The Fresnel volume and transmitted waves

Abstract

In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high‐frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite‐frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite‐frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band‐limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite‐frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite‐frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat's theorem in the context of geometric ray theory is a special case of the finite‐frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

Estimation of the effect of nonisotropically distributed energy on the apparent arrival time in correlations

“…for positive correlation time, the square root of the ratio between wavelength and distance between receivers: the higher the frequency, the narrower the end-fire lobes (Roux and Kuperman, 2004;Larose, 2005;Spetzler and Snieder, 2004). In the asymptotic limit considered, coherent reconstruction is then controlled by θ approaching 0 (for positive correlation time).…”

Detection of moving capillary front in porous rocks using X-ray and ultrasonic methods

“…Our results show that the P wave amplitude starts to decrease sharply when the capillary front is at a distance between 3 (for Savonnières) and 6 mm (for SaintMaximin) from the ultrasonic sensors. Interestingly this distance is of the same order of magnitude as the size of the first Fresnel zone defined in wave propagation theory [21,22]. This zone can be approximated for homogeneous and isotropic media by an ellipsoid with a major axis equal to the distance between the sensors D and two equivalent minor axes corresponding approximately to the wavelength λ of the P wave propagating in our rocks (λ ∼ 6 mm).…”

Borehole seismic monitoring of CO2 storage within a saline aquifer at Ketzin, Germany

“…"The region with the highest sensitivity corresponds to the first Fresnel volume, through which the main energy of the wave travels to the receiver around the geometric ray" (Liu et al (2009)), therefore the sensitivity kernel can be used to monitor the properties of the Fresnel zone (Chap. 3.5, Jensen et al (2000), Spetzler and Snieder (2004)). When calculating the Fresnel zones, the degree of cancellation in Fresnel zones beyond the first is set to α = 1 (Eq.…”

“…For sensitivity kernels for a multi-frequency wave field the negative and positive Fresnel zones tend to cancel except for the first Fresnel zone, and therefore the Fréchet kernel is almost confined to the first Fresnel zone" (Jensen et al (2000)). "The region with the highest sensitivity corresponds to the first Fresnel volume, through which the main energy of the wave travels to the receiver around the geometric ray" (Liu et al (2009)), therefore the sensitivity kernel can be used to monitor the properties of the Fresnel zone (Jensen et al (2000); Spetzler and Snieder (2004)). …”