Time-Magnitude Correlations and Time Variation of the Gutenberg–Richter Parameter in Foreshock Sequences

Abstract

The time dependence of the parameter of the Gutenberg–Richter (GR) magnitude distribution is identified for foreshock sequences of earthquakes, correlated with the main shock, by using the geometric-growth model of earthquake focus, the magnitude distribution of correlated earthquakes and the time-magnitude correlations, derived recently. It is shown that this parameter decreases in time in the foreshock sequence, from the background values down to the main shock. If correlations are present, this time dependence and the time-magnitude correlations provide a tool of monitoring the foreshock seismic activity. We analyze the relevance of such a procedure for the occurrence moment and the magnitude of a main shock. The limitations of such an analysis are discussed.

Appendix

1.1 Geometric-Growth Model

A typical earthquake is characterized by a small focal region (Apostol, 2006). Since the relevant distance scale is much larger than the focal dimension, we may view the focus as a point in an elastic body. The seismic energy is accumulated in the focal by the movement of the tectonic plates. This energy accumulation is described by the continuity equation

$$\frac{\partial E}{\partial t}=-{\varvec{v}}gradE, $$

(14)

where E is the energy, t denotes the time and ${\varvec{v}}$ is an accumulation velocity. Since the focal region is localized, the derivatives in this equation may be replaced by ratios of small, finite differences. As an example, we write $\Delta E/\Delta x$ instead of $\partial E/\partial x$ for the coordinate x. At the borders of the focus the energy tends to zero, such that $\Delta E=-E$. We may assume that the coordinates of the borders move uniformly, according to $\Delta x=u_{x}t$, etc, where we denote by ${\varvec{u}}$ a small velocity of the medium. By using these assumptions, we get from Eq. (14)

$$\frac{\partial E}{\partial t}=\left( \frac{v_{x}}{u_{x}}+\frac{v_{y}}{u_{y}}+\frac{v_{z}}{u_{z}}\right) \frac{E}{t}. $$

(15)

For a uniform motion the two velocities are equal (${\varvec{v}}={\varvec{u}}$), and we get a coefficient 3 in Eq. (15). If the motion is one dimensional, the coefficient is 1. Therefore, the above equation can be written as

$$\frac{\partial E}{\partial t}=\frac{1}{r}\frac{E}{t}, $$

(16)

where the parameter r varies in the range 1/3–1. For a shearing fault we have $u_{x}=v_{x}$ and $u_{y}=2v_{y}$, $v_{z}=0$, because, apart form the x-direction, the energy is accumulated also along two opposite perpendicular directions (y-directions), in order to conserve the mass. We get r = 2/3, which corresponds to the mean value $\beta =br=2.3$ (b = 3.45), accepted as the reference value (see the main text). Therefore, the parameter r is a statistical parameter, related, mainly, to the effective number of dimensions of the focus. The above model is called the geometric-growth model of seismic energy accumulation.

In order to integrate Eq. (16) we need a cutoff energy and a cutoff time. Therefore, a small amount of energy $E_{0}$ is accumulated in a short time $t_{0}$. This energy may be lost in the next time $t_{0}$, or the accumulation process may continue. These processes are called fundamental processes, or $E_{0}$-seismic events. From equtaion (16) we get the law of energy accumulation

$$t/t_{0}=(E/E_{0})^{r}. $$

(17)

The energy E may be released in an earthquake, which occurs after time t.

1.2 Gutenberg–Richter Law

It is well known that seismic moment ${\overline{M}}$ is related to the magnitude M through the Hanks-Kanamori empirical law

$$\ln {\overline{M}}=const+bM, $$

(18)

where b = 3.45 ($\frac{3}{2}$ for base 10). The seismic moment can be defined as ${\overline{M}}=\left( \sum _{ij}M_{ij}^{2}\right) ^{1/2}$, where $M_{ij}$ is the tensor of the seismic moment; it is related to the energy through ${\overline{M}}=2\sqrt{2}E$ (Apostol, 2019), such that Eq. (18) can be written as

$$\ln E=const+bM $$

(19)

(by using another const). This relation can be cast in the form

$$E/E_{0}=e^{bM}, $$

(20)

where $E_{0}$ is a cutoff energy. Now, we can make use of Eq. (17), and get

$$t=t_{0}e^{brM}=t_{0}e^{\beta M}, $$

(21)

where $\beta =br$. From this equation we derive the useful relations $dt=\beta t_{0}e^{\beta M}dM$, or $dt=\beta tdM$. In the well-known Gutenberg–Richter distribution we have

$$dP=\beta e^{-\beta M}dM. $$

(22)

On the other hand, from Eq. (21) we get $dt=\beta t_{0}e^{\beta M}dM$, or $dt=\beta tdM$. By using this result, Eq. (22) becomes

$$dP=\beta \frac{t_{0}}{t}\frac{1}{\beta t}dt=\frac{t_{0}}{t^{2}}dt. $$

(23)

This is the time distribution of independent earthquakes; it gives the probability $dP=t_{0}dt/t^{2}$ for an earthquake to occur between t and t + dt. Since t is the accumulation time, this earthquake has energy E and magnitude M, related by the above formulae. An equivalent derivation of the time probability can be obtained from the definition of the probability of the fundamental $E_{0}$-seismic events ($dP=-\frac{\partial }{\partial t}\frac{t_{0}}{t}dt$; Apostol (2021)).

