# User:Ageary/Derivation

The concept of a reflection coefficient is fundamental to reflection seismology. The definition of a reflection coefficient (RC) at normal incidence is:

${\displaystyle RC={\frac {\rho _{2}V_{2}-\rho _{1}V_{1}}{\rho _{2}V_{2}+\rho _{1}V_{1}}}}$

where ${\displaystyle \rho _{2}}$ and ${\displaystyle V_{2}}$ are the density and compressional wave velocity of the medium below a reflecting interface and ${\displaystyle \rho _{1}}$ and ${\displaystyle V_{1}}$ are the density and compressional wave velocity of the medium above a reflecting interface. This formula is well known to all practicing geophysicists, but its derivation from first principles may not be so well known. The following treatment is an expanded version of the derivation given by Dix (p.344-7), in which the direction of wave propagation is shown as horizontal (in the x direction - see the diagram below).[1] This description is simply for convenience, since the actual direction of propagation is vertical (in the z direction) with normal incidence at a horizontal reflecting boundary.

Notation according to Dix[1]

Critical conditions are that both normal stresses and normal displacements are continuous across the reflecting interface (at x = 0) for all time, that is,

 ${\displaystyle (X_{x})_{1}=(X_{x})_{2}\ \ {\text{for all t}}}$ (1)

 ${\displaystyle (u)_{1}=(u)_{2}\ \ {\text{for all t}}}$ (2)

The key to this derivation is recognizing that these conditions mean the particle displacement above (with reference to Dix’s figure, to the left of) the reflecting interface, which is the sum of the particle motions of the incident and reflected pulses, must be equal to the particle displacement below (with reference to Dix’s figure, to the right of) the reflecting interface, which is the particle motion of the transmitted pulse.

 ${\displaystyle {\text{Dilatation}}\ \Delta \sim {\frac {\partial u}{\partial x}}+{\frac {\partial w}{\partial z}}}$ (3)

 ${\displaystyle {\text{Dilatation}}\ \Delta _{x}\sim {\frac {\partial u}{\partial x}}\ \ {\text{in the x-direction}}}$ (4)

From the general form of Hooke’s Law for small strains (Dix, p.304)[1],

 ${\displaystyle X_{x}=2\mu ({\frac {\partial u}{\partial x}})+\lambda ({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}})}$ (5)

where ${\displaystyle \mu }$ and ${\displaystyle \lambda }$ are Lame’s first and second elastic parameters (${\displaystyle \mu }$ is also known as the shear or rigidity modulus). From equations (5) and (4), strain (dilatation) in the x-direction only is

 ${\displaystyle X_{x}=(2\mu +\lambda )({\frac {\partial u}{\partial x}})=(2\mu =\lambda )\Delta _{x}}$ (6)

## Incident, transmitted and reflected pulses

Assume the incident, transmitted and reflected pulses have the following forms, and note with reference to Dix’s original figure that the incident and transmitted pulses travel to the right and the reflected pulse travels to the left:

${\displaystyle {\text{Incident pulse is}}\ \ \ \ \ \ \ \ ({\frac {\partial _{u}}{\partial x}})_{inc}=\Delta _{inc}(x,t)\ \ \ =f(t-{\frac {x}{V_{1}}})}$

${\displaystyle {\text{Transmitted pulse is}}\ ({\frac {\partial _{u}}{\partial x}})_{trans}=\Delta _{trans}(x,t)=af(t-{\frac {x}{V_{1}}})}$

${\displaystyle {\text{Reflected pulse is}}\ \ \ \ \ \ ({\frac {\partial _{u}}{\partial x}})_{refl}=\Delta _{refl}(x,t)\ \ =bf(t+{\frac {x}{V_{1}}})}$

where a and b are scalars for partitioning of energy at the reflecting interface
At x=0,

 ${\displaystyle {\text{Incident pulse is}}\ \ \ \ \ \ \ ({\frac {\partial _{u}}{\partial x}})_{inc}=\Delta _{inc}(0,t)\ \ \ =f(t)}$ (7)

 ${\displaystyle {\text{Transmitted pulse is}}\ ({\frac {\partial _{u}}{\partial x}})_{trans}=\Delta _{trans}(0,t)=af(t)}$ (8)

 ${\displaystyle {\text{Reflected pulse is}}\ \ \ \ \ \ ({\frac {\partial _{u}}{\partial x}})_{refl}=\Delta _{refl}(0,t)\ \ =bf(t)}$ (9)

From (1) ${\displaystyle (X_{x})_{1}=(X_{x})_{2}\ {\text{at}}\ x=0}$, so from (6)

 ${\displaystyle (2\mu _{1}+\lambda _{1})\Delta _{inc}(0,t)+(2\mu _{1}+\lambda _{1})\Delta _{refl}(0,t)=(2\mu _{2}+\lambda _{2})\Delta _{trans}(0,t)}$ ${\displaystyle (2\mu _{1}+\lambda _{1})(1+b)f(t)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =(2\mu _{2}+\lambda _{2})af(t)}$ (10)

## Velocity of displacement

Equation (2) implies that ${\displaystyle ({\frac {\partial u}{\partial t}})_{1}=({\frac {\partial u}{\partial t}})_{2}}$, which means that both displacement and velocity of displacement are continuous across the reflecting interface for all time. The velocity of displacement is related to dilatation by:

