On acoustic reciprocity theorems and the construction of

On acoustic reciprocity theorems and the construction of transmission response from reflection data Bogdan G. Nita *University of Houston M-OSRP Annual Meeting 20 -21 April, 2005 University of Houston

Outline • • Motivation One-way wavefield decomposition Reciprocity theorems Reconstructing the phase from amplitude information • Minimum phase condition • Summary and Conclusions

Outline • • Motivation One-way wavefield decomposition Reciprocity theorems Reconstructing the phase from amplitude information • Minimum phase condition • Summary and Conclusions

Motivation • Virtual source – Shell • Seismic interferometry • Deep earth seismology • Model type independent imaging

Inverse scattering imaging • Inverse scattering imaging subseries method has shown tremendous value for 1 D and 2 D acoustic media (Shaw, Liu) • H. Zhang leads the efforts to identify the subseries for imaging in a 1 D elastic medium • Model type independent method

Internal multiple attenuation subseries = G 0 =D • The attenuation algorithm requires three reflection data sets to build up an internal multiple = Imaged Data

Leading order imaging sub-series Linear 2 nd Order 3 rd Order 4 th Order = G 0 + = V 1 + A subseries of the inverse series + + …

Data requirements for model type independent imaging Reflection data Transmission data

Methods for obtaining transmission data • Measure/record it (e. g. VSP) • Determine it from reflection data using reciprocity theorems Inverse scattering series constructs the transmission response order by order from reflection data •

Outline • • Motivation One-way wavefield decomposition Reciprocity theorems Reconstructing the phase from amplitude information • Minimum phase condition • Summary and Conclusions

Seismic experiment FS At any depth, the total wavefield has an up-going and a down-going component

Two way wavefield reciprocity FS • Acoustic response does not change if the source and receiver are interchanged

Two way wavefield reciprocity FS • Acoustic response does not change if the source and receiver are interchanged

Why do we need one-way wavefields • Migration • Deghosting • To be able to define reflection and transmission responses

One-way wavefields • Reciprocity is not obvious for one way wavefields • One way wavefield decomposition is not unique

Up-down wavefield decomposition • Pressure normalized one-way wavefields – Widely used – Do not satisfy the reciprocity theorem • Flux normalized one-way wavefields – Satisfy the reciprocity theorem M. V. De Hoop 1996, Wapenaar 2004, 2005

Pressure normalized up-down decomposition Acoustic pressure Particle velocity

Pressure normalized up-down decomposition 1 D medium Continuity of P and Vz at the interface Reciprocity is not satisfied!

Flux normalized up-down decomposition Acoustic pressure Particle velocity

Flux normalized up-down decomposition 1 D medium Continuity of P and Vz at the interface Reciprocity is satisfied!

Medium dependence • The one-way wavefield decompositions only depend on the medium where the data is collected

Conclusions: one-way wavefield decomposition • Decomposition is not unique • Pressure normalized one-way wavefields do not satisfy reciprocity • Flux-normalized one-way wavefields satisfy reciprocity • The two decompositions only depend on the medium where the data is collected

Outline • • Motivation One-way wavefield decomposition Reciprocity theorems Reconstructing the phase from amplitude information • Minimum phase condition • Summary and Conclusions

Reciprocity theorems • Two-way wavefields – Convolution type – Correlation type • One way wavefields – Convolution type – Correlation type Fokkema and van den Berg 1990

One-way wavefield theorem of the correlation type Independent acoustic states • The region between and is source free • Valid only for lossless media with evanescent waves neglected

Transmission from reflection • Use the one way reciprocity of the correlation type • Same experiments and Substitute into the one-way reciprocity theorem of correlation type and divide by the source wavelet

Transmission from reflection • relation between the amplitude of reflection data and that of transmission data • all the phase information is lost and there is no unique way of recovering it • phase reconstruction requires one additional relation which is sometimes provided by the minimum phase condition • minimum phase property for a wavefield depends on the medium that the wave propagates through • for general 3 D acoustic and elastic media the wavefield usually has mixed phase

Conclusions for reciprocity theorems • One way reciprocity theorem of correlation type provides a relation between the amplitude of the reflection data and that of the transmission data • To recover the phase one needs one additional relation which is sometimes provided by the minimum phase condition

Outline • • Motivation One-way wavefield decomposition Reciprocity theorems Reconstructing the phase from amplitude information • Minimum phase condition • Summary and Conclusions

A real signal • For arbitrary functions there is no connection between X and Y

Causal signals Causal Is analytic in the upper half complex plane Causal Related through Hilbert transforms

Causal signals • A causal signal can be fully reconstructed from its frequency domain real or imaginary parts

Amplitude and phase relations

Amplitude and phase relations • When F contains no zeroes in the upper complex-frequency half plane

Amplitude and phase relations • F is analytic and has no zeros implies is analytic and hence its real and imaginary parts are related through Hilbert transforms • phase is constructed from amplitude

Amplitude and phase relations • F is analytic in the upper complex-frequency half plane Causality • F has no zeroes in the upper complex-frequency half plane – Minimum phase condition

Conclusions: Reconstructing the phase from amplitude information • The phase can be reconstructed from amplitude information only if the signal is – Causal – Satisfies the minimum phase condition

Outline • • Motivation One-way wavefield decomposition Reciprocity theorems Reconstructing the phase from amplitude information • Minimum phase condition • Summary and Conclusions

Minimum phase condition • A signal is minimum phase if it has no zeroes in the upper complex-frequency half plane • The inverse has no poles hence it is analytic • Zeroes create phase-shifts • Passing beneath a zero causes a phase-shift of • Minimum phase-shift Complex frequency plane

Minimum phase condition in time domain Eisner (1984) Output energy • the output energy of a minimum phase signal integrated up to time T is greater than that of a non-minimum phase signal with the same frequency-domain magnitude • Hence a minimum phase signal has more energy concentrated at earlier times than any other signal sharing its spectrum

Minimum phase reflectors • A minimum phase reflector has the property of reflecting the acoustic energy faster than any non-minimum phase reflector • In a minimum phase medium the perfect velocity transfer condition is satisfied: the wave that enters the medium and the one that exits it have the same propagation speed • This holds for normal incident intramodal reflection – more general situations (e. g. converted waves) are presently under investigation

Outline • • Motivation One-way wavefield decomposition Reciprocity theorems Reconstructing the phase from amplitude information • Minimum phase condition • Summary and Conclusions

Summary and conclusions • Model type independent ISS imaging requires both reflection and transmission data • One-way reciprocity theorem of the correlation type relates amplitude of the reflection data and transmission data • To recover the phase an additional condition – minimum phase condition – is necessary • Seismic arrivals are presently under investigation to determine their phase properties

Acknowledgements • Co-author: Arthur B. Weglein. • Support: M-OSRP sponsors. • Collaboration with Gary Pavlis and Chengliang Fan, Indiana University