1.3 Correlations: Time-Magnitude Correlations

Two (or more) earthquakes may depend on one another, by various mutual influences. We say that those earthquakes are correlated. We limit ourselves to two-earthquake (pair) correlations, which bring the main contribution. Two earthquakes may share their energy; then we have time-magnitude correlations, as shown below. Also, two earthquakes may share their accumulation time. Then, we have time correlations (also called purely dynamical correlations), as shown in the next Appendix (Apostol, 2021). The statistical distributions are affected by both these correlations. Other types of (statistical) correlations may appear, due, for instance, to additional constraints imposed upon the statistical variables (for instance, the magnitudes of the accompanying events be smaller than the magnitude of the main shock).

An energy E, accumulated in time t, may be released in two successive earthquakes, with energies $E_{1,2}$. The two earthquakes share the seismic energy. We may write $E=E_{1}+E_{2}$ and

$$\begin{aligned} t/t_{0}&=(E/E_{0})^{r}=\left( E_{1}/E_{0}+E_{2}/E_{0}\right) ^{r}=\\ & =(E_{1}/E_{0})^{r}(1+E_{2}/E_{1})^{r} \end{aligned}$$

(24)

(Eq. 17). This equation may bewritten as

$$t=t_{1}\left[ 1+e^{b(M_{2}-M_{1})}\right] ^{r}, $$

(25)

where $t_{1}=t_{0}(E_{1}/E_{0})^{r}$ is the accumulation time of the earthquake with energy $E_{1}$ and magnitude $M_{1}$, and $M_{2}$ is the magnitude of the earthquake with energy $E_{2}$. Equation (25) leads to

$$b(M_{2}-M_{1})=\ln \left[ \left( 1+\tau /t_{1}\right) ^{1/r}-1\right] , $$

(26)

where $t=t_{1}+\tau$, $\tau$ being the time elapsed from the occurrence of the earthquake 1 until the occurrence of the earthquake 2. In foreshock–main shock–aftershock sequence $\tau /t_{1}\ll 1$, such that we get from the above equation

$$M_{2}=\frac{1}{b}\ln \frac{\tau }{\tau _{0}},\,\,\tau _{0}==rt_{0}e^{-b(1-r)M_{1}}. $$

(27)

We can see that $\tau$ given by this equation differs from the accumulation time of the $M_{2}$-earthquake (compare with Eq. 21). The difference arises from parameters which depend on the $M_{1}$-earthquake, as expected for correlated earthquakes. We may view the $M_{1}$-earthquake as a main shock and the $M_{2}$-earthquake as a foreshock or an aftershock. These accompanying earthquakes are correlated to the main shock. These are the time-magnitude correlations.

1.4 Time Correlations

Two earthquakes may share their accumulation time $t=t_{1}+t_{2}$, such that an earthquake appears in time $t_{1}$, followed by another which appears in time $t_{2}$. From Eq. (23) the probability density of such an event is given by

$$-\frac{\partial }{\partial t_{2}}\frac{t_{0}}{(t_{1}+t_{2})^{2}}=\frac{2t_{0}}{(t_{1}+t_{2})^{3}} $$

(28)

(where $t_{0}<t_{1}<+\infty$, $0<t_{2}<+\infty$). We may pass in this formula to magnitude distributions ($t_{1,2}=t_{0}e^{\beta M_{1,2}}$), and get the probability

$$d^{2}P=4\beta ^{2}\frac{e^{\beta (M_{1}+M_{2})}}{\left( e^{\beta M_{1}}+e^{\beta M_{2}}\right) ^{3}}dM_{1}dM_{2} $$

(29)

(where $0<M_{1,2}<+\infty$, corresponding to $t_{0}<t_{1,2}<+\infty$, which introduces a factor 2 in Eq. (28)). This is a pair, bivariate statistical distribution (Apostol, 2021). By integrating with respect to $M_{2}$, we get the so-called marginal distribution, i.e. the distribution of a correlated earthquake,

$$dP=\beta e^{-\beta M_{1}}\frac{2}{\left( 1+e^{-\beta M_{1}}\right) ^{2}}dM_{1}; $$

(30)

By integrating further this distribution from $M_{1}=M$ to $+\infty$, we get the correlated cumulative distribution

$$P(M)=\int _{M}^{\infty }dP=e^{-\beta M}\frac{2}{1+e^{-\beta M}}. $$

(31)

For $M\gg 1$ the correlated cumulative distribution becomes $P(M)\simeq 2e^{-\beta M}$ and $\ln P(M)\simeq \ln 2-\beta M$. Therefore, the slope $\beta$ of the logarithm of the independent cumulative distribution (Gutenberg–Richter, standard distribution $e^{-\beta M}$) is not changed (for moderate and large magnitudes); the distribution is only shifted upwards by $\ln 2$. For small magnitudes ($M\ll 1$) the slope of the correlated cumulative distribution becomes $\beta /2$ (by the series expansion $P(M)\simeq 1-\frac{1}{2}\beta M+\cdots$ of Eq. 31); this result differs from the slope $\beta$ of the standard Gutenberg–Richter distribution ($e^{-\beta M}\simeq 1-\beta M+\cdots$). The correlations modify the slope of the Gutenberg–Richter standard distribution for small magnitudes. This is the roll-off effect referred to in the main text.