${\displaystyle {\text{Incident pulse:}}}$

${\displaystyle ({\frac {\partial u}{\partial x}})_{inc}\ \ \ \ \ =-({\frac {1}{V_{1}}}){\frac {\partial u}{\partial t}}}$

${\displaystyle -V_{1}({\frac {\partial u}{\partial x}})_{inc}={\frac {\partial u}{\partial t}}}$

${\displaystyle {\text{Transmitted pulse:}}}$

${\displaystyle ({\frac {\partial u}{\partial x}})_{trans}\ \ \ \ \ =-({\frac {1}{V_{2}}}){\frac {\partial u}{\partial t}}}$

${\displaystyle -V_{2}({\frac {\partial u}{\partial x}})_{trans}={\frac {\partial u}{\partial t}}}$

${\displaystyle {\text{Reflected pulse:}}}$

${\displaystyle ({\frac {\partial u}{\partial x}})_{refl}\ \ \ =({\frac {1}{V_{1}}}){\frac {\partial u}{\partial t}}}$

${\displaystyle V_{1}({\frac {\partial u}{\partial x}})_{refl}={\frac {\partial u}{\partial t}}}$

By equation (2),

 ${\displaystyle -V_{1}({\frac {\partial u}{\partial x}})_{inc}+V_{1}({\frac {\partial u}{\partial x}})_{refl}=-V_{2}({\frac {\partial u}{\partial x}})_{trans}}$ (11)

Substituting into equation (11) from equations (7) through (9),

 ${\displaystyle -V_{1}f(t)+V_{1}bf(t)=-V_{2}af(t)}$ ${\displaystyle (-V_{1}+V_{1}b)f(t)=-V_{2}af(t)}$ (12)

Divide equations (10) and (12) by ${\displaystyle f_{t}{\text{for}}f_{t}\neq 0}$,

 ${\displaystyle (2\mu _{1}+\lambda _{1})(1+b)=(2\mu _{2}+\lambda _{2})a}$ (13)

 ${\displaystyle -V_{1}+V_{1}b=-V_{2}a}$ (14)

By definition of elastic parameters ${\displaystyle (2\mu +\lambda )=\rho V^{2}}$, so equation (13) is

 ${\displaystyle \rho _{1}V_{1}^{2}(1+b)=\rho _{2}V_{2}^{2}a}$ (15)

## Solving for the reflection coefficient for normal incidence

Equations (14) and (15) are simultaneous in a and b and so can be solved for b, which is the reflection coefficient for normal incidence. Solving equation (14) for a

 ${\displaystyle -V_{1}+V_{1}b\ \ \ \ \ =-V_{2}a}$ ${\displaystyle {\frac {(-V_{1}+V_{1}b)}{-V_{2}}}\ \ =a}$ ${\displaystyle ({\frac {V_{1}}{V_{2}}})(1-b)\ \ \ =a}$ (16)

Solving equation (15) for b,

 ${\displaystyle \rho _{1}V_{1}^{2}(1+b)=\rho _{2}V_{2}^{2}a}$ ${\displaystyle b={\frac {(\rho _{2}V_{2}^{2}a-\rho _{1}V_{1}^{2})}{\rho _{1}V_{1}^{2}}}}$ (17)

Substituting for a from equation (16) into equation (17),

 ${\displaystyle b={\frac {(\rho _{2}V_{2}^{2}[({\frac {V_{1}}{V_{2}}})(1-b)])-\rho _{1}V_{1}^{2})}{\rho _{1}V_{1}^{2}}}}$ (18)

Solving equation (18) for b,

 ${\displaystyle b={\frac {(\rho _{2}V_{1}V_{2}-b\rho _{2}V_{1}V_{2}-\rho _{1}V_{1}^{2})}{\rho _{1}V_{1}^{2}}}}$ ${\displaystyle b+{\frac {(b\rho _{2}V_{1}V_{2})}{\rho V_{1}^{2}}}={\frac {(\rho _{2}V_{1}V_{2}-\rho _{1}V_{1}^{2})}{\rho _{1}V_{1}^{2}}}}$ ${\displaystyle {\frac {b(\rho _{1}V_{1}^{2}+\rho _{2}V_{1}V_{2})}{\rho _{1}V_{1}^{2}}}={\frac {(\rho _{2}V_{1}V_{2}-\rho _{1}V_{1}^{2})}{\rho _{1}V_{1}^{2}}}}$ ${\displaystyle b(\rho _{1}V_{1}^{2}+\rho _{2}V_{1}V_{2})=(\rho _{2}V_{1}V_{2}-\rho _{1}V_{1}^{2})}$ ${\displaystyle b={\frac {(\rho _{2}V_{1}V_{2}-\rho _{1}V_{1}^{2})}{(\rho _{1}V_{1}^{2}+\rho _{2}V_{1}V_{2})}}}$ ${\displaystyle b={\frac {(V_{1}(\rho _{2}V_{2}-\rho _{1}V_{1}))}{(V_{1}(\rho _{1}V_{1}+\rho _{2}V_{2}))}}}$ ${\displaystyle b={\frac {(\rho _{2}V_{2}-\rho _{1}V_{1})}{(\rho _{2}V_{2}+\rho _{1}V_{1})}}}$ (19)

Equation (19) is the definition of a reflection coefficient for normal incidence as stated at the outset.

## References

1. Dix, C. H. (1952), Seismic prospecting for oil: Harper & Brothers.