b value in Seismology

The b value probability density function

The Gutenberg-Richter law (GR Law) [1] describes the relationship between earthquake quanities and earthquake magnitudes with the following relationship

\[log[N(M)] = a - bM\]

where N denotes earthquake quantity with magnitudes no less than M, a and b are two constants.

The above equation could be written as,

\[N(M) = 10^{a-bM}\]

which is a cumulative distritbuion function (CDF). Differentiate and normalize the CDF function can give the probability distribution function (PDF).

The differentiation process is

\[\begin{split}\begin{align} \frac{dN}{dM}&=(10^{a-bM})'\\ &=(-b)\ln(10)10^{a-bM} \end{align}\end{split}\]

The PDF function increases from \(M_{max}\) to \(M_{min}\), which leads to a negative value, therefore, the negative sign should be removed.

For an earthquake catalog with a magnitude range of \([M_{min},M_{max}]\), the total earthquake quantity \(N_{total}=10^{a-bM_{min}}-10^{a-b{M_{max}}}\), the normalization process gives the probability density function (PDF)

\[\begin{split}\begin{align} f(M) &= (b)\ln(10)\frac{10^{a-bM}}{10^{a-bM_{min}}-10^{a-b{M_{max}}}}\\ &= b\ln(10)\frac{10^{-bM}}{10^{-bM_{min}}-10^{-b{M_{max}}}} \end{align}\end{split}\]

Maximum likelihood estimation

By ignoring the \(M_{max}\) conponent, which is a small value if \(M_{max}>>M_{min}\), the PDF function could be written as

\[\begin{split}\begin{align} f(M,b)&=\frac{b}{\log_{10}e}e^{\frac{-b}{\log_{10}e}(M-M_0)}\\ f(M,b')&=b'e^{-b'(M-M_0)} \end{align}\end{split}\]

where \(b'={b}/{\log_{10}e}\).

We define two parameters,

\[\begin{split}\begin{align} y_i&=\frac{\partial}{\partial{b'}}\log f(M_i,b')\\ Y&=\sum_{i=1}^ny_i \end{align}\end{split}\]

For a large \(n\) value, the distributionf of \(Y\) will be Gaussian.

The mean value of \(y_i\) is 0 because

\[\begin{split}\begin{align} E(y)=\int_{M_0}^\infty yf(M,b')dM&=\int_{M_0}^\infty (1+M_0b'-Mb')e^{-b'(M-M_0)}dM\\ &=\int_{M_0}^\infty M_0b'e^{-b'(M-M_0)} dM +\int_{M_0}^\infty (1-Mb')e^{-b'(M-M_0)}dM\\ &=\int_{M_0}^\infty -M_0de^{-b'(M-M_0)} +\int_{M_0}^\infty dMe^{-b'(M-M_0)}\\ &=(0-(-M_0)) +(0-M_0)\\ &=0 \end{align}\end{split}\]

Therefore, for \(Y\), we have

\[E(Y)=E(\sum_{i=1}^ny_if(M_i,b'))=0\]

We then can get

\[\begin{split}\begin{align} 0&=\sum_{i=1}^{n}(\frac{1}{b'}+M_0-M_i)\\ &=\frac{n}{b'}+nM_0-\sum_{i=1}^{n}M_i\\ b'&=\frac{1}{\sum{M_i}/n-M_0} \end{align}\end{split}\]

Using the relationship \(b=b'\log_{10}e\) we can get the b value.

The standard error

The variance of \(y'\) could be obtained by

\[E(y^2)=\int_{M_0}^{\infty}y^2f(M,b')dM=\int_{M_0}^{\infty}(1+M_0-M)^2e^{-b'(M-M0)}dM=\frac{1}{b'^2}\]

References