, for example, compare the third and fourth of Zoeppritz’s equations, equations (3.2h) and (3.2i); nevertheless symmetry may still be of value; for example, if in deriving equation (3.2f) we obtained the term , we should be suspicious because of the lack of symmetry with equation (3.2e).
We must be on the lookout for singularities [places where a function becomes infinite, as approaches 0.5]. What do singularities mean in a “physical sense”? What happens in the real world? Singularities cause computer programs to bomb, so programs must always be analyzed to make certain that they do not involve any potential singularities.
Most problems are deterministic, that is, they have a definite answer (or answers in some cases); this is so whenever the number of unknowns equals the number of independent equations . However, when the number of equations is less than the number of unknowns , the unknowns are “underdetermined” and the best we can do is to find relations between the unknowns. In the “overdetermined” case, where , only approximate “best-fit” solutions are possible. As an example, when we try to find a velocity function that represents a set of time-depth data, we often seek a least-squares solutions (see also Sheriff and Geldart, 1995, Section 9.5.5 and in this book, problem 9.33).
Frequently the physics of a situation provides the equation. If we are asked to define the boundary conditions which govern the behavior of waves generated at the boundary between a fluid and a solid, we know that both P- and S-waves will exist in the solid but only a P-wave in the fluid. Therefore, a wave incident on the boundary will in general give rise to three waves involving three unknowns (the amplitudes of these three waves) and to fix these we will need exactly three boundary conditions obtained by applying physical principles, in this case the continuity of normal stresses and strains and the vanishing of shear stress at the boundary (see problem 2.10).
Merely substituting numerical values into an equation may produce ambiguity as to the dimensions of the answer. Including the dimensions when substituting solves this problem. Thus, suppose we wish to calculate the acoustic impedance, , given that and , we write
Because the numerators and denominators of the three multiplying factors are equal, each has the value of unity, and multiplying by one does not change a value. Multiplying by one also provides a means of changing from one measurement system to another. Thus if we are given that and , we have
Appendix K of The Encyclopedic Dictionary of Applied Geophysics (Sheriff, 2002) lists conversion factors that often occur in geophysics.