Reciprocity and power balance for partially coupled interfaces Kees Wapenaar Evert Slob Jacob Fokkema Centre for Technical Geoscience Delft University of Technology

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

A B A B ‘State A’ ‘State B’

A B A B ‘State A’ ‘State B’

‘State B’ ‘State A’ kA, rA kB, rB V n PB QB QA PA State A State B PA, Vk,A QA,Fk,A kA, rA PB, Vk,B QB,Fk,B kB, rB Wave fields Sources Medium

V n PB QB QA PA ‘State B’ ‘State A’

‘State B’ ‘State A’ V n PB QB QA PA Convolution-type reciprocity theorem: forward problems

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

‘State B’ ‘State A’ V n PB QB QA PA Correlation-type reciprocity theorem

‘State B’ ‘State A’ V n Q Q P P Power-flux through boundary Power dissipated in medium Power radiated by sources

‘State B’ ‘State A’ V n PB QB QA PA Correlation-type reciprocity theorem: inverse problems

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

‘State B’ ‘State A’ V n PB QB QA PA Convolution-type reciprocity theorem

Unified notation (convolution type):

Unified notation (convolution type):

Unified notation (convolution type):

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

Acoustic: Poroelastic: Elastodynamic: Seismoelectric: Electromagnetic:

‘State B’ ‘State A’ V n PB QB QA PA Correlation-type reciprocity theorem

V n PB QB QA PA ‘State B’ ‘State A’

Unified notation (convolution type): Unified notation (correlation type):

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

‘State B’ ‘State A’ n n V V PB QB QA PA Perfectly coupled interfaces: No consequences for reciprocitytheorems of convolution type and correlation type Next: consider partially coupled interfaces

Review of linear slip model Displacement jump:

Review of linear slip model of Schoenberg

Review of linear slip model of Pyrak-Nolte et al.

Review of linear slip model of Pyrak-Nolte et al. Frequency domain

Review of linear slip model of Pyrak-Nolte et al.

Review of linear slip model Schoenberg, Pyrak-Nolte et al: diagonal Nakagawa et al.: full matrix, with Generalization:

Horizontal interface: Arbitrary interface:

Generalized boundary condition:

Acoustic: Poroelastic: Elastodynamic: Seismoelectric: Electromagnetic:

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

n n V V PB QB QA PA ‘State B’ ‘State A’

n n V V PB QB QA PA ‘State B’ ‘State A’

‘State B’ ‘State A’ n n V V PB QB QA PA Convolution-type reciprocity theorem: forward problems

‘State B’ ‘State A’ n n V V PB QB QA PA Correlation-type reciprocity theorem

‘State B’ ‘State A’ n n V V Q P Q P Power-flux through boundary Power dissipated in medium Power radiated by sources Power dissipated by interfaces

‘State B’ ‘State A’ n n V V PB QB QA PA Correlation-type reciprocity theorem: inverse problems

Review of reciprocity theorems Convolution type Correlation type Unified notation Acoustic Elastodynamic Electromagnetic Poroelastic Seismoelectric Review of boundary conditions Extension of reciprocity theorems Conclusions

Unified reciprocity theorems have been formulated of the convolution and correlation type

Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves

Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves Boundary condition for imperfectly coupled interface:

Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves Boundary condition for imperfectly coupled interface: No effects on source-receiver reciprocity

Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves Boundary condition for imperfectly coupled interface: No effects on source-receiver reciprocity Imaginary part of accounts for dissipation by interfaces