According to Huygenss Principle, What Is the Explanation for Boundary Behaviors?

MEASUREMENT METHODS | Electrochemical: Quartz Microbalance

F. Wudy , ... H.J. Gores , in Encyclopedia of Electrochemical Power Sources, 2009

Continuing moving ridge

Huygen's principle describes the reflection of a moving ridge front at the boundary of two materials of different densities, in this case the quartz crystal sensor and the surrounding medium. A continuing moving ridge is established past the relationship between the thickness of the quartz crystal oscillator and the wavelength, which reveals the simple postulation that the thickness (δ _Q) of the quartz crystal including the sparse electrodes equals one-half of the wavelength (λ) (compare Figure four), expressed every bit the following equation for the showtime harmonic oscillation:

Figure 4. Resonance condition, the thickness (δ _Q) of the quartz crystal with its electrodes (gray) equals half the length of the wave.

[1] ${\delta }_{\text{Q}}=\frac{1}{2}\lambda$

The propagation velocity v _Q of the wave front in quartz crystals is well known and is dependent on the resonance frequency f ₀:

[2] ${\upsilon }_{\text{Q}}=\lambda f_0=3340\text{g}{\text{s}}^{-1}$

with

[3] $f_0=\frac{{\upsilon }_{\text{Q}}}{2{\delta }_{\text{Q}}}$

Typical thickness–frequency value pairs can be found in Table 1. With about thirty   MHz and a thickness of 56   μm, the sensor devices would get extremely thin and therefore nearly unmanageable. To reach higher frequencies, the oscillators are operated at the 2d, 3rd, or an even higher harmonic frequency.

Tabular array 1. Typical values for the thickness of a quartz crystal sensor and the resulting fundamental frequencies

δ Q (μm)	f 0 (MHz)
334	5
278	6
167	10
56	thirty

Read full chapter

URL:

https://www.sciencedirect.com/science/commodity/pii/B9780444527455000794

The Omega-Theory

Jure Žalohar , in Developments in Structural Geology and Tectonics, 2018

The Twinning Effect

In Capacity 15 and 19 , we showed that seismic activation of a fault corresponds to emission of the CSV strain waves perpendicular in both directions to the fault airplane. This mechanism was recognized as like to the Huygens principle in the physics of waves; therefore nosotros chosen it the Cosserat-Huygens principle. We likewise showed that because of the Cosserat-Huygens principle the B-spectral theorem is not just linked to the powers of the Golden ratio and the Pi-constant, only also to the half of the powers of the Gilded ratio and the Pi-constant. In this second case the earthquake happens when two CSV strain waves (fronts) "collide" within the block divisional by parallel fault planes within the Ω-prison cell (run into Fig. 19.ane in Chapter nineteen). In summary, nosotros can say that whenever the effective value of Båth's parameter is related to the one-half of the powers of the Aureate ratio and the Pi-constant, each earthquake associated with the mistake planes in the Ω-cell is twinned with the earthquake acquired by CSV strain wave "collisions" between the error planes in the same Ω-cell. According to this principle the number of earthquakes in the Ω-sequence is doubled; we will call this the twinning principle. Phenomenological observations hold with theoretical predictions, just interestingly, only for small-scale effective values of Båth'south parameter. In this example, we have:

(49) $N_{\max }=12.$

Indeed, nosotros observed Ω-sequences composed of a larger number of sequence events. 2 such examples are illustrated in Chapter 17 for the case of the Kraljevo (2010) earthquake. The starting time example (Fig. 17.ix in Chapter 17) is the Ω₁ C ₂₃₄₅₆₇  ↓ DSR Ω-sequence that was based on the effective value of the Båth's parameter B _eff   =   ane.089 and was composed of 9 sequence events. The second example is illustrated in Fig. 17.10 in Chapter 17, which shows the Ω₁ C ₂₃₄₇  ↑ Ω-sequence defined by the effective value of the Båth'southward parameter B _eff   =   i.229. This sequence contained x sequence events.

The Ω-limitation law is well in agreement with previous studies on the repeating convulsion sequences (RES). These studies show that the number of events in repeating earthquake sequences (RES) ranges betwixt three and seven (Chen et al., 2007, 2009) in some cases up to 9 (e.g., Matsuzawa et al., 2002).

Read full chapter

URL:

https://www.sciencedirect.com/scientific discipline/commodity/pii/B9780128145807000204

Earthquake Seismology

K. Satake , in Treatise on Geophysics, 2007

4.17.6.2 Ray-Theoretical Approach

If the tsunami wavelength is much smaller than the calibration of heterogeneity in propagation velocity, or the alter in water depth, then we can utilize the geometrical ray theory of optics. The wavefronts of propagating tsunami can be drawn on the basis of Huygens' principle. Such a diagram is called a 'refraction diagram'. Refraction diagrams can be prepared for major tsunami sources and used for tsunami warning; as soon as the epicenter is known, the tsunami arrival times tin can be readily estimated. Figure 1 shows the refraction diagram from the 2004 Andaman–Sumatra earthquake with wavefronts at each 60 minutes. It is shown that the tsunami was expected to get in at Thai and Indian coasts in 2   h and at Africa in seven   hs.

The refraction diagrams tin also be drawn backwards from coasts. Such a diagram is called an 'inverse refraction diagram' and used to estimate the tsunami source area. In this example, the wavefronts or rays are traced backwards for the corresponding travel times (Miyabe, 1934). The traced wavefronts from each observed station bound the seismic sea wave source surface area.

Read full chapter

URL:

https://world wide web.sciencedirect.com/scientific discipline/article/pii/B978044452748600078X

Seismic Waves and Rays in Elastic Media

In Handbook of Geophysical Exploration: Seismic Exploration, 2003

vii.3 Eikonal equation

In order to obtain ψ(x), we turn our attention to equations (7.10), from which we can factor out the components of vector A(x). Hence, we tin can write

(7.11) $\begin{array}{cc}{\displaystyle \sum _{k=1}^3\left({\displaystyle \sum _{j=1}^3{\displaystyle \sum _{l=1}^3{c}_{ijk\mathrm{fifty}}\left(\text{ten}\right)\frac{\partial \psi }{\partial x_j}\frac{\partial \psi }{\partial x_l}-\rho \left(\text{x}\right){\delta }_{i\mathrm{grand}}}}\right)}{A}_k\left(\text{x}\right)=0,& i\in \left\{\mathrm{ane},2,3\right\}.\end{array}$

In view of expression (6.65), allow us denote

(7.12) $\begin{array}{cc}{\mathcal{p}}_j:=\frac{\partial \psi }{\partial {\mathrm{ten}}_j},& j\in \left\{\mathrm{ane},2,\mathrm{iii}\right\}\end{array},$

where p is the phase-slowness vector, which describes the slowness of the propagation of the wavefront.

