TY - JOUR
T1 - Formulation of the spatial autocorrelation (SPAC) method in dissipative media
AU - Nakahara, Hisashi
N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.
PY - 2012/9
Y1 - 2012/9
N2 - Spatial autocorrelation (SPAC) method is formulated in dissipative media for one-, two- and three-dimensional (1-D, 2-D and 3-D) scalar wavefields based on the generalized wave equation or the generalized telegraph equation. A rather straightforward derivation is possible by using a close mathematical relation between the SPAC method and seismic interferometry, though a model set-up should be modified by including attenuation. For 3-D cases, the normalized cross spectrum of a scalar wavefield in a dissipative medium is found to be different from that in a non-dissipative medium merely by an exponentially damping term. However, expressions for 1-D and 2-D cases are not as simple as 3-D cases meaning that not only amplitude but also phase in the normalized cross-spectrum are modified by attenuation. The SPAC expressions derived are considered to be applied to rather homogeneous distribution of attenuation. For the estimation of attenuation, the SPAC method needs to be used for larger station separations than the traditional SPAC method does. Analysis of the SPAC expressions for 2-D cases shows that the conjecture of Prieto (2009) is not strict but approximately good for small attenuation. This study will provide a theoretical basis to estimate not only phase velocity but also attenuation from analysis of ambient noises.
AB - Spatial autocorrelation (SPAC) method is formulated in dissipative media for one-, two- and three-dimensional (1-D, 2-D and 3-D) scalar wavefields based on the generalized wave equation or the generalized telegraph equation. A rather straightforward derivation is possible by using a close mathematical relation between the SPAC method and seismic interferometry, though a model set-up should be modified by including attenuation. For 3-D cases, the normalized cross spectrum of a scalar wavefield in a dissipative medium is found to be different from that in a non-dissipative medium merely by an exponentially damping term. However, expressions for 1-D and 2-D cases are not as simple as 3-D cases meaning that not only amplitude but also phase in the normalized cross-spectrum are modified by attenuation. The SPAC expressions derived are considered to be applied to rather homogeneous distribution of attenuation. For the estimation of attenuation, the SPAC method needs to be used for larger station separations than the traditional SPAC method does. Analysis of the SPAC expressions for 2-D cases shows that the conjecture of Prieto (2009) is not strict but approximately good for small attenuation. This study will provide a theoretical basis to estimate not only phase velocity but also attenuation from analysis of ambient noises.
KW - Seismic attenuation
KW - Wave propagation
UR - http://www.scopus.com/inward/record.url?scp=84865296787&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84865296787&partnerID=8YFLogxK
U2 - 10.1111/j.1365-246X.2012.05591.x
DO - 10.1111/j.1365-246X.2012.05591.x
M3 - Article
AN - SCOPUS:84865296787
VL - 190
SP - 1777
EP - 1783
JO - Geophysical Journal International
JF - Geophysical Journal International
SN - 0956-540X
IS - 3
ER -