Diffraction, Interference, and Depolarization of Elastic ... - Springer Link

Journal of Mathematical Sciences, Vol. 127, No. 6, 2005

DIFFRACTION, INTERFERENCE, AND DEPOLARIZATION OF ELASTIC WAVES. CAUSTIC AND PENUMBRA A. P. Kiselev, V. O. Yarovoy, and E. A. Vsemirnova

UDC 550.3; 534.2; 517.9

A theoretical study of polarization-spectral anomalies of wave fields, or their deviations from the predictions based on simple plane-wave models, is presented. A simple unified method of numerical simulation of anomalies of nonstationary wave fields near a caustic and in the penumbra is described. It uses both the leading and correcting terms in asymptotic expansions. Examples of calculation of displacements and average polarization ellipses are given. Qualitative properties of wave fields are discussed. A review of earlier research on polarization anomalies of elastic waves is given. Bibliography: 38 titles.

Dedicated to L. A. Molotkov on the occasion of his 70th birthday 1. Introduction The polarization of the plane wave is commonly accepted (especially in the seismic context) as standard or “normal” polarization. Anomalous polarization or depolarization is a deviation from the plane-wave standard. The theory of homogeneous plane waves (for example, see [1]) predicts the linear polarization of P waves. However, a detailed analysis of VSP field data shows that nonlinear polarization is typical of even first arrivals. Proper processing of PM VSP data reliably fixes a nonlinearity of polarization of even few percent (e.g., see [2–5]). In the 1980s (see [2]) it was expected that mass introducion of three component measurements in the vertical seismic profiling (VSP) would yield qualitatively new information about the structure of the surrounding medium. These expectations, which seem to us very reasonable, still remain to be realized. Theorists are, at least partly, responsible for that, because, first, they did not yet complete the analysis of all important specific machineries of depolarization and, second, they did not succeed in sharing their knowledge with seismic exploration audience. In this paper, we attempt to make a step forward in both directions. Main and additional components. In the cases where the wave can be reliably traced, the component of the displacement directed “as a respective plane wave” is said to be the main or “normal” component, while the components orthogonal to it are called anomalous or additional. Generally, the waveforms are different in them. The difference in waveforms is determined by the reasons why polarization anomalies arise. The spectral and correlation characteristics of all components carry some information about details of the wave process and the structure of the surrounding medium (and E. I. Galperin’s expectations [2] of three-component data analysis were seemingly based on this). These characteristics have earlier been analyzed in more or less detail for various models of the medium and the wave process. Diffraction in the context of ray theory. The term “additional component” is borrowed from the classical paper on ray theory [6], which generalized homogeneous plane waves. Analysis (see [7]) showed that the notion of wave itself is based on ray theory. Within its framework, the main and the additional component arise as the leading and the first correcting term of an asymptotic expansion. In the case of a smoothly inhomogeneous medium (in a homogeneous medium, in particular), for a regular field of rays when the wavefront can be locally approximated by its tangent plane, the presence of additional components is naturally explained by the transverse diffusion of the wave field by a large-scale, smooth inhomogeneity (see [8]). In this case, additional components were discussed by several authors (e.g., see [9–12, 14] and others). It was shown that the waveform in the additional component is an integral of the waveform in the main component. For this reason, consideration of polarization anomalies requires little complication in the computation of the wave field. The additional components in these models can easily be traced because of the stability of their waveforms and amplitudes. Interference. Several authors (e.g., see [15, 16]) studied polarization anomalies of elastic waves near interfaces of large size, where anomalies are due to interference. Effects of scattering by small isolated inclusions and their periodic compositions were investigated (see [17]). It was observed that in scattering by smooth small inclusions Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 136–153. Original article submitted March 10, 2003. c 2005 Springer Science+Business Media, Inc. 1072-3374/05/1276-2413


