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

.. math::
    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,

.. math::
    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

.. math::
    \begin{align}
        \frac{dN}{dM}&=(10^{a-bM})'\\
        &=(-b)\ln(10)10^{a-bM}
    \end{align}

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

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

.. math::
    \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}


Maximum likelihood estimation
-----------------------------------
By ignoring the :math:`M_{max}` conponent, which is a small value if :math:`M_{max}>>M_{min}`,  the PDF function could be written as

.. math::
   \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}

where :math:`b'={b}/{\log_{10}e}`.

We define two parameters,

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

For a large :math:`n` value, the distributionf of :math:`Y` will be Gaussian.

The mean value of :math:`y_i` is 0 because

.. math::
   \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}

Therefore, for :math:`Y`, we have

.. math::
   E(Y)=E(\sum_{i=1}^ny_if(M_i,b'))=0

We then can get 

.. math::
   \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}

Using the relationship :math:`b=b'\log_{10}e` we can get the *b* value.

The standard error
~~~~~~~~~~~~~~~~~~~~~~~~
The variance of :math:`y'` could be obtained by

.. math::
   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
---------------------------------
.. [1] Gutenberg, B. and C. Richter (1954). *Seismicity of the Earth and Associated Phenomena*, 2nd Edition, pp. 310.
.. [2] Aki, Keiiti (1965). *17. Maximum Likelihood Estimate of b in the Formula logN=a-bM and its Confidence Limits*. Bulletin of the Earthquake Research Institude