More statistical tools for maximum possible earthquake magnitude estimation

Abstract

In this paper, we introduce additional statistical tools for estimating the maximum regional earthquake magnitude, $m_{\max}$, as complement to those already introduced by Kijko and Singh (Acta Geophys 59(4):674–700, 2011). Four new methods are introduced and investigated, with regard to their applicability and performance. We present an example of application and a comparison that includes the methods introduced earlier by the previous authors. A condition for the existence of the Tate–Pisarenko estimate and a proof of the asymptotic equivalence of the Tate–Pisarenko and Kijko–Sellevoll estimates are presented in the two appendices.

Appendix 1: Condition of existence of the Tate–Pisarenko solution of m max

In practice, it has been noticed that when the Tate–Pisarenko method was applied in an iterative fashion and it diverges on some occasions. In this Appendix, we present a condition for the convergence (and divergence) of this method.

Let m denote the unknown, true maximum magnitude. Assume that m is estimated by means of iteration, which means

$$ m_{i + 1} = m_{0} - \frac{{1 - \exp \left[ { - \beta (m_{i} - m_{\hbox{min} } )} \right]}}{{n\beta \exp \left[ { - \beta (m_{i} - m_{\hbox{min} } )} \right]}}, $$

(20)

or

$$ m_{i + 1} = m_{0} - \frac{1}{n\beta } + \frac{1}{n\beta }\exp \left[ { - \beta (m_{i} - m_{\hbox{min} } )} \right]. $$

(21)

Claim

There exists $m$ such that

$$ m = m_{0} - \frac{1}{n\beta } + \frac{{\exp \left[ { - \beta (m_{i} - m_{\hbox{min} } )} \right]}}{n\beta }. $$

(22)

Now, let $m' = \beta \left( {m - m_{\hbox{min} } } \right)$; therefore, $m' = m'_{0} - 1/n + \exp (m')/n$ (for iteration, $m'_{i + 1} = m'_{0} - 1/n + \exp (m'_{i} )/n$).

Consider subsequently

$$ m' = m_{0} ' - \frac{1}{n} + \exp (m'). $$

(23)

Taking the shape of LHS and RHS into consideration (Fig. 1), note that for Eq. (23) to have exactly one root, it must be that LHS = RHS precisely, where the first derivatives of LHS and RHS coincide; that is, where the derivative of RHS = 1. This turns out to be where $m' = \ln (n)$. Therefore, for one root, it must be the case that

$$ \ln (n) = m_{0} ' - \frac{1}{n} + \frac{{\exp \left[ {\ln (n)} \right]}}{n}, $$

(24)

that is

$$ \ln (n) = m_{0} ' - \frac{1}{n} + 1. $$

(25)

After further graphical consideration, we find that the equation has no roots if (the curve moved vertically upward in Fig. 1)

$$ \ln (n) < m_{0} ' - \frac{1}{n} + 1, $$

(26)

and

$$ \therefore \ln (n) < \beta \left( {m_{\hbox{max} }^{\text{obs}} - m_{\hbox{min} } } \right) - \frac{1}{n} + 1. $$

(27)

Fig. 1

Thick curve represents the curve $(m') = m_{0} - 1/n + \exp (m')/n$, and the thin line represents the unit function $g(m') = m'$

For large n,

$$ \ln (n) < \beta \left( {m_{\hbox{max} }^{\text{obs}} - m_{\hbox{min} } } \right) + 1. $$

(28)

Equation (20) has exactly two roots when (the curve moved vertically downward in Fig. 1)

$$ \ln (n) > m_{0} ' - \frac{1}{n} + 1, $$

(29)

or equivalently

$$ \therefore \ln (n) > \beta \left( {m_{\hbox{max} }^{\text{obs}} - m_{\hbox{min} } } \right) - \frac{1}{n} + 1. $$

(30)

For large n,

$$ \ln (n) > \beta \left( {m_{\hbox{max} }^{\text{obs}} - m_{\hbox{min} } } \right) + 1. $$

(31)

If a fixed point (solution) does exist, it would usually indicate two fixed points. In addition, the following can be stated:

• 1. It is possible for the smaller fixed point to be less than zero, and, therefore, smaller than $m_{\hbox{max} }^{\text{obs}}$. In either event, the smaller fixed point is neither appropriate nor can it be evaluated easily, which is a possible point of convergence for iteration. If convergence does take place, it must converge to the larger fixed point.