2413

(e.g., see [18] and references therein) and by sharp small and middle-scale inclusions (see [19, 20]), the additional component is chaotic in form and amplitude and is therefore poorly correlated. In the present paper, we deal with model examples in which diffraction and interference intertwine. Here, waveforms in the main and additional components vary in a complicated way, and therefore the employment of the Fourier transform with respect to time (which is not necessary when ray theory is applicable) is inevitable for calculating nonstationary wave fields. Boundary layers. We consider wave processes associated with the situation where the regularity of geometric ray fields does not hold near certain singular lines in the vicinity of which the wavefronts do not resemble plane ones. Near these lines, separate wave arrivals merge together, and one can say that interfering waves ”enter into diffractional interaction.” When the regularity of rays does not take place, diffraction and interference intertwine. Small regions where the above phenomena occur were named boundary layers (see, e.g., [21]). Boundary layers associated with caustics and penumbral areas are of most interest for seismic studies. In both cases, there exist regions where particular wave arrivals can be identified and traced, as well as narrow boundary layers in the strict sense where the phases of waves arrivals coincide and their individuality is lost. Caustic. A caustic, i.e., an envelope of rays, may arise, e.g., upon reflection from a concave surface. In the illuminated area, at a distance from the caustic, interfering incident and outgoing waves can be distinguished. In the shadow area no waves exist. When passing from the illuminated area to the shadow one, in a narrow boundary layer around the caustic one observes, first, a confluence of interfering waves and, next, the gradual disappearance of the wave field. A caustic corresponds to a fold of a wavefront, and there the wavefront is quite unlike a plane one. For this reason it is natural that the wave field near a caustic is not described by ray theory.

Fig. 1. Rays near a caustic. Leading-term formulas for a scalar wave field near a simple, smooth caustic have long been known (see [22, 23]). We also mention recent paper [24]. In the seismic context, the case of elasticity theory was discussed, e.g., in [25–27]. Penumbra. Another example of the diffraction boundary layer, important for seismic studies, is a penumbral zone which arises in diffraction by a large-scale obstacle, either smooth or having a sharp edge. In the case of a sharp edge (see Fig. 2a), the incident and diffracted edge waves interfere inside the illuminated area; inside the shadow area, a diffracted edge wave exists. In the boundary layer around the line separating the illuminated and shadow areas, the wave field has a specific Fresnel-type structure. For a convex, smooth obstacle, the incident and reflected waves interfere inside the illuminated area; in the shadow, the wave field is described by the Friedlander–Keller modes, which travel along the surface and then slide off, see [21] (Fig. 2b). Near the geometric boundary of the illuminated and shadow areas, the wave field has a specific boundary-layer structure. For both cases of smooth and edged bodies, the leading term of the asymptotic wave field in the boundary layer is the same and involves the Fresnel integral. The solution of the scalar problem for an edged body was described by Fresnel; for the smooth case this was done by Fock [28, 29]. Relevant formulas for the leading term in elastodynamics were obtained simultaneously for a smooth, convex body in [30–32] and for an edged body in [33] (in the spirit of the boundary layer approach [28, 21] and by the techniques akin to the Kirchhoff method [23], respectively). For a particular angular obstacle, namely, for a half-infinite crack, a Fresnel integral was 2414

derived from the explicit solution of the diffraction problem in [34]. For discussion of the leading term of the asymptotic expansion, see also [35, 36]. For the case of an isotropic homogeneous medium, formulas with Fresnel integrals can be derived by a very simple procedure [13, 14] on the basis of asymptotic expressions for the potentials given in standard textbooks on diffraction theory [22, 23]. The “normally” polarized wave field is described by the leading term of the asymptotic expansion, and anomalous polarization is given by the leading correcting term. This trick is also convenient (see [14]) for boundary layers of a different nature, for example, for a caustic (and not necessarily simple and smooth one).

