Checking solutions

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 1 1 - 6 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

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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 ${\displaystyle V^{2}}$ belongs—or even that it is ${\displaystyle V^{2}}$ instead of ${\displaystyle V}$. Thus we may want to decide which form of the four equations below is correct:

{\displaystyle {\begin{aligned}\mathrm {(i)} {\frac {\partial ^{2}\psi }{\partial x^{2}}}=V^{2}{\frac {\partial ^{2}\psi }{\partial t^{2}}},&\quad \mathrm {(ii)} {\frac {\partial ^{2}\psi }{\partial x^{2}}}=V{\frac {\partial ^{2}\psi }{\partial t^{2}}},\\\mathrm {(iii)} V^{2}{\frac {\partial ^{2}\psi }{\partial x^{2}}}={\frac {\partial ^{2}\psi }{\partial t^{2}}},&\quad \mathrm {(iv)} V{\frac {\partial ^{2}\psi }{\partial x^{2}}}={\frac {\partial ^{2}\psi }{\partial t^{2}}}.\end{aligned}}}

Both sides of all equations have ${\displaystyle \partial ^{2}\psi }$ so we ignore this factor. Denoting dimensions of length and time by ${\displaystyle L}$ and ${\displaystyle T}$, ${\displaystyle V}$ has dimensions ${\displaystyle L/T}$, so the first equation equates ${\displaystyle \mathrm {L} ^{-2}}$ to ${\displaystyle \mathrm {L} ^{2}\mathrm {T} ^{-4}}$, hence cannot be correct. The second equation equates ${\displaystyle \mathrm {L} ^{-2}}$ to ${\displaystyle \mathrm {LT} ^{-3}}$, while the fourth equates ${\displaystyle \mathrm {L} ^{-1}\mathrm {T} ^{-1}}$ to ${\displaystyle \mathrm {T} ^{-2}}$, hence both are incorrect. The third equation has dimensions ${\displaystyle \mathrm {T} ^{-2}}$ 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?

{\displaystyle {\begin{aligned}\mathrm {(a)} \,t=t_{i}+Vx;\quad \mathrm {(b)} \,t=t_{i}+x^{2}/V;\quad \mathrm {(c)} \,t=t_{i}+x/V.\end{aligned}}}

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:

{\displaystyle {\begin{aligned}\alpha ={\rm {sin}}^{-1}\left(\Delta t/\Delta x\right).\end{aligned}}}

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 ${\displaystyle T/L}$.

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:

{\displaystyle {\begin{aligned}(t/t_{0})^{2}=1+(Vt_{0}/x)^{2}\quad \mathrm {or} \quad (t/t_{0})^{2}=1+(x/Vt_{0})^{2}?\end{aligned}}}

In the first equation the time ${\displaystyle t}$ becomes smaller as the distance ${\displaystyle x}$ increases, which is not reasonable, hence the equation must be incorrect. In the second equation, ${\displaystyle t}$ increases as ${\displaystyle x}$ 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 ${\displaystyle x'}$ to the depth of a refractor is incorrect:

{\displaystyle {\begin{aligned}x'=\left(2h/V_{1}\right)(V_{1}^{2}-V_{2}^{2})^{1/2}\quad \mathrm {or} \quad x'=\left(2h/V_{1}\right)(V_{2}^{2}-V_{1}^{2})^{1/2}.\end{aligned}}}

Because ${\displaystyle V_{2}>V_{1}}$ for a head wave to exist, ${\displaystyle (V_{1}^{2}-V_{2}^{2})^{1/2}}$ 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:

