2:30 PM - 2:45 PM
[SSS12-04] Representation of differential equations for comprehension of aftershock decay and afterslips
Keywords:modified Omori formula, aftershock decay, afterslip, differential equation
The modifid Omori formula of an aftershock decay is represented as n(t)=K(t+c)-p ---(1) (Utsu, Geophys. mag., 1961). The cumulative number N(t) for this equation is represented by N(t)=K{c1-p-(t+c)1-p}/(p-1) (Utsu et al., JPE,1992). In this study, we differentiate eq.(1) with time t. Then we get dn(t)/dt=-pK(t+c)-(p+1)=-p(t+c)-1K(t+c)-p and rewrite as
dn(t)/dt=-p(t+c)-1n(t)---(2).
For comparing, we show a differential equation of radioactive material decay
dN(t)/dt=-λN(t)---(3).
Both equations show that the larger products of the coefficient parts and n(t) or N(t), the higher the rate of decrease, respectively. Comparing both equations, it shows λ~p(t+c)-1---(4). The coefficient in eq.(3) is constant and it means the rate of nuclear decay is time independent, however the coefficient part of the eq.(2) becomes smaller with time. These temporal changes give an image of how, after the main shock, the fault and the surrounding space change from in loose a state, where earthquakes are likely to occur, to in tight a state. Furthermore, the coefficient part is large among shorter from the mainshock, which means that an aftershock productivity is high for one aftershock, and as time passes, the aftershock productivity decreases. This could be thought of as including the effect that aftershocks produce aftershocks such as the larger aftershocks that occur more in a short period from the mainshock produce the more aftershocks. However, it is not a principal mechanism that the aftershocks produce aftershocks, we think rather that essentially the same causes related to afterslip deformation are causing aftershocks.
One model of the afterslip is represented as D(t)=aln(1+t/b)+c+dln(1+t/e)-fexp(-gt)+Vt (Tobita, EPS, 2016). Here, we treat the first term as the main part of the formula. We differentiate with t and get dD(t)/dt=a/b(1+t/b)-1. Put as dD/dt=v, once again we differentiate v(t)=a/b(1+t/b)-1 and reform dv/dt=-1/b(1+t/b)-1a/b(1+t/b)-1 and then get
dv/dt=-1/b(1+t/b)-1v---(5).
It can be seen that the afterslip also shows a similar behavior to the aftershock decay.
By the way, a comparison with the radioactive materials decay that led to the relationship of eq.(4) is introduced in Seismic Studies: A comprehensive Review (Utsu, 1999) as the coefficient μ(t)=(p-1)(t+c)-1. This discrepancy of eq.(4) arises because we compared simply the differential equation for the number of aftershocks per unit time and the number of radioactive materials. Now we think dNa(t)dt=-μ(t)Na(t) as Na(t) numbers of aftershocks occurred after time t. Then the numbers of aftershocks are written as Na(t)=∫n(τ)dτ(integration from t to ∞, same below) and put in the above equation. Then we can get the μ(t) from followings.
left side=d/dt∫n(τ)dτ=∫dn(τ)/dtdτ=n(∞)-n(t)=-n(t)
right side=-μ(t)∫K(τ+c)-pdτ=-μ(t)K/(1-p)(∞+c)1-p+μ(t)K/(1-p)(t+c)1-p=-μ(t)(t+c)/(p-1)n(t)
As another expression of the differential equation, the Omori formula was treated as a solution of namely the truncated Bernoulli differential equation dn/dt+σn2=0 (e.g. Zavyalov et al., Appl. ci., 2022). Guglielmi (Izvestiya, 2016) discussed the σ was time-dependent. However, we believe that a variables separation type differential equation makes it easier to intuitively understand that the coefficient becomes smaller over time.
dn(t)/dt=-p(t+c)-1n(t)---(2).
For comparing, we show a differential equation of radioactive material decay
dN(t)/dt=-λN(t)---(3).
Both equations show that the larger products of the coefficient parts and n(t) or N(t), the higher the rate of decrease, respectively. Comparing both equations, it shows λ~p(t+c)-1---(4). The coefficient in eq.(3) is constant and it means the rate of nuclear decay is time independent, however the coefficient part of the eq.(2) becomes smaller with time. These temporal changes give an image of how, after the main shock, the fault and the surrounding space change from in loose a state, where earthquakes are likely to occur, to in tight a state. Furthermore, the coefficient part is large among shorter from the mainshock, which means that an aftershock productivity is high for one aftershock, and as time passes, the aftershock productivity decreases. This could be thought of as including the effect that aftershocks produce aftershocks such as the larger aftershocks that occur more in a short period from the mainshock produce the more aftershocks. However, it is not a principal mechanism that the aftershocks produce aftershocks, we think rather that essentially the same causes related to afterslip deformation are causing aftershocks.
One model of the afterslip is represented as D(t)=aln(1+t/b)+c+dln(1+t/e)-fexp(-gt)+Vt (Tobita, EPS, 2016). Here, we treat the first term as the main part of the formula. We differentiate with t and get dD(t)/dt=a/b(1+t/b)-1. Put as dD/dt=v, once again we differentiate v(t)=a/b(1+t/b)-1 and reform dv/dt=-1/b(1+t/b)-1a/b(1+t/b)-1 and then get
dv/dt=-1/b(1+t/b)-1v---(5).
It can be seen that the afterslip also shows a similar behavior to the aftershock decay.
By the way, a comparison with the radioactive materials decay that led to the relationship of eq.(4) is introduced in Seismic Studies: A comprehensive Review (Utsu, 1999) as the coefficient μ(t)=(p-1)(t+c)-1. This discrepancy of eq.(4) arises because we compared simply the differential equation for the number of aftershocks per unit time and the number of radioactive materials. Now we think dNa(t)dt=-μ(t)Na(t) as Na(t) numbers of aftershocks occurred after time t. Then the numbers of aftershocks are written as Na(t)=∫n(τ)dτ(integration from t to ∞, same below) and put in the above equation. Then we can get the μ(t) from followings.
left side=d/dt∫n(τ)dτ=∫dn(τ)/dtdτ=n(∞)-n(t)=-n(t)
right side=-μ(t)∫K(τ+c)-pdτ=-μ(t)K/(1-p)(∞+c)1-p+μ(t)K/(1-p)(t+c)1-p=-μ(t)(t+c)/(p-1)n(t)
As another expression of the differential equation, the Omori formula was treated as a solution of namely the truncated Bernoulli differential equation dn/dt+σn2=0 (e.g. Zavyalov et al., Appl. ci., 2022). Guglielmi (Izvestiya, 2016) discussed the σ was time-dependent. However, we believe that a variables separation type differential equation makes it easier to intuitively understand that the coefficient becomes smaller over time.