Note that the significant of p can be seen by examining expression (7.half dozen) and because a three-dimensional continuum. Therein, ψ is a part relating position variables, ten _ane, x _two and x ₃, to the traveltime, t. Thus, since ψ has units of time, p_j := ∂ψ/∂ten_j has units of slowness and the level sets of ψ (x) tin be viewed as wavefronts at a given fourth dimension t. Consequently, in view of properties of the gradient operator, p = ∇ (x) is a vector whose direction corresponds to the wavefront normal and whose magnitude corresponds to the wavefront slowness.

In view of note (7.12), we can write equations (7.eleven) as

(7.xiii) $\begin{array}{cc}{\displaystyle \sum _{k=\mathrm{ane}}^{\mathrm{iii}}\left({\displaystyle \sum _{j=1}^3{\displaystyle \sum _{\mathrm{50}=1}^{\mathrm{iii}}{c}_{ijkl}\left(\text{ten}\right){\mathcal{p}}_j{\mathcal{p}}_{\mathrm{50}}-\rho \left(\text{ten}\right){\delta }_{i\mathrm{one\; thousand}}}}\right)}{A}_{\mathrm{yard}}\left(\text{x}\right)=0,& i\in \left\{1,2,3\right\}.\end{array}$

Equations (7.thirteen) are referred to as Christoffel'southward equations.

In Chapter 10, we discuss equations (7.13) in the context of the particular symmetries of continua, which were introduced in Chapter 5. Therein, we also show that the eigenvalues resulting from these equations are associated with the velocity of the wavefront while the corresponding eigenvectors are the displacement directions. Herein, we written report the general form of equations (vii.13).

Nosotros know from linear algebra that equations (vii.13) accept nontrivial solutions if and only if

(7.14) $\begin{array}{cc}\det \left[{\displaystyle \sum _{j=1}^3{\displaystyle \sum _{l=1}^3{c}_{ij\mathrm{one\; thousand}l}\left(\text{x}\right){\mathcal{p}}_j{\mathcal{p}}_l-\rho \left(\text{10}\right){\delta }_{ik}}}\right]=0,& i,\mathrm{yard}\in \left\{\mathrm{i},2,3\right\}.\end{array}$

Assuming that p ^two ≠ 0, we can write determinant (7.14) equally

(7.xv) $\begin{array}{cc}{\left({\mathcal{p}}^2\right)}^3\det \left[{\displaystyle \sum _{j=1}^3{\displaystyle \sum _{l=1}^3{c}_{ijk\mathrm{50}}\left(\text{10}\right)\frac{{\mathcal{p}}_j{\mathcal{p}}_l}{{\mathcal{p}}^2}-\frac{\rho \left(\text{10}\right)}{{\mathcal{p}}^{\mathrm{ii}}}{\delta }_{ik}}}\right]=0,& i,\mathrm{yard}\in \left\{1,2,\mathrm{iii}\right\}.\end{array}$

Note that p ² = 0 would mean that the slowness of the propagation of the wavefront is zero. This would imply the velocity to be space, which is a nonphysical situation. Besides, in view of determinant (7.14), p ² = 0 would result in det [ρ (ten) δ _ik ] = 0, which would imply ρ (x) = 0.

Expression (7.15) is a polynomial of caste 3 in p ², where the coefficients depend on the direction of the phase-slowness vector, p. Whatever such polynomial can be factored out as

(vii.sixteen) $\left[{\mathcal{p}}^{\mathrm{ii}}-\frac{\mathrm{ane}}{{\mathrm{five}}_1^2\left(\text{x,p}\right)}\right]\left[{\mathcal{p}}^{\mathrm{two}}-\frac{\mathrm{one}}{v_2^{\mathrm{ii}}\left(\text{x,p}\right)}\right]\left[{\mathcal{p}}^2-\frac{1}{v_3^2\left(\text{x,p}\right)}\right]=0,$

where i/υ_i ² are the roots of polynomial (7.xv).

The conditions imposed on the c_ijkl by the stability conditions — discussed in Department four.three — imply that the three roots of polynomial (seven.15) are existent and positive. These properties are further discussed in Department 10.1. The existence of three roots implies the existence of three types of waves, which can propagate in anisotropic continua.

Now, allow united states of america consider a given root of equation (vii.16). Each root is the eikonal equation for a given type of wave, namely,

(7.17) $\begin{array}{cc}{\mathcal{p}}^2=\frac{1}{v_i^{\mathrm{ii}}\left(\text{x,p}\right)},& i\in \left\{1,2,\mathrm{iii}\right\}\end{array}.$

Allow us examine the pregnant of this equation. ^two

Since p ² = p · p is the squared magnitude of the slowness vector, which is normal to the wavefront, and then — in view of the wavefronts being the loci of constant phase — υ is the role describing phase velocity. This velocity is a role of position, x, and the direction of p. Hence, equation (seven.17) applies to anisotropic inhomogeneous continua and can be viewed as an extension of equation (six.71), which is valid for isotropic inhomogeneous continua.

Because 2 adjacent wavefronts, we can view equation (7.17) as an minute formulation of Huygens' principle. ^three

Notation that function υ is homogeneous of degree 0 in the p_i. In other words, the orientation of a wavefront is described by the direction of p and is independent of the length of p. Hence, in equation (7.17) we could besides write υ _i = υ _i (ten, n), where n is a unit of measurement vector in the direction of p. Notably, we will utilise this notation in Chapter 10.

Furthermore, as shown explicitly in Chapter 10, the phase-velocity function can be expressed in terms of the properties of the continuum, namely, its mass density and elasticity parameters. Thus, the eikonal equation relates the magnitude of the slowness with which the wavefront propagates to the properties of the continuum through which it propagates.

In the mathematical context, the eikonal equation is a differential equation. Recalling expressions (vii.12), nosotros can rewrite equation (seven.17) as

(7.eighteen) ${\left[\nabla \psi \left(\text{x}\right)\right]}^2=\frac{1}{v^2\left(\text{x,p}\right)}.$

In full general, the eikonal equation is a nonlinear, first-club, partial differential equation in x to exist solved for the eikonal function, ψ (10). Information technology belongs to the Hamilton-Jacobi class of differential equations. ⁴

Equation (7.nine) is the transport equation. This transport equation possesses a vectorial form that is valid for anisotropic inhomogeneous continua. Information technology is analogous to the scalar send equation (6.74), which is valid for isotropic inhomogeneous continua.

Read full affiliate

URL:

https://www.sciencedirect.com/science/article/pii/S0950140103800563

Earthquake Seismology

Chiliad. Satake , in Treatise on Geophysics (2d Edition), 2015

4.19.vi.2 Ray-Theoretical Approach

If the tsunami wavelength is much smaller than the scale of heterogeneity in propagation velocity, or the change in h2o depth, so the geometrical ray theory of optics can exist applied ( Chapter 1.05 ). The wave fronts of a propagating tsunami can be fatigued on the basis of Huygens' principle. Such a diagram is called a refraction diagram. Refraction diagrams can be prepared for major tsunami sources and can exist used for seismic sea wave warnings; as before long as the epicenter is known, the tsunami inflow times can exist readily estimated. Some examples can be establish at the NOAA NGDC Seismic sea wave Database. Figure 2 shows the refraction diagrams for the 2004 Andaman–Sumatra and 2011 Tohoku earthquakes and the moving ridge fronts for each hour. The 2004 Indian Body of water tsunami was expected to arrive at the Thai and Indian coasts inside 2   h and at the African coast inside 7   h of the earthquake, while the 2011 Tohoku seismic sea wave was expected to arrive at the Due north and South American coasts within 8 and 18   h, respectively.