(a)

(b) Fig. 2. Rays in a penumbral area: (a) a large-scale smooth convex body; (b) a large-scale edge. The goal of the present paper. Prospective generalizations. The above-mentioned papers concerning the caustic and penumbral wave fields discuss either the “normal” polarization or the harmonic in time problem. Here, we suggest a simple unified method of numerical simulation of the polarization of a nonstationary elastic wave field near a smooth (i.e., having no cusps) caustic and in the penumbra. The method allows us to obtain qualitative and quantitative information about the spectra of the components of the wave field. We consider anomalies of P waves in the simple case of a homogeneous, isotropic medium. The anomalies are found to be rather intensive. Consideration of S waves in such media, based on the formulas presented in [14], is quite similar, and the characteristics of additional components appear qualitatively the same, but their contribution to the total wave field increases (similar results were recorded (e.g., see [11]) in the cases where ray theory was applicable). More complicated caustics in homogeneous media can be considered in much the 2415

same way. Consideration of a smooth inhomogeneity and anisotropy complicates significantly the mathematical machinery without modifying qualitative conclusions. 2. Nonstationary wave fields and their spectra Potentials. A nonstationary displacement field u = u(r, t) in an isotropic elastic medium, where r is an observation point and t is time, is described by the elastodynamics equation a2 grad div u − b2 rot rot u −

∂ 2u = 0, ∂t2

(1)

where a and b stand for the velocities of P and S waves. The field of P waves can be represented as u = grad ϕ,

(2)

where ϕ = ϕ(r, t) is a scalar potential that satisfies the scalar wave equation ∆ϕ −

1 ∂2ϕ =0 a2 ∂t2

(3)

with ∆ denoting the Laplacian. For simplicity, we assume in the sequel that the wave process is independent of the Cartesian coordinate z, whence r = (x, y) and ∆ = ∂ 2 /∂x2 + ∂ 2 /∂y2 . We use classical formulas for the spectra of nonstationary signals, which are easily described in terms of the spectrum Φ = Φ(r, ω) of the potential ϕ(r, t). They are connected by the relation +∞


Φ(r, ω)e−iωt dω.

ϕ(r, t) =

(4)

−∞

Here, for ω > 0, the variable ω has the meaning of the circular frequency. Obviously, Φ(r, ω) satisfies the Helmholtz equation ω2 (5) ∆Φ + 2 Φ = 0, a and the spectrum of a displacement vector (which is a displacements field harmonic in time) U = U(r, ω) can be represented as U(r, ω) = grad Φ(r, ω).

(6)

A consequence of the fact that the displacement field is real. The nonstationary displacement wave field u(r, ω) is real and therefore the potential ϕ is real as well. However the spectra of wave fields in physically interesting problems are usually complex. The fact that ϕ is real immediately implies (e.g., see [37]) the following way of continuation of the complex spectrum to negative values of ω: Φ(−ω) = Φ∗ (ω) with

∗

(7)

standing for complex conjugation. Equations (4) and (7) imply that
∞ [ReΦ(ω) cos ωt − ImΦ(ω) sin ωt] dω,

ϕ(t) = 0

where Re and Im are the real and imaginary parts. Henceforth we omit the dependence on r. 2416

(8)

3. The wave field near a caustic Spectrum of the caustic potential. Coordinates convenient for the description of the wave field in the vicinity of a caustic are the normal n to the caustic, directed to the illuminated area, and the arc length s of the caustic; in these coordinates, ∂ 1 ∂ grad = en + es . (9) n ∂n 1+ R ∂s Here en and es are unit vectors along the coordinate curves and R is the curvature radius of the caustic. By the main component we mean the component tangent to the caustic, us = (u · es ). The description of the spectrum near the caustic will be based on the following formula, which is asymptotic for large values of ωR/a: (10) Φ(ω) = eiωs/av (−N ) Ψ(ω), where 1 v(M ) = √ π


