Equation 3.1 can be simplified to the following equation
γ(t,m;m_m )=e^(α-βm)/〖(t+c)〗^p (3.2)
Where a_0=a+bm_m , α=a_0 ln10 and β=b ln10 are defined.
|γ_m (t,m;m_m )|=|∂γ(t,m;m_m )/∂m|=e^α/〖(t+c)〗^p βe^(-βm) (3.3)
Where |γ_m (t,m;m_m )| represents the absolute value of the partial derivative of γ(t,m;m_m ), and it is the instantaneous daily rate density of aftershocks of magnitude m at time t following a main-shock of magnitude m_m. e^α/〖(t+c)〗^p denotes the mean instantaneous daily rate of aftershocks at time t following the main-shock of magnitude m_m. βe^(-βm) is the exponential probability density function of aftershock magnitudes. Another form of |γ_m (t,m;m_m )| may be more useful. First we need to define the lower bound and upper bound of magnitude of aftershock (m_l and m_u). Then the exponential part in equation 3.3 can be transferred.
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The value is typically chosen as 5 because the magnitudes smaller than 5 can be ignored in practical engineering. On the other hand, m_u denotes the maximum magnitude of aftershock, usually taken to be the main-shock magnitude m_m. We can transfer equation 3.4 to |γ_m (t,m;m_m )|=μ(t;m_m ) f_M (m;m_m ) m_l≤m≤m_m (3.5) Where μ(t;m_m ) denotes the instantaneous daily rate density of aftershocks with magnitudes between m_l and m_m at time t following a main-shock of magnitude m_m. It is equal to (〖10〗^(a+b(m_m-m_l))-〖10〗^a)/〖(t+c)〗^p . f_M (m;m_m )=(βe^(-β(m-m_l)))/(1-e^(-β(m_m-m_l)) ) denotes the truncated exponential probability density function of aftershock magnitudes (m_l and m_m). μ^* (t,T;m_m )=∫_t^(t+T)▒〖μ(τ;m_m ) □(24&dτ)〗=(〖10〗^(a+b(m_m-m_l))-〖10〗^a)/(p-1)[(t+c)^(1-p)-(t+T+c)^(1-p)]