The refraction diagrams tin can be drawn from a coastal point. If the wave fronts or rays are traced backward from the coastal observation points for the respective travel times, then they encompass the tsunami source area (Miyabe, 1934). This type of diagram is chosen an changed refraction diagram and has been used to estimate the tsunami source area. The tsunami source extent of the 2004 Sumatra–Andaman earthquake was estimated from inverse refraction diagrams (Lay et al., 2005; Neetu et al., 2005) ( Section 4.19.two.1 ).

Read full affiliate

URL:

https://www.sciencedirect.com/science/article/pii/B9780444538024000865

Modern Global Seismology

In International Geophysics, 1995

3.6 Partitioning of Seismic Energy at a Boundary

We accept seen in the previous sections of this chapter that when a body moving ridge encounters a boundary or discontinuity at which the seismic velocity changes, the wave will reflect or refract. Equally we volition show, when a P or SV wave impinges on a boundary, four derivative waves consequence, every bit shown in Effigy 3.23: (1) P', the refracted or transmitted P moving ridge (note that P head waves are a subset of P'), (2) SV', the refracted SV (it is possible to have P waves generate a SV caput wave if β₂ > α_one), (3)P, the reflected P moving ridge, and (iv) SV, the reflected SV wave. The ray geometry of these derived waves is governed by Snell's police. By Snell'south law, all of the rays must have the same ray parameter, p, since all the waves must move along the boundary with the aforementioned apparent velocity:

Figure iii.23. Ray for a P wave incident on a solid-solid boundary and the rays for waves generated at the interface.

(three.83) $\begin{array}{l}\left(sini\right)/{\alpha }_{\mathrm{ane}}=\left(sin\gamma \right)/{\beta }_1=\left(sin{\gamma }^{\prime }\right)/{\beta }_{\mathrm{two}}\\ =\left(sin{i}^{\prime }\right)/{\alpha }_{\mathrm{ii}}.\end{array}$

When an SH moving ridge encounters a discontinuity surface parallel to the SH move, merely two waves are generated: (1) SH, reflected, and (two) SH', refracted. (SH' tin be a head moving ridge.) The being of multiple waves derived from a single incident wave implies that the energy of the incident wave must be partitioned. Although Snell's constabulary and ray theory can predict the geometry of the wave interaction, nosotros must return to a wavefield representation to determine the amplitude partitioning.

In Figure 3.23, the interface separates two materials of distinct elastic properties. Within either half-infinite the equations of motion for homogeneous media are valid. The physics that govern the wave propagation crave that stresses and displacements exist "transmitted" across the interface. Thus a stress imbalance propagating in layer i will outcome in a stress imbalance in layer ii, giving rise to a wavefield. There are several types of interface. If the interface is between two solids, all components of stress at the interface and all components of deportation are continuous. This is called a welded interface. If the interface is between a solid and a perfect fluid, the fluid may skid along the interface, since information technology has no rigidity. Thus, the tangential displacements are not continuous, and the tangential tractions must vanish. In addition, the normal traction and normal displacements at the fluid–solid contact are continuous. At a costless surface, all the tractions must be zero, and no explicit restriction is placed on the displacements. Note that these conditions are on tractions, not stresses. For example, if the x ₁ x ₂ plane is a costless surface, then σ₃₁ = σ₃₂ = σ₃₃ = 0, but the other components of stress are not constrained.

These conditions on continuity of displacement and stress are the basis for predicting the partitioning of energy. Now return to Figure 3.23. Why does the P moving ridge produce both a reflected and refracted P wave and a reflected and refracted SV moving ridge? Information technology makes sense that no SH wave will be produced because the particle motion of the incident P wave is confined to the x ₁ 10 ₃ aeroplane, and no "refraction" of the P wave at a horizontal boundary will produce move in the x ₂ plane. Refraction of the P wave will cause particle displacements that are not parallel on reverse sides of the interface (see Effigy iii.24). Thus, the P-wave displacements alone exercise non combine to give continuous displacements or tractions beyond the welded interface. The boosted particle move required to make the fields continuous results in SV-wave-blazon motion, which is also bars to the x ₁ x _iii plane. Call back, simply P- and Southward-wave motions exist as propagating disturbances. In a fluid, where no S waves exist, the P waves reflect and transmit purely as P waves because only normal displacements and normal tractions need to remain continuous at the purlieus.

FIGURE three.24. P-wave particle motions for the incident, reflected, and refracted P waves. Annotation that if this is a solid-solid boundary, the shear stress in the two layers will non match at the purlieus, requiring the generation of SV motion in both media.

We can quantify the free energy segmentation by using the potentials introduced in Section 2.iv for plane waves. The P-wave and SV-wave potentials for the various wave components are represented past

(3.84) $\begin{array}{l}{\phi }_{\left(\text{layer 1}\right)}^{-}={\phi }_{\text{incident ray}}+{\phi }_{\text{reflected ray}}\\ {\phi }_{\left(\text{layer two}\right)}^{+}={\phi }_{\text{refracted}}\\ {\psi }^{-}={\psi }_{\text{reflected}}\\ {\psi }^{+}={\psi }_{\text{refracted}},\end{array}$

where ϕ and ψ are the P and SV potentials, respectively. The plane-wave potentials are of the course

(3.85) ${\phi }_{\text{incident}}=A_{\mathrm{i}}exp\left[i\omega \left(p{x}_1+{\eta }_{{\alpha }_1}{x}_3-t\right)\right].$

Recall that (KX ₁)/west = (sin i)/α = p. Similarly, kx _three /westward = ηα_one. Nosotros can write similar equations for the other potentials in (3.84):

(iii.86) $\begin{array}{l}{\phi }_{\text{reflected}}=A_{\mathrm{two}}exp\left[i\omega \left(p{\mathrm{10}}_1+{\eta }_{{\alpha }_1}{x}_3-t\right)\right]\\ {\phi }_{\text{refracted}}=A_{\mathrm{three}}exp\left[i\omega \left(p{x}_1+{\eta }_{{\alpha }_2}{\mathrm{10}}_3-t\right)\right]\\ {\phi }_{\text{reflected}}=B_{2}exp\left[i\omega \left(p{x}_1+{\eta }_{{\beta }_1}{\mathrm{ten}}_{\mathrm{three}}-t\right)\right]\\ {\phi }_{\text{refracted}}=B_{3}exp\left[i\omega \left(p{\mathrm{10}}_1+{\eta }_{{\beta }_2}{x}_{\mathrm{three}}-t\right)\right].\end{array}$

The various vertical slownesses are for the associated velocities. Note that the sign of the x ₃ term changes, depending on whether the ray is refracted or reflected. This indicates the direction (down or upward) in which the ray is traveling.