∞ cos(βM + β 3 /3)dβ 0

is the Airy–Fock function (see [21, 29]) and



2ω2 a2 R is the dimensionless normal. Here, the assumption usual for time-harmonic diffraction theory that ω > 0 is made. The only difference with the classical formula, which can be found, e.g., in [21–23], is the factor Ψ(ω), which should be determined depending on the waveform in the incident wave. N =n

3

An asymptotic wave field in the illuminated area. In the illuminated area described by N > 0, for distances from the caustic such that N is large (and the incident and outgoing waves are therefore distinguishable), we use the asymptotic formula (see [21, 29]) v(−N ) ≈

 sin( 23 N 3/2 + π/4) 1  −i 2 N 3/2 +iπ/4 + 23 iN 3/2 −iπ/4 3 √ √ = e + e , 4 4 N 2 N

N → +∞.

This enables us to represent the spectrum of the potential as the sum of an incident and an outgoing wave Φ− and Φ+ , Φ∼ = Φ− + Φ+ , where Φ− =

− A Ψ(ω)eiωτ +iπ/4 , 1/6 ω

Φ+ =

Here,

+ A Ψ(ω)eiωτ −π/4 , 1/6 ω

 12

A = A(r) = is real and independent of ω, τ

−

1 = a



2 s− 3

ω > 0.

(11)

a2 R 2

√ 24n 

2n3 R



is the geometric travel time for the incident wave, and    1 2 2n3 + τ = s+ a 3 R is the travel time for the outgoing wave (see [14, 21]). In accordance with (9) and (11), the main component of the incident elastic wave field for small n and large N equals u− s ≈

∂Φ− iω − iωA ≈ Φ ≈ 1/6 eiπ/4 Ψ(ω), ω > 0. ∂s a ω

(12)

2417

Determination of the spectrum of the penumbral wave field by the waveform of the incident wave. The elastic wave field in the illuminated area is represented, for large N , as the sum of the incident wave u− and the outgoing wave u+ . We assume that the displacement field in the main component of the incident − wave u− s = (u · es ) near a caustic but for large N is expressed in the ray form − − u− s (r, t) = h(r)f (t − τ )

√

(13)

with a given real function f − describing its waveform; here h(r) = ψ/ J, J stands for the geometric spreading of rays, ψ is the so-called “initial data” characterizing the transverse distribution of the field intensity in the main component (e.g., see [6, 21]). The waveform f − (t) and its spectrum F −(ω) are related by the Fourier inversion formulas +∞ +∞

1 − − −iωτ − f (τ ) = F (ω)e dω, F (ω) = f − (p)eiωp dp. (14) 2π −∞

−∞

The comparison of (12) with the spectrum of the main component of the incident wave yields F −(ω) , ω > 0, (15) ω5/6 where C = C(r) lim (h/A) =  0 is a real, smoothly varying factor, which can approximately be viewed as a n→0 constant. Formulas (2), (8), (10), (14), and (15) allow for numerical modelling of both the main and the additional component of the wave field near a caustic. The employment of asymptotic expressions for the Airy functions (e.g., [21, 29]) furnishes a basis for a qualitative analysis of their spectra. Ψ(ω) = C e−i

3π 4

Qualitative properties of wave fields and their spectra. Equations (9) and (10) imply that exactly on the caustic the main component has frequency dependence ∼ iωΨ(ω), and the additional component is ∼ ω2/3 Ψ(ω). The main component on the caustic (n = 0) has the most high-frequency spectral contents (with the spectrum ∼ iωΨ(ω)). The other waveforms, in decreasing order of asymptotic intensity at high frequencies, are as follows: the additional component on the caustic (∼ ω2/3 Ψ(ω)), the main and additional components in the incident wave (ω5/6 Ψ(ω) and ω−1/6 Ψ(ω), respectively). In this case, as in the cases where ray theory can be applied, the frequency dependence in the additional component differs from that in the main component by the factor 1/(−iω) (for ω > 0). In the shade, where n < 0, with distance from the caustic the spectra of both components rapidly shift to the low-frequency region with their intensities rapidly decreasing. Relashionship between the waveforms of the incident and outgoing waves. Now we explicitly express the time dependence f + (t) in the main component of the outgoing wave in terms of the time dependence f − (t) of the incident wave. Consider their spectra +∞
± f (t) = F ± (ω)e−iωt dω. −∞

Equation (11) implies that for ω > 0,

F + (ω) = −iF − (ω).

