New insights into the seismic time term method for heterogeneous upper mantle slowness structures

Abstract

The seismic time term method of Gardner (Geophysics 4:247–259, 1939) has become popular in the context of seismic $P_n$ studies, since it provides a possibility to estimate not only the Earth’s crustal thickness, but also the P-wave slowness of the uppermost mantle. In the last decades an extended form of this Time Term Method has been extensively used in order to predict a heterogeneous isotropic or anisotropic upper mantle slowness function. One of the main goals of this study is to formalize the mathematical construction for which the Time Term Equations are indeed linearized versions of the Travel Time Equations for such a complex laterally varying slowness function. For this purpose, an alternative definition of Time Terms is given that allows to consider them as parts of the exact travel time equation. In the case of constant upper mantle slowness functions these terms coincide with the classical Time Terms of Gardner. As a consequence, an alternative derivation of the extended Time Term Method can be given for heterogeneous (isotropic) upper mantle slowness functions.

Appendix

Proof of Lemma 1

Along with $A(\eta )=0$ for $\eta \ne 0$ also $A(1)=\frac{1}{\eta }A(\eta \cdot 1)=0$. Let h be a bounded, continuous function on ${\mathfrak {M}}$ with $\inf := \inf _{{\mathfrak {M}}}h$. With h also $h-\inf$ is bounded with $\sup _{{\mathfrak {M}}} h-\inf <\infty$. As a consequence

$$\begin{aligned} 0\le A(h - \inf )\le & {} A(\sup _{{\mathfrak {M}}}(h-\inf )\cdot 1)\\ {}= & {} \sup _{{\mathfrak {M}}}(h-\inf )\cdot A(1) = 0. \end{aligned}$$

Hence, $A(h)=A(h-\inf )+A(\inf )=0$. $\square$

Proof of Lemma 2

As above, let $T_x$ be the crustal travel time field of a signal emitted in x and traveling with the fixed slowness $S_{{\mathcal {C}}}^0$. Let h be a function of the relevant function space with $h(x)\ge 0$ for all $x\in {\mathfrak {M}}$. Obviously, A (from Lemma 1) is just the Fréchet derivative of $f(S_{{\mathcal {M}}}):=T_a(a'(S_{{\mathcal {M}}}))-T_{{\hat{a}}}^{S_{{\mathcal {M}}}^0}(a'(S_{{\mathcal {M}}}))$ in $S_{{\mathcal {M}}}^0$. It suffices to study the difference quotient

$$\begin{aligned} \frac{1}{\gamma }(f(S_{{\mathcal {M}}}^0+\gamma h)-f(S_{{\mathcal {M}}}^0)) \end{aligned}$$

(16)

for $\gamma >0$. Consider the travel time of a signal travelling from a to a hypothetical receiver x on the refractor with slowness $S_{{\mathcal {M}}}^0$. The impinging point $a'$ of the critically refracted ray is determined by Fermat’s Principle. If the distance between x and a is larger than the one between $a'$ and a, the minimal travel time from a to x is realized by the head wave, i.e. $T_a(a') + T_{a'}^{S_{{\mathcal {M}}}^0}(x)\le T_a(x)$. Taking $x= a'(S_{{\mathcal {M}}}^0+\gamma h)$ for $h,\gamma >0$, this is indeed the case, again for sake of simplicity assuming $a' = a'(S_{{\mathcal {M}}}^0)$ and $a'(\gamma ) = a'(S_{{\mathcal {M}}}^0+\gamma h)$: $T_a(a') + T_{a'}^{S_{{\mathcal {M}}}^0}(a'(\gamma )) \le T_a(a'(\gamma ))$ and hence

$$\begin{aligned} T_a(a'(\gamma )) - T_a(a') \ge T_{a'}^{S_{{\mathcal {M}}}^0}(a'(\gamma )). \end{aligned}$$

But from Fermat’s Principle we also have

$$\begin{aligned} T_{{\hat{a}}}^{S_{{\mathcal {M}}}^0}(a'(\gamma )) \le T_{{\hat{a}}}^{S_{{\mathcal {M}}}^0}(a') + T_{a'}^{S_{{\mathcal {M}}}^0}(a'(\gamma )). \end{aligned}$$

This implies

$$\begin{aligned} T_a(a'(\gamma )) - T_a(a') \ge T_{{\hat{a}}}^{S_{{\mathcal {M}}}^0}(a'(\gamma )) - T_{{\hat{a}}}^{S_{{\mathcal {M}}}^0}(a'). \end{aligned}$$

The non-negativity of the difference quotient (16) follows immediately. $\square$