The ratios of the postinteraction amplitudes (A ₂, A _three, B _two, B ₃) over the incident amplitude (A ₁) are called the reflection and transmission coefficients. These coefficients control the division of amplitude among the potentials. The tractions and displacements can exist calculated from the potentials by taking the derivatives with respect to ten ₁ and ten ₃, which preserves the exponential character of the potentials.

In general, the purlieus conditions in a welded interface require significant algebraic manipulation (encounter Table 3.ane), then we will consider a simplified example. A P moving ridge incident on a fluid-fluid interface generates no S waves, so nosotros need only consider reflected and refracted P waves. From (iii.85) and (iii.86) we tin write downwardly equations for the P-moving ridge potential:

TABLE 3.1. Displacement Reflection and Transmission Coefficients

Coefficient	Formula
Solid-gratis surface (P–SV)
RPP	{–[(one/β2)–2p2]2 + 4p2ηαηβ}/A
RPS	{four(α/β)pηα[1/β2) – 2p2]}/A
RSP	{4(β/α)pηβ[(ane/β2) – 2pii]}/A
Rss	{−[(1/β2) – 2p2]2 + 4piiηαηβ}/A
RSS (SH)	one
Solid–solid (P–SV)
RPP	[(bηα1 – cηα2 )F – (a + dηα1ηβ2)Hp2 ]/D
R PS	–[2ηα1(ab+cdηα2ηβ2)p(αone/β1)]/D
T PP	[2ρoneηα1F(α1/αii)]/D
T PS	[2ρ1ηα1Hp(αone/β2)]/D
RSS	– [bηβ1 – cηβ2]E – (a + bηα2ηβ1)Gptwo]/D
RSP	– [2ηβ1(ab + cdηα2ηβ2)p(β1/α1)]/D
R SS (SH)	$\frac{{\mu }_{\mathrm{ane}}{\eta }_{{\beta }_{\mathrm{ane}}}-{\mu }_2{\eta }_{{\beta }_2}}{{\mu }_1{\eta }_{{\beta }_{\mathrm{one}}}+{\mu }_{\mathrm{ii}}{\eta }_{{\beta }_2}}$
T SS (SH)	$\frac{2{\mu }_1{\eta }_{{\beta }_1}}{{\mu }_{\mathrm{i}}{\eta }_{{\beta }_1}+{\mu }_2{\eta }_{{\beta }_{\mathrm{ii}}}}$
a = ρ2(1 – 2β2 2 p 2) – ρ1(1 – 2β1 2 p ii)	E = bηα1 + cηα2
b = ρ2(1 – 2βii 2 p 2) – 2ρiβ1 2 p 2	F = bηβ1 + cηβ2
c = ρ1(1 – 2βone 2 p 2) + 2ρ2β2 two p ii	Yard = a – dηα1ηβ2
d = two(ρ2β2 2–ρaneβi ii)	H = a – dηα2ηβ1
D = EF + GHp 2
A = [(1/β ii)– 2p ii]2 + 4p 2ηα1ηβ1

(3.87) $\begin{array}{ll}\mathrm{medium\; 1}:\hfill & {\phi }_1=A_{1}exp\left[i\omega \left(p{\mathrm{10}}_{\mathrm{i}}+{\eta }_{\mathrm{i}}{\mathrm{10}}_3-t\right)\right]\hfill \\ \hfill & +A_{2}exp\left[i\omega \left(p{x}_{\mathrm{ane}}-{\eta }_1{x}_3-t\right)\right]\hfill \\ \mathrm{medium\; ii}:\hfill & {\phi }_2=A_{3}exp\left[i\omega \left(p{x}_1+{\eta }_{\mathrm{ii}}{x}_{\mathrm{three}}-t\right)\right].\hfill \end{array}$

The P displacements are related to the potentials by Eq. (2.91):

(3.88) $\text{u}=\frac{\partial \phi }{\partial {\mathrm{10}}_1}{\text{x}}_1+0{\text{x}}_2+\frac{\partial \phi }{\partial {\mathrm{10}}_3}{\text{10}}_3.$

The advisable boundary conditions for the fluid-fluid boundary are continuity of normal stress and displacement (σ₃₃ and u _three). Mathematically, the deportation status is given by

(3.89) ${\frac{\partial {\phi }_1}{\partial x_3}=\frac{\partial {\phi }_2}{\partial {\mathrm{ten}}_3}|}_{x_3=0}.$

Substituting (3.87) into this equation yields

(three.ninety) $\begin{array}{l}i\omega {\eta }_1\left(A_1-A_{\mathrm{two}}\right)e^{i\omega \left(p{\mathrm{ten}}_{\mathrm{i}}-t\right)}\\ =i\omega {\eta }_{\mathrm{ii}}{A}_3{e}^{i\omega \left(p{x}_{\mathrm{ane}}-t\right)}\end{array}$

or

(three.91) ${\eta }_1\left(A_1-A_2\right)={\eta }_2{A}_{\mathrm{iii}}.$

The condition of stress continuity is given by

(3.92) ${\sigma }_{33}^{-}=\lambda \nabla \text{u}+\mathrm{two}\mu {\varepsilon }_{33}={\sigma }_{33}^{+},$

just μ = 0 in a fluid. Thus

(3.93) ${\lambda }_1{\nabla }^{\mathrm{two}}{\phi }_1={\lambda }_{\mathrm{ii}}{\nabla }^2{\phi }_2.$

Nosotros tin simplify (3.93) past using the fact that ϕ satisfies the wave equation:

(3.94) ${\nabla }^{\mathrm{ii}}\phi =\frac{1}{{\alpha }^2}\frac{{\partial }^{\mathrm{ii}}\phi }{\partial t^2}=\frac{-{\omega }^2}{{\alpha }^2}\phi .$

Therefore, for ten _iii = 0,

(3.95) $\frac{{\lambda }_1}{{\alpha }_1^2}\left(A_{\mathrm{ane}}+A_2\right)=\frac{{\lambda }_{\mathrm{two}}}{{\alpha }_2^{\mathrm{ii}}}{A}_{\mathrm{three}}.$

Now, for a fluid, ${\lambda }_{\mathrm{ane}}={\rho }_{\mathrm{one}}{\alpha }_{\mathrm{ane}}^2$ and ${\lambda }_2={\rho }_2{\alpha }_{\mathrm{ii}}^2$ , so we tin can rewrite (3.91) and (three.95) as a system of equations:

(3.96) $\begin{array}{l}{A}_1-A_2=\frac{{\eta }_2}{{\eta }_1}{A}_{\mathrm{iii}}\\ A_1-A_{\mathrm{two}}=\frac{{\rho }_2}{{\rho }_1}{A}_{\mathrm{iii}}.\end{array}$