The functions f ± (t) are real, and therefore, in accordance with (7), for all real ω we have F + (ω) = −isgn (ω)F − (ω), where sgn(ω) = 1 for ω > 0 and sgn(ω) = −1 for ω < 0. Since +∞
1 eiωs sgn (ω) = v.p. ds, π s

(16)

(17)

−∞

where v.p. means the principal value of the integral, the convolution theorem implies that +∞
1 f − (σ) + f (t) = v.p. dσ. π σ −t

(18)

−∞

Thus, the waveform in the main component of the outgoing wave is the Hilbert transform [37] of the waveform in the main component of the incident wave. The relationship between their additional components (which are integrals of f ± at a sufficient distance from the caustic) is precisely the same. 2418

An example of numerical simulation. In Fig. 3, results of simulation of the main and the additional component of the wave field in the vicinity of a caustic are presented. We observe the partition of the wave field into incident and outgoing waves as the observer moves away from the caustic into the illuminated area. The waveforms are seen to agree with the above qualitative theory.

Fig. 3. Synthetic seismograms and simulated trajectories for P waves in a neighborhood of a caustic. Caustic’s radius is R = 3Λ, where Λ is the characteristic wavelength in the main component of the incident wave. Solid and dotted lines show the main and the additional component, respectively. In the frame: (1) the signal waveform in the main component of the incident wave far from a caustic, (2) its Hilbert transform. For values of parameters that are typical of seismic exploration, the intensities of additional components may be very high. 4. The wave field in the penumbra Spectrum of the penumbral potential. Coordinates convenient for the description of wave fields in the penumbra are r and ϑ: the distance from the edge and the direction measured from the direction of the incident field, respectively (see Fig. 2b). In these variables, ∂ 1 ∂ + eϑ , (19) ∂r r ∂ϑ where er and eϑ are unit coordinate vectors. The main component of the wave field in the penumbra is the radial component ur = (u · er ). The angular component uϑ = (u · eϑ ) is additional. The asymptotic expression for the spectrum of the potential for small ϑ and large ωr/a is grad = er

Φ = ei

ωr a

cos ϑ

F(Θ)Ψ(ω),

(20) 2419

where


Θ

e−i 4 F(Θ) = √ π

π

 exp iα2 dα

−∞

is the Fresnel integral and

 Θ=ϑ

ωr . 2a

This differs from the classical formula (see [22, 23]) only by the factor Ψ(ω), which we choose by comparing the wave field described by potential (20) and that of the incident wave. Determination of the spectrum of the penumbral potential by the incident wave. As the observation point r moves away into the illuminated area so that the argument Θ of the Fresnel integral is large, the asymptotics (see [29]) iΘ2 −3iπ/4 e √ , Θ  1, F(Θ) ≈ 1 + e 2 πΘ together with the assumption that the diffraction angle is small, yields  ∂Φ iω iω iωx/a a 1 iωr/a √ −i π 4 ur = ≈ Φ≈ e Ψ(ω) + e e ωΨ(ω), ∂r a a 2πr ϑ

ω > 0.