• 2. If the larger fixed point is less than $m_{\hbox{max} }^{\text{obs}}$, iteration would not converge, as it is clear that a point of convergence for the iteration would have to be larger than $m_{\hbox{max} }^{\text{obs}}$.

• 3. It can be verified graphically that if the smaller fixed point was much smaller than $\ln (n)$, the larger fixed point would be much larger than $\ln (n)$.

Appendix 2: Asymptotic equivalence of the Cooke–Kijko and Tate–Pisarenko estimates

It is known (Kijko and Singh 2011) that both the generic equation of Cooke and the Tate–Pisarenko estimates produce asymptotically unbiased estimates, which strongly suggests a measure of equivalence between the two estimators. In this section, we show that Cooke’s so-called generic equation, used with the truncated Gutenberg–Richter frequency–magnitude relation, and the Tate–Pisarenko method from Kijko and Singh (2011) are asymptotically equivalent for a large number of earthquakes. In most of the estimators that Kijko and Singh (2011) introduced, they made use of an equation in the form:

$$ m_{\hbox{max} } = m_{\hbox{max} }^{\text{obs}} + \Delta . $$

(32)

Specifically, Cooke’s generic equation has

$$ \Delta = \int\limits_{{m_{\hbox{min} } }}^{{m_{\hbox{max} } }} {\left[ {F\left( {x|m_{\hbox{max} } } \right)} \right]^{n} {\text{d}}x} , $$

(33)

and the Tate–Pisarenko estimate has

$$ \Delta = \frac{1}{{n \times f\left( {m_{\hbox{max} } |m_{\hbox{max} } } \right)}}, $$

(34)

where $f( \cdot |m_{\hbox{max} } )$ and $F( \cdot |m_{\hbox{max} } )$, respectively, are the density and distribution functions of earthquake magnitudes, given the maximum magnitude.

Let $F(m|m_{\hbox{max} } )$ and $f(m|m_{\hbox{max} } )$ take the form of the Gutenberg–Richter frequency–magnitude distribution:

$$ F\left( {m|m_{\hbox{max} } } \right) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {m \le m_{\hbox{min} } } \hfill \\ {\frac{{1 - \exp \left[ {\beta \left( {m - m_{\hbox{min} } } \right)} \right]}}{{1 - \exp \left[ {\beta \left( {m_{\hbox{max} } - m_{\hbox{min} } } \right)} \right]}},} \hfill & {m_{\hbox{min} } \le m \le m_{\hbox{max} } } \hfill \\ {1,} \hfill & {m \ge m_{\hbox{max} } } \hfill \\ \end{array} } \right., $$

(35)

$$ f\left( {m|m_{\hbox{max} } } \right) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {m \le m_{\hbox{min} } } \hfill \\ {\frac{{ - \beta \exp \left[ {\beta \left( {m - m_{\hbox{min} } } \right)} \right]}}{{1 - \exp \left[ {\beta \left( {m_{\hbox{max} } - m_{\hbox{min} } } \right)} \right]}},} \hfill & {m_{\hbox{min} } \le m \le m_{\hbox{max} } } \hfill \\ {0,} \hfill & {m \ge m_{\hbox{max} } } \hfill \\ \end{array} } \right.. $$

(36)

Subsequently, using Eq. (33) with Cramer’s approximation for large values of $n$, we have the following (Kijko and Singh 2011):

$$ \Delta = \int\limits_{{m_{\hbox{min} } }}^{{m_{\hbox{max} } }} {\exp \left[ { - n\left( {1 - F\left( {x|m_{\hbox{max} } } \right)} \right)} \right]{\text{d}}x} . $$

(37)

Laplace’s approximation (Compson 2004) leads to an asymptotic evaluation of the integral (37),

$$ \Delta = \frac{{\exp \left[ { - n\left( {1 - F\left( {m_{\hbox{max} } |m_{\hbox{max} } } \right)} \right)} \right]}}{{n \times f\left( {m_{\hbox{max} } |m_{\hbox{max} } } \right)}} = \frac{1}{{n \cdot f\left( {m_{\hbox{max} } |m_{\hbox{max} } } \right)}}, $$

(38)

which is the Tate–Pisarenko estimate in Eq. (34).