Thus we can solve for ratios of the amplitudes

(iii.97) $\begin{array}{l}\frac{A_3}{A_{\mathrm{i}}}=\Im =\frac{2{\rho }_1{\eta }_{\mathrm{one}}}{{\rho }_1{\eta }_2+{\rho }_2{\eta }_1}\\ \frac{A_{\mathrm{two}}}{A_1}=\Re =\frac{{\rho }_{\mathrm{ii}}{\eta }_{\mathrm{i}}-{\rho }_1{\eta }_2}{{\rho }_1{\eta }_2+{\rho }_2{\eta }_1}.\end{array}$

I and R are referred to every bit the transmission and reflection coefficients, respectively. Note that I and R depend on 17, which is (cos i)/α. Thus the partitioning of potential amplitudes depends on the angle at which the ray strikes the purlieus. Consider the case of vertical incidence (p = 0, η_i = 1/α_ane η₂ = 1/α_two):

(3.98) ${\Re }_{i=0}=\frac{{\rho }_{\mathrm{two}}/{\alpha }_1-{\rho }_{\mathrm{i}}/{\alpha }_2}{{\rho }_1/{\alpha }_{\mathrm{two}}-{\rho }_2/{\alpha }_1}=\frac{{\rho }_2{\alpha }_2-{\rho }_{\mathrm{one}}{\alpha }_1}{{\rho }_{\mathrm{ane}}{\alpha }_{\mathrm{one}}+{\rho }_2{\alpha }_{\mathrm{two}}}$

(3.99) ${\Im }_{i=0}=\frac{2{\rho }_1/{\alpha }_1}{{\rho }_1/{\alpha }_2+{\rho }_2/{\alpha }_1}=\frac{\mathrm{two}{\rho }_1{\alpha }_2}{{\rho }_1{\alpha }_1+{\rho }_2{\alpha }_{\mathrm{ii}}}.$

Now at this point, the reflection and manual coefficients are for potential, non displacement. We can obtain deportation terms past recalling u ₃ = ∂ϕ/∂ten _three:

(3.100) $\begin{array}{l}\frac{u_{\text{reflected}}}{u_{\text{incident}}}=\frac{-i\omega {\eta }_1}{i\omega {\eta }_1}\frac{A_2}{A_1}=\frac{{\rho }_{\mathrm{i}}{\alpha }_1-{\rho }_{\mathrm{ii}}{\alpha }_2}{{\rho }_{\mathrm{ane}}{\alpha }_1+{\rho }_2{\alpha }_2}\\ =R=-{\Re }_{i=0}\end{array}$

(3.101) $\begin{array}{l}\frac{u_{\text{reflected}}}{u_{\text{incident}}}=\frac{-i\omega {\eta }_2}{i\omega {\eta }_1}\frac{A_{\mathrm{three}}}{A_{\mathrm{one}}}=\frac{1/{\alpha }_2}{\mathrm{ane}/{\alpha }_{\mathrm{one}}}\frac{2{\rho }_{\mathrm{i}}{\alpha }_{\mathrm{ii}}}{{\rho }_{\mathrm{i}}{\alpha }_{\mathrm{i}}+{\rho }_2{\alpha }_2}\\ =T=\frac{{\alpha }_1}{{\alpha }_2}{\Im }_{i=0}.\end{array}$

The R and T derived here, which are the vector displacement transmission and reflection coefficients, take extensive utilise in geophysics despite being derived for fluids and vertical incidence. 1 must be careful to go along track of the vector displacement with respect to the direction the wave is propagating in defining the sign of the movement. These reflection and transmission coefficients also concur for solid-solid interfaces at near-vertical incidence. The energy is partitioned quite simply: T – R = 1. The quantity pa is known as audio-visual impedance, and depending on how acoustic impedance changes across the boundary, the reflection coefficient tin can take values of –1 to +i. Similarly, the range of the manual coefficient is 0 to 2. A free-surface boundary will have a vertical-incidence reflection coefficient of –ane (the displacement reverses direction with respect to the direction of progpagation). The amplitude of transmitted displacement is zilch.

If nosotros return to the general class of I and R (nonvertical incidence), we can investigate the behavior of the system as the angle of incidence varies. If α₂ < α ₁ and ρ₂α₂ > ρ ₁α_one, and so R will be a positive value for normal incidence. As i increases, R will decrease, reaching zero at an angle of incidence called the intramission angle:

(3.102) $\frac{{\rho }_2}{{\rho }_1}=\frac{\sqrt{{\left({\alpha }_1/{\alpha }_{\mathrm{ii}}\right)}^{\mathrm{two}}-{sin}^{\mathrm{two}}i}}{\sqrt{1-{sin}^{2}i}}.$

Beyond the intramission angle, the reflection coefficient decreases to a value of –1 at grazing incidence i = 90°). If α_two > α ₁ and ρ₂α_ii > ρ ₁α_i ^, the reflection coefficient is e'er negative and equals –1 for grazing incidence.

If α₂ < α ₁, a caput wave is produced at the critical bending, i _c = sin⁻ⁱ(α_2/α₁). At incident angles greater than the critical angle, no P waves volition propagate in the lower medium. This is because p = (sin i)/α _i = 1/c (where c is the apparent velocity) becomes greater than l/α_two. Thus η₂ = [(1/α₂ ²) – p ²]^½ become imaginary.

We tin can write ${\eta }_{\mathrm{two}}=i{\hat{\eta }}_{\mathrm{two}}=\pm \iota {\left[p^2-\left(\mathrm{one}/{\alpha }_2^{\mathrm{ii}}\right)\right]}^{1/2}$ , where we cull the positive sign such that the amplitude of the refracted potential (3.87) decreases exponentially abroad from the boundary. This keeps the wave energy bounded. Effigy iii.25 illustrates the head wave with exponentially decomposable displacements in the half-space. The manual coefficient is circuitous, and to keep the ray parameter abiding, angle i ₂ becomes complex.

Effigy 3.25. Exponential decay of the particle movement of a caput wave propagating along the boundary.

We can rewrite the postcritical reflection coefficient in (iii.97) equally

(iii.103) $\Re =\frac{{\rho }_{\mathrm{ii}}{\eta }_{\mathrm{one}}-{\rho }_{\mathrm{i}}i{\hat{\eta }}_2}{{\rho }_{\mathrm{two}}{\eta }_1+{\rho }_{\mathrm{i}}i{\hat{\eta }}_2}.$

Now R is a circuitous number divided by its conjugate. This implies that the magnitude of R is 1, but there is a phase shift of θ

(3.104) $\Re ={\mathrm{due\; east}}^{i\theta }$

(3.105) $\theta =2{tan}^{-1}\left(\frac{{\rho }_1{\hat{\eta }}_2}{{\rho }_2{\eta }_1}\right).$

Since the modulus of the reflection coefficient is one, the postcritical reflection is referred to as full reflection, simply it volition carry differently than precritical reflections. Figure 3.26 shows a synthetic seismogram contour generated for increasing angles of incidence (increasing altitude). Beyond 60 km, the reflected inflow has an angle of incidence that is greater than i_c. This is the distance at which a caput wave first occurs and begins to move out from the reflected arrival. At 450 km the reflected moving ridge is incident on the purlieus at near-grazing incidence; the reflected waveform is very similar to that at 50 km, except the polarity is completely reversed.