(21)

The last term on the right-hand side describes the spectrum of the edge wave radiated from the tip. The next to the last term must coincide, in the illuminated area, with the spectrum of the incident wave. The incident wave giving rise to a penumbral wave field (20) is a plane wave coming to the obstacle from the left (see [22, 23]). Suppose it has the form u− = ex f − (t − x/a) ≈ er f − (t − x/a). (22) Here x = r cos ϑ is a Cartesian coordinate along the direction in which the incident plane wave propagates, and ex is its unit coordinate vector. Comparing the “normally” polarized component of the wave field described by the next to the last term in (21) and the given spectrum of the x component of the incident wave, we obtain Ψ(ω) =

aF −(ω) , iω

ω > 0,

(23)

where F − and f − are related by inversion formulas (14). Qualitative properties of wave fields and their spectra. The spectrum of the main component of the P wave on the shadow line (which is the same as the spectrum in the main component of the incident √ wave) is of order iωΨ(ω). The additional component on the shadow line at high frequencies is smaller (∼ ωΨ(ω)). In √ the main (radial) component of the edge wave, the order of the spectrum is ∼ ωΨ(ω), and in its additional √ component it is ∼ Ψ(ω)/ ω. Average polarization ellipse. In processing field data, an important polarization characteristic is an average polarization ellipse (in 3D, a polarization ellipsoid should be considered) (see [4, 5, 38]). Consider the covariation matrix T of the wave field, the elements Tij (r) of which at a point r are defined by the formula t
2 (r)

Tij (r) =

ui(r, t)uj (r, t)dt,

(24)

t1 (r)

where ui and uj are the components of the displacement field in an orthogonal coordinate system. The integral is taken over a time window in which the wave is localized. The matrix T is obviously symmetric and nonnegative. In the two-dimensional case, we denote its eigenvalues by λ1 (r) and λ2 (r), and its eigenvectors by e1 (r) and e2 (r), so that Te1,2 = λ1,2 e1,2 . Obviously, e1 (r) and e2 (r) are orthogonal. The eigenvalues characterize the distribution of the energy carried by the displacement field in the directions of the eigenvectors. The average polarization ellipse (to be more precise, 2420

the average amplitude polarization ellipse)

at a point r is the ellipse the axes of which are directed along e1 (r) and e2 (r) and have lengths λ1 (r) and λ2 (r), respectively. The lengths and orientations of the axes of the polarization ellipse are independent of a coordinate system. The polarization ellipse is characterized at each point r by the lengths of its axes, and by the direction of one of them, for example, of the largest one. It gives a good visual impression of the polarization of the displacement field at a given point. The larger the ratio of the lengths of the axes, the closer the polarization to the linear one. For waves harmonic in time (in which case integration in (24) proceeds over the period (see [38])), the polarization ellipse coincides with the trajectory of a particle. For the plane P wave, the polarization ellipse degenerates into a straight segment.

Fig. 4. Average polarization ellipses for the P wave in the penumbra. The waveform in the main component is the same as is shown in Fig. 3; Λ is its characteristic wavelength. In Fig. 4 we present simulated ellipses of polarization of the P wave field in the penumbra, where the incident and diffracted waves are physically indistinguishable. The numerical simulation was based on formulas (8), (19), (20), and (23). The waveform in the main component is the same as in Fig. 3. It is seen that the polarization tends to a linear one rather slowly with distance from the obstacle. 5. Conclusions 1. We presented a unified approach to numerical simulation of polarization anomalies of elastic waves near a caustic and in the penumbral area. 2. Numerics exemplifying the method show typical features of polarization anomalies in boundary layers such as • very high intensity of additional components whose amplitudes are comparable with those in the main components; • the presence of waveforms new and noticeably different from the signal form in the main component and its integral, which is typical of areas with a regular ray structure. 3. In boundary layers where the wave field cannot be divided into distinct arrivals, each of the components has a higher frequency content than in the “far zone,” and the dominant frequencies in the main component are always higher than those in the additional one. 4. Both the main and the additional component are traceable reliably along the boundary layer and poorly in a crosswise direction. 5. The above qualitative conclusions concerning the anomalies of polarization remain valid in the case of smoothly inhomogeneous and anisotropic media. The authors are indebted to T. B. Yanovskaya for interesting discussion of this research. Translated by A. Kiselev. 2421

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