{\displaystyle {\begin{aligned}&&\varepsilon _{xy}={\dfrac {\partial v}{\partial x}}+{\dfrac {\partial u}{\partial y}},\quad \varepsilon _{yz}={\dfrac {\partial w}{\partial y}}+{\dfrac {\partial v}{\partial z}},\quad \varepsilon _{zx}={\dfrac {\partial u}{\partial z}}+{\dfrac {\partial w}{\partial x}},\\&&\sigma _{xx}=\lambda \Delta +2\mu \varepsilon _{xx},\quad \sigma _{yy}=\lambda \Delta +2\mu \varepsilon _{yy},\quad \sigma _{zz}=\lambda \Delta +2\mu \varepsilon _{zz},\\&&{\dfrac {\sin \theta _{1}}{\alpha _{1}}}={\dfrac {\sin \delta _{1}}{\beta _{1}}}={\dfrac {\sin \theta _{2}}{\alpha _{2}}}={\dfrac {\sin \delta _{2}}{\beta _{2}}},\\&&R\left(\omega \right)=X\left(\omega \right)*\left(1/j\omega \right),\quad X\left(\omega \right)=-R\left(\omega \right)*\left(1/j\omega \right)\\\end{aligned}}}

(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 ${\displaystyle -B_{1}{\rm {\;sin\;}}\delta _{1}}$, 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, ${\displaystyle 1/\left(1-2\sigma \right)}$ as ${\displaystyle \sigma }$ 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 ${\displaystyle n}$ equals the number of independent equations ${\displaystyle m}$. However, when the number of equations is less than the number of unknowns ${\displaystyle (n>m)}$, the unknowns are “underdetermined” and the best we can do is to find ${\displaystyle \left(n-m\right)}$ relations between the unknowns. In the “overdetermined” case, where ${\displaystyle m>n}$, 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, ${\displaystyle Z=\rho V_{P}}$, given that ${\displaystyle \rho =1.0\,\mathrm {g/cm} ^{3}}$ and ${\displaystyle V_{P}=2.0\,\mathrm {km/s} }$, we write

{\displaystyle {\begin{aligned}Z&=\rho V_{P}={\frac {1.0\,\mathrm {g} }{\mathrm {cm} ^{3}}}\times {\frac {2.0\,\mathrm {km} }{\mathrm {s} }}={\frac {2.0\,\mathrm {g\,km} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\\&={\frac {2.0\,\mathrm {g\,km} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\times {\frac {10^{3}\,\mathrm {m} }{1\,\mathrm {km} }}\times \left({\frac {10^{2}\,\mathrm {cm} }{\mathrm {m} }}\right)^{3}\times {\frac {1\,\mathrm {kg} }{10^{3}\,\mathrm {g} }}={\frac {2.0\times 10^{6}\,\mathrm {kg} }{\mathrm {m} ^{3}\,\mathrm {s} }}.\end{aligned}}}

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 ${\displaystyle V_{P}=6000\,\mathrm {ft/s} }$ and ${\displaystyle \rho =1\,\mathrm {g/cm} ^{3}}$, we have

{\displaystyle {\begin{aligned}Z&={\rho }V_{P}={\frac {1.0\,\mathrm {g} }{\mathrm {cm} ^{3}}}\times {\frac {6000\,\mathrm {ft} }{\mathrm {s} }}={\frac {6000\,\mathrm {g\,ft} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\\&={\frac {6000\,\mathrm {g\,ft} }{\mathrm {cm} ^{3}\,\mathrm {s} }}\times {\frac {1\,\mathrm {km} }{3281\,\mathrm {ft} }}\times {\frac {10^{3}\,\mathrm {m} }{1\,\mathrm {km} }}\times \left({\frac {10^{2}\,\mathrm {cm} }{\mathrm {m} }}\right)^{3}\times {\frac {1\,\mathrm {kg} }{10^{3}\,\mathrm {g} }}\\&={\frac {6000}{3281}}\times {\frac {10^{6}\,\mathrm {kg} }{\mathrm {m} ^{3}\,\mathrm {s} }}={\frac {1.8\times 10^{6}\,\mathrm {kg} }{\mathrm {m} ^{3}\,\mathrm {s} }}.\end{aligned}}}

Appendix K of The Encyclopedic Dictionary of Applied Geophysics (Sheriff, 2002) lists conversion factors that often occur in geophysics.