Figure 3.26. The modify in reflected pulse shape (phase) every bit the incidence angle exceeds the critical angle. For the model shown, the head wave outset appears at ∼ sixty km. A comparing of seismograms at 50 and 450 km shows that the polarity has been reversed.

It is clear from Figure three.26 that the reflected wave changes shape as the source-receiver distance increases. Although the stage shift in Eq. (3.105) explains this shape change, it is instructive to return to the equation for the reflection potential. Noting that A ₂ =A ₁ R = A ₁ eastward ^iθ, nosotros can write the potential for the postcritical reflected arrival as

(3.106) $\phi =A_{1}exp\left[i\theta \right]exp\left[i\omega \left(p{x}_1-{\eta }_1{x}_{\mathrm{iii}}-t\right)\right].$

At present consider the behavior of θ:

$\begin{array}{ll}\theta =0\hfill & \text{if}i=i_{\text{c}}\hfill \\ \theta <0\hfill & \text{for}i>i_{\text{c}}\hfill \\ \theta =-\pi \hfill & i=\pi /2.\hfill \end{array}$

Nosotros showtime rewrite (3.106) as

(3.107) $\phi =A_{1}exp\left[i\omega \left(p{\mathrm{ten}}_{\mathrm{i}}-{\eta }_1{\mathrm{ten}}_3-t+\left(\theta /\omega \right)\right)\right].$

Now θ/ω is explicitly a new or additional phase term. If nosotros apply the constant-phase argument to rail the behavior of a item wavefront, we have

(iii.108) $p{\mathrm{10}}_{\mathrm{ane}}-{\eta }_1{\mathrm{ten}}_3-t+\left(\theta /\omega \right)=\text{abiding}\text{.}$

The term $-t+\left(\theta /\omega \right)=-\left(t-\theta /\omega \right)=\hat{t}$ is an credible time that at present depends on frequency. Thus, the position of the wavefront is frequency dependent; lower frequencies (smaller ω) will have earlier arrival times than high frequencies (retrieve θ < 0). As ω → ∞, $\hat{t}=t$ . This implies that the wavefront is "spread out" for a postcritical reflection, each harmonic term having a separate plane wave. This behavior is chosen dispersion, a phenomenon we will become very familiar with in the next chapter. A consequence of the dispersion is that the strongest reflection coefficient occurs exactly at i _c (R = one, θ = 0, and wavefronts do non degrade).

Reflection and transmission at a welded interface are much more complicated than at a fluid-fluid interface. However, the SH system remains fairly simple considering interaction with the purlieus does not produce any P or SV energy, so nosotros will briefly consider this case. Every bit with the fluid-fluid case, there are ii boundary weather condition: (ane) continuity of tangential deportation (5 _ii ⁺ = Five ₂ ⁻) and (2) continuity of shear stress (σ₂₃ ⁺ = σ₂₃ ⁻). Applying these conditions yields SH-deportation reflection and transmission coefficients:

(3.109) $\begin{array}{l}T=\frac{\mathrm{two}{\mu }_1{\eta }_{{\beta }_1}}{{\mu }_1{\eta }_{{\beta }_1}+{\mu }_2{\eta }_{{\beta }_{\mathrm{two}}}}\\ R=\frac{{\mu }_1{\eta }_{{\beta }_1}-{\mu }_2{\eta }_{{\beta }_2}}{{\mu }_1{\eta }_{{\beta }_1}+{\mu }_{\mathrm{ii}}{\eta }_{{\beta }_2}}.\end{array}$

These equations are nearly identical to Eqs. (3.97), and if we consider the example of vertical incidence, then (3.109) reduces to

(iii.110) $\begin{array}{l}T=\frac{2{\rho }_{\mathrm{one}}{\beta }_{\mathrm{ane}}}{{\rho }_1{\beta }_1+{\rho }_{\mathrm{two}}{\beta }_{\mathrm{two}}}\\ R=\frac{{\rho }_1{\beta }_{\mathrm{one}}-{\rho }_{\mathrm{ii}}{\beta }_{\mathrm{ii}}}{{\rho }_1{\beta }_{\mathrm{i}}+{\rho }_{\mathrm{two}}{\beta }_{\mathrm{two}}}.\end{array}$

Box 3.4

Seismic Diffraction

The analogy between seismic ray theory and optics extends to the concept of diffraction. Diffraction is defined as the manual of energy by nongeometric ray paths. In optics, the archetype example of diffraction is light "leaking" around the edge of an opaque screen. In seismology, diffraction occurs whenever the radius of curvature of a reflecting interface is less than a few wavelengths of the propagating wave. Figure 3.B4.1a shows a plane wave incident upon an opaque (acoustic impedance is space) boundary. Ray theory requires that waves arriving at seismometers at points F and Yard have identical amplitudes; no free energy is transmitted to the right of point G. In fact, the edge of the boundary acts similar a secondary source (Huygens' principle) and radiates free energy forward in all directions. These diffractions tin can be understood from the standpoint of Fresnel zones, a concept that states that waves reflect from a large region rather than merely a point. Thus, the Fresnel zone causes the ray traveling to F to "see" the edge of the reflector, although the geometric raypath conspicuously misses the boundary. The outset Fresnel zone may exist thought of as a cone with the edge of the reflector as its apex. For a receiver that is a altitude d beyond the reflector, the cone's radius is given by r = d + ½λ, where A is the wavelength of the seismic wave. Effigy 3.B4.1b shows the aamplitude variation predicted for the experiment given in 3.B4.1a.

Effigy three.B4.1. (a) Rays incident on a grating. Energy is diffracted around the edge. (b) Amplitude of energy every bit a function of altitude into the diffraction zone.

(From Doornbos, 1989). Copyright © 1989

Diffraction is present at many scales within the Globe and has occasionally led to erroneous interpretations of structure. Effigy three.B4.2 shows an instance from reflection seismology. Here, a high-velocity layer is sandwiched between half-spaces, and the layer is kickoff by a normal mistake. The seismograms shown are for a source and receiver placed at each successive distance point. At x = 2000, the seismogram is fabricated upwardly of two pulses, of opposite polarity, representing reflections off the tiptop and bottom of the layer. Every bit x increases, later arrivals begin to appear, forming a parabola known as a "diffraction frown."

FIGURE 3.B4.2. A synthetic reflection seismic department for the construction at the top of the figure. A mistake offsets a high-velocity bed. B1, B2, and B3 are diffracted arrivals.

(From Waters, K. H. "Reflection Seismology: A Tool for Energy Resource Exploration." Copyright © 1981 John Wiley &amp; Sons.) Copyright © 1981 John Wiley &amp; Sons

The quantity ρβ is called the shear impedance. The SH critical-bending behavior for β_two > β _one is very analogous to that described for the acoustic (fluid) instance.

