所謂的全波形反演的伴隨狀態法

07-31

所謂的全波形反演的伴隨狀態法

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假設全波形反演的目標函數具有如下形式：

$Jleft(m ight)=frac{1}{2}int_{0}^{T}fleft(u,m,t ight)dt$

常見的 $fleft(u,m,t ight)$ 的選取方式有：

$fleft(u,m,t ight)=sum_{i=0}^{k}|uleft(x_{r_i},t ight)-d_{r_i}left(t ight)|^2$

$frac{delta J}{delta m}=int_{0}^{T}frac{delta f}{delta u}frac{delta u}{delta m}dt$

波場u滿足波動方程：

$egin{equation} left{ egin{aligned} & m^2frac{partial^2 u}{partial t^2}=Delta u, xin Omega=H imes R^2\ & uleft(x,0 ight)=0 \ & frac{partial u}{partial t}left(x,0 ight)=0 \ & uleft(x,t ight)Big|_{partialOmega}=0 end{aligned} ight. end{equation}$

其中 $H=[0,infty)$

然後呢，這個 $frac{delta u}{delta m}$ 就要滿足下面的微分方程：

$egin{equation} left{ egin{aligned} & m^2frac{partial^2 }{partial t^2}frac{delta u}{delta m}-Deltafrac{delta u}{delta m}=-2mfrac{partial^2 u}{partial t^2}, xin Omega=H imes R^2\ & frac{delta u}{delta m}left(x,0 ight)=0 \ & frac{partial }{partial t}left(frac{delta u}{delta m} ight)left(x,0 ight)=0 \ & frac{delta u}{delta m}left(x,t ight)|_{partialOmega}=0 end{aligned} ight. end{equation}$

$egin{equation} egin{aligned} frac{delta J}{delta m}&=int_{0}^{T}sum_{i=0}^{k}left(u(r_{i},t)-d_{i}left(t ight) ight)frac{delta u}{delta m}dt \ &=int_{0}^{T}int_{Omega}frac{delta u}{delta m}sum_{i=0}^{k}left(uleft(x,t ight)-dleft(r_i,t ight) ight)delta(x-r_i)dVdt end{aligned} end{equation}$

計算FWI的梯度需要用到Green公式。就簡單推一下Green公式應該怎麼寫吧。首先考慮Stokes公式

$int_{partial Omega}omega=int_{Omega}domega$

根據這個Stokes定理，可以推出下面這個式子成立

$egin{equation} egin{aligned} &int_{Omega}left(vfrac{partial^2 u}{partial (x^{i})^2}-ufrac{partial^2 v}{partial (x^{i})^2} ight)dx^{1}wedge dx^{2}wedge dots wedge dx^{n} \ =&int_{partial Omega}(-1)^{i-1}left(vfrac{partial u}{partial x^{i}}-ufrac{partial v}{partial x^{i}} ight)dx^{1}wedge dx^{2}wedgedots dx^{i-1}wedge dx^{i+1}wedge dotswedge dx^{n} end{aligned} end{equation}$

這個其實就是Green公式。通過微分形式的運算規則可以驗證上面的微分關係，於是式子就成立了。

$egin{equation} egin{aligned} & dleft(left(vfrac{partial u}{partial x^{i}}-ufrac{partial v}{partial x^i} ight)dx^1wedge dx^2wedgedotswedge dx^{i-1}wedge dx^{i+1}wedgedotswedge dx^{n} ight) \ =&(-1)^{i-1}left(frac{partial v}{partial x^i}frac{partial u}{partial x^i}+vfrac{partial^2 u}{partial (x^i)^2}-frac{partial u}{partial x^i}frac{partial v}{partial x^i}-ufrac{partial^2 v}{partial (x^i)^2} ight)dx^1wedge dx^2wedgedotswedge dx^n \ =&(-1)^{i-1}left(vfrac{partial^2 u}{partial (x^i)^2}-ufrac{partial^2 v}{partial (x^i)^2} ight)dx^1wedge dx^2wedgedotswedge dx^n end{aligned} end{equation}$

有了Green公式，在 $frac{delta u}{delta m}$ 滿足的微分方程兩端同時乘以一個函數w。這裡w是一個輔助函數，只是為了計算梯度，後面可以推出w需要滿足的方程。

$m^2wfrac{partial^2 }{partial t^2}frac{delta u}{delta m}=wDelta frac{delta u}{delta m}-2mwfrac{partial^2 u}{partial t^2}$

對上面的等式兩邊積分，在，然後推啊推啊

$egin{equation}egin{aligned} &int_{0}^Tint_{Omega}wDelta vdxdydzdt\ =&int_0^Tint_{Omega}wleft(frac{partial^2 v}{partial x^2}+frac{partial^2 v}{partial y^2}+frac{partial^2 v}{partial z^2} ight)dxdydzdt\ =&int_0^Tint_{Omega}vleft(frac{partial^2 w}{partial x^2}+frac{partial^2 w}{partial y^2}+frac{partial^2 w}{partial z^2} ight)dxdydzdt\ +&int_0^Tint_{partial Omega}left(left(vfrac{partial w}{partial x}-wfrac{partial v}{partial x} ight)dydz \ +left(-vfrac{partial w}{partial y}+wfrac{partial v}{partial y} ight)dxdz \ +left(vfrac{partial w}{partial z}-wfrac{partial v}{partial z} ight)dxdy ight)dt\ =&int_0^Tint_{Omega}vDelta wdxdydzdt+int_0^Tint_{partial Omega}left(vfrac{partial w}{partial z}-wfrac{partial v}{partial z} ight)dxdydt \ =&int_0^Tint_{Omega}vDelta wdxdydzdtend{aligned}end{equation}$

$egin{equation}egin{aligned} &int_0^Tint_{Omega}wfrac{partial^2 v}{partial t^2}dxdydzdt\ =&int_0^Tint_{Omega}vfrac{partial^2 w}{partial t^2}dxdydzdt+int_{Omega}left(wfrac{partial v}{partial t}Big|_{t=0}^{t=T}-vfrac{partial w}{partial t}Big|_{t=0}^{t=T} ight)dxdydz end{aligned}end{equation}$

你會發現當w滿足一定條件時，下面這個式子會成立

$int_0^Tint_{Omega}frac{delta u}{delta m}left(m^2frac{partial^2}{partial t^2}-Delta ight)wdxdydzdt=int_0^Tint_{Omega}-2mwfrac{partial^2 u}{partial t^2}dxdydzdt$

w滿足什麼條件呢，也就是滿足以下微分方程時，上面那個式子成立

$egin{equation} left{ egin{aligned} & m^2frac{partial^2 w}{partial t^2}-Delta w=sum_{i=0}^{k}left(uleft(x,T-t ight)-dleft(x,T-t ight) ight)deltaleft(x-r_i ight), xin Omega=H imes R^2\ & wleft(x,T ight)=0 \ & frac{partial w}{partial t}left(x,T ight)=0 \ & wleft(x,t ight)Big|_{partialOmega}=0 end{aligned} ight. end{equation}$

然後人們就把這個w叫做伴隨波場，這個方法就叫做伴隨狀態法，可以利用這個方法比較方便地在數值上計算FWI的梯度。

如果目標函數的形式發生變化，只需要將第一個微分方程的右端項修改為相應的形式，就可以得到對應的梯度計算方法了。

其實波場的正向傳播，然後是殘差的反向傳播，非常像人工神經網路里的BP演算法，BP也是先feed forward，然後誤差反向傳播，而且關鍵是兩者都是在計算梯度。我覺得這兩個東西如果從某個更深刻的角度來看，應該會有聯繫。