Accuracy

From the series definition of a matrix function, we see that the matrix exponential can be represented as follows:

$$\exp(\Delta z M)= \sum_{k=0}^{\infty}{(\Delta z M)^k\over k!}.$$ (11)

Richardson, et al. approximate this matrix exponential by splitting the given matrix into pieces:

$$\exp(\Delta z M)=\exp(\Delta z ( M_e + M_o)).$$ (12)

Equation (12) can be represented with the series

$$\exp(\Delta z ( M_e + M_o)) = I + \Delta z ( M_e + M_o) + {{\Delta z^2 ( M_e + M_o)^2}\over 2!} + \cdots$$ (13)

$$= I + \Delta z ( M_e + M_o) + {{\Delta z^2 ( M_e^2 + M_eM_o + M_oM_e + M_o^2)}\over 2!} + \cdots.$$ (14)

An approximate calculation of equation (13) is

$$\exp(\Delta z M_e)\exp(\Delta z M_o)$$ (15)

$$= \left(I + \Delta z (M_e) + {{\Delta z^2 (M_e)^2}\over 2!} ... ...Delta z (M_o) + {{\Delta z^2 (M_o)^2}\over 2!} + \cdots \right)$$ (16)

$$= I + \Delta z (M_e + M_o) + {{\Delta z^2 (M_e^2 + 2M_eM_o + M_o^2)}\over 2!} + \cdots.$$ (17)

Equation (14) and (15) are identical if M_e and M_o are commutative, i.e., 2M_eM_o = M_eM_o + M_oM_e. However, in our case, M_e and M_o are not commutative, thus approximation (15) will have errors on the order of $\epsilon(\Delta z^2)$.By careful arrangement of two different split matrices, we can improve the accuracy as follows. We define

$$M = {M_e\over 2}+M_o+{M_e\over 2}$$ (18)

which has an error on the order of $\epsilon(\Delta z^3)$, and

$$M = {M_e\over 4}+{M_o\over 2}+{M_e\over 2}+{M_o\over 2}+{M_e\over 4}$$ (19)

with an error on the order of $\epsilon(\Delta z^4)$.

Figure shows the improvement in accuracy when we use a higher order approximation. On the left impulse response, we used the first order approximation with split as equation (18) and on the right impulse response, we used the third order approximation as equation (19). We can see a decrease in the dispersion at higher order approximation. Comparing with the impulse response of implicit scheme in Fig , the higher order explicit scheme shows the comparable accuracy as we can see in Fig .

 

fig2
Figure 2 Impulse responses of explicit 15-degree migration with the split matrices (a) M = M_e+M_o, (b) M = M_e/4+M_o/2+M_e/2+M_o/2+M_e/4.