The P–SV system requires using every potential term in Eq. (3.84). In full general, 4 derivative waves exist for each incident P or SV wave. The velocities may allow both P and SV head waves for incident P or 5 waves. For the welded interface, σ_3i., u _one, and u ₃ must be continuous (used for boundary weather condition). For the case of an incident P moving ridge, the displacement boundary weather [using (3.86)], give (u ₁, continuous)

(3.111) $p\left(A_{\mathrm{ane}}+A_{\mathrm{two}}\right)+{\eta }_{{\beta }_{\mathrm{ane}}}{B}_{\mathrm{ane}}=p{A}_3-{\eta }_{{\beta }_{\mathrm{two}}}{B}_2$

and (u₃ continuous)

(3.112) ${\eta }_{{\alpha }_{\mathrm{i}}}\left(A_1+A_2\right)+p{B}_{\mathrm{one}}={\eta }_{{\alpha }_{\mathrm{ii}}}{A}_3-B_{\mathrm{ii}}.$

The continuity of stress (σ₃₃ continuous) gives

(3.113) $\begin{array}{l}{\lambda }_{\mathrm{one}}{p}^{\mathrm{two}}\left(A_1+A_{\mathrm{ii}}\right)+{\lambda }_{\mathrm{i}}p{\eta }_{{\beta }_1}{B}_{\mathrm{ane}}+\left({\lambda }_{\mathrm{i}}+2{\mu }_{\mathrm{i}}\right)\\ \times \left[{\eta }_{{\alpha }_1}^2\left(A_1+A_2\right)-{\eta }_{{\beta }_1}p{B}_1\right]\\ ={\lambda }_{\mathrm{ii}}{p}^{\mathrm{two}}{A}_3-p{\eta }_{{\beta }_{\mathrm{two}}}{\lambda }_2{B}_2+\left({\lambda }_{\mathrm{ii}}+2{\mu }_2\right)\\ \times \left({\eta }_{{\alpha }_2}^{\mathrm{two}}{A}_3+{\eta }_{{\beta }_2}^2{A}_2\right)\end{array}$

and (σ₃₁ continuous)

(three.114) $\begin{array}{l}{\mu }_1\left[\mathrm{two}p{\eta }_{{\alpha }_{\mathrm{i}}}\left(A_1-A_2\right)+p^{\mathrm{ii}}{B}_1-{\eta }_{{\beta }_1}^2{B}_{\mathrm{i}}\right]\\ ={\mu }_2\left[\mathrm{ii}p{\eta }_{{\alpha }_2}{A}_{\mathrm{three}}+p^{\mathrm{ii}}{B}_{\mathrm{two}}-{\eta }_{{\beta }_{\mathrm{two}}}^{\mathrm{two}}{B}_{\mathrm{two}}\right].\end{array}$

Thus we have iv equations with five unknowns. Information technology is sufficient to determine the ratios with respect to A _ane, thus obtaining R _pp , R _PS , T _pp , and T _PS . The algebra required to obtain these coefficients is all-encompassing, and we get out it to the reader as an practise to obtain the concluding values given in Table 3.i. Table 3.one lists the standard reflection and transmission coefficients for solid-solid and solid-air (free-surface reflections) interfaces.

Figures 3.27 and three.28 bear witness the reflection and transmission coefficients for P waves incident from below and to a higher place a welded interface. In the showtime example, the wave is going from a fast- to a boring-velocity cloth, and there are no critical angles. The energy sectionalisation is dominated by R _pp and T _PP from 0° to approximately 20°. Over this range, R _PP and T _PP are nearly identical to what would exist obtained from the acoustic impedance mismatch [Eqs. (3.100) and (3.101)]. When the P wave is incident from the low-velocity medium, the critical angle is 38.5°. The P transmission coefficient is 0 beyond this angle. Every bit the bending of incidence approaches 38.5°, the coefficients vary apace. In particular, T_pp gets very large before going to goose egg. This tin be explained by a simple geometric argument, as shown in Effigy 3.29. Considering the aamplitude of the pulse is proportional to the square root of energy per surface area, as expanse goes to null, the aamplitude becomes large.

Effigy 3.27. Reflection and refraction coefficients for a P wave incident on a purlieus from a high-velocity region. For most-vertical incidence (bending =0°), the reflected and refracted P-wave amplitudes approximately equal those predicted by acoustic-impedance mismatches [(Eqs. three.100) and (3.101)]. There are no critical angles in this case.

FIGURE 3.28. Reflection and refraction coefficients for a P wave incident on a boundary from a low-velocity region. i _c for the P wave occurs at 38.v°. Since the S velocity in the lower medium is lower than the upper P velocity, the refracted S moving ridge never reaches a disquisitional angle.

Effigy three.29. Schematic of ray bundles striking a purlieus between low- and loftier-velocity material. The amplitude of the pulse is inversely proportional to the area dA. Every bit i approaches the disquisitional angle i _c, dA ₂ goes to zero, and the amplitude of the refracted wave becomes very big.

The partitioning of a wave into four new waves at each purlieus in the Earth results in seismograms that are rich in arrivals. We refer to the partitioning of P waves into SV waves or SV waves into P waves as mode conversions. Manner conversions provide of import data well-nigh Earth structure. Figure 3.30 shows a seismogram from a deep crustal earthquake in the Mississippi embayment. A converted phase Sp is generated at a sediment-boulder interface. This arrives ahead of Due south by a fourth dimension proportional to the depth of the interface and the v _P /v _{due south} ratio in the crust. Other examples of reflected and converted phases are described in chapter 7.

FIGURE three.30. Instance of mode conversion at a boundary. The SV moving ridge is converted to a P moving ridge at a sediment-bedrock interface, giving rise to the S _p precursor to S on the vertical (ten) seismogram, while P→S conversions (P_s ) are seen on the horizontals.

(Courtesy of W. Mooney.)

Read total chapter

URL:

https://world wide web.sciencedirect.com/science/article/pii/S007461420580004X

Deep Earth Seismology

5.F. Cormier , in Treatise on Geophysics (Second Edition), 2015

i.06.vii.2 Hybrid Methods

To save computation, it is sometimes necessary to combine two methods, a computationally cheaper method in a large region and a computationally more expensive method in a small region, to report structural complexity near a boundary or region of rapid velocity or density transition. Normally, the computationally more expensive method is a numerical method, such as finite difference or spectral element. The usual fashion in which the ii methods are connected is to compute a Kirchhoff integral. The integral is carried out on a surface or purlieus that separates the structurally circuitous region from the simple region, employing Huygens principle to connect the wave fields in the 2 regions by assuasive each betoken on a wave front to propagate every bit a new bespeak source. Kirchhoff integrals can also be used to calculate the effects of scattering by inclusions or the frequency-dependent reflection or manual beyond a curved interface when the radius of curvature is on the order of a wavelength. A hybrid method of this type was used past Wen and Helmberger (1998) to report heterogeneity in the lowermost mantle using a finite deviation method, connecting information technology to GRT in a radially symmetrical overlying mantle. The near detailed treatment of fully elastic integrands for P, SV, and SH waves can be found in Frazer and Sen (1985). Probably due to the need for intense customization for specific problems, codes for Kirchhoff integrals are not generally distributed. A skillful starting point for whatsoever awarding are the acoustic issues described in Shearer (1999, pp. 138–140), which can then exist generalized using the elastic formulas in Frazer and Sen (1985).

Capdeville et al. (2003a) developed a hybrid method that allows modal solutions in large homogeneous or weakly heterogeneous regions to be coupled to the numerical spectral element (SEM) solutions in strongly heterogeneous regions. This hybrid method has been applied to a thin, strongly heterogeneous, D″ region at the base of the mantle sandwiched between a homogeneous core and mantle (Capdeville et al., 2003b; To et al. 2005). A similar approach might also be viable at higher frequencies and local and regional ranges by coupling locked surface wave modes to SEM solutions.

Read full affiliate

URL:

https://world wide web.sciencedirect.com/science/commodity/pii/B9780444538024000051

Seismology and the Structure of the Earth

V.F. Cormier , in Treatise on Geophysics, 2007

1.05.vii.ii Hybrid Methods

To salve computation, it is sometimes necessary to combine two methods, a computationally cheaper method in a large region, and a computationally more than expensive method in a small region to study structural complexity near a purlieus or region of rapid velocity or density transition. Usually the computationally more expensive method is a numerical method, such every bit finite deviation or spectral element. The usual style in which the ii methods are connected is to compute a Kirchhoff integral. The integral is carried out on a surface or boundary that separates the structurally circuitous region from the simple region, employing Huygens principle to connect the wavefields in the two regions by assuasive each point on a wave front end to propagate as a new signal source. Kirchhoff integrals tin as well be used to calculate the effects of handful past inclusions or the frequency-dependent reflection or transmission across a curved interface when the radius of curvature is on the order of a wavelength. A hybrid method of this blazon was used past Wen et al. (1998) to study heterogeneity in the lowermost drape using a finite difference method, connecting it to GRT in a radially symmetric overlying drape. The nigh detailed treatment of fully elastic integrands for P, SV, and SH waves can be found in Frazer and Sen (1985). Probably due to the need for intense customization for specific problems, codes for Kirchhoff integrals are not more often than not distributed. A good starting point for any application are the acoustic problems described in Shearer (1999, pp. 138–140), which tin can then be generalized using the elastic formulas in Frazer and Sen (1985).

Capdeville et al. (2003a) developed a hybrid method that allows modal solutions in big homogeneous or weakly heterogeneous regions to be coupled to the numerical SEM solutions in strongly heterogeneous regions. This hybrid method has been applied to a thin, strongly heterogeneous, D″ region at the base of operations of the mantle sandwiched between a homogeneous core and curtain (Capdeville et al., 2003b; To et al., 2005; see Affiliate 1.18). A like approach might also be feasible at higher frequencies and local and regional ranges by coupling locked surface-wave modes to SEM solutions.

Read full affiliate

URL:

https://world wide web.sciencedirect.com/science/article/pii/B9780444527486000055

Body-waves and ray theory – amplitudes

Charles J. Ammon , ... Terry C. Wallace , in Foundations of Modern Global Seismology (2nd Edition), 2021

13.5 Trunk-wave energy flux factors

The steady-state reflection and transmission amplitude ratios insure that the boundary conditions at elastic boundaries are satisfied. To compute the aamplitude of the waves transmitting through, or reflecting off boundaries requires that we also consider the energy flux at the boundary. No free energy is trapped at the boundary – so the free energy of the reflected and transmitted waves must balance that of the incidence wave. If the displacement of a P-wave is $u(x_1,x_3,t)=A\cos \omega (p{x}_{\mathrm{i}}+{\eta }_{\alpha }{x}_{\mathrm{iii}}-t)$ and then the energy flux across a unit area of the wavefront is $\rho \alpha A^{\mathrm{ii}}{\sin }^2\omega (p{x}_1+{\eta }_{\alpha }{x}_3-t)$ . If the wave strikes the interface with an angle i, we account for the wavefront projection onto the horizontal boundary past multiplying by ratio of the incident wave expanse to the boundary area, $\cos i$ (Fig. 13.17). For an Southward-wave incident with angle j, we must multiply by a cistron of $\cos j$ .

Effigy thirteen.17. (left) The ratio of the incoming ray bundle to its projection onto the boundary. (right) Change in area of a transmitted moving ridge ray tube. To conserve energy, nosotros must account for this modify in expanse and multiply the deportation aamplitude ratios by the ratio of the surface area change to preserve energy.

Box 13.3 Seismic diffraction

The analogy between seismic ray theory and optics extends to the concept of diffraction. Diffraction is defined as the transmission of energy by nongeometric ray paths. In eyes, the archetype example of diffraction is light "leaking" around the edge of an opaque screen. In seismology, diffraction occurs whenever the radius of curvature of a reflecting interface is less than a few wavelengths of the propagating moving ridge. Fig. B13.3.1 shows a plane moving ridge incident upon an opaque (acoustic impedance is infinite) purlieus. Ray theory requires that waves arriving at seismometers at points F and G have identical amplitudes; no free energy is transmitted to the right of point G. In fact, the edge of the boundary acts like a secondary source (Huygens' principle) and radiates free energy frontwards in all directions. These diffractions tin be understood from the standpoint of Fresnel zones, a concept that states that waves reflect from a large region rather than but a signal. Thus, the Fresnel zone causes the ray traveling to F to "see" the border of the reflector, although the geometric raypath clearly misses the boundary. The first Fresnel zone may be thought of as a cone with the edge of the reflector as its apex. For a receiver that is a distance d across the reflector, the cone's radius is given past $r=d-\frac{1}{\mathrm{two}}\lambda$ , where A is the wavelength of the seismic wave. Fig. B13.3.2 shows the amplitude variation predicted for the experiment given in Fig. B13.3.i.

Figure B13.3.1. (A) Rays incident on a grating. Energy is diffracted effectually the edge. (B) Amplitude of energy as a function of distance into the diffraction zone (from Doornbos, 1989).

Diffraction is present at many scales within the World and has occasionally led to erroneous interpretations of structure. Fig. B13.3.2 shows an example from reflection seismology, which is a stacked section of a constructed model containing five layers and iv small lenses that simulate diffractors (Dell and Gajewski, 2011). The layer velocities are constant, with the 4th layer containing four small lenses with a lateral extension of 200 meters. The diffractors class a parabola known as a diffraction frown.

Effigy B13.3.2. Constructed case with 4 small lenses of 200 thou lateral extension simulating diffractions. (A) Stacked department of the recorded wavefield and (B) diffraction-only data. Lateral extension of the seismic line is 6250 m. (modified from Dell and Gajewski, 2011).

For the energy expression, we must include the density and wave-speed in the expressions. If nosotros stand for the speed of the incoming wave as $v_{in}$ and that of the reflected or transmitted wave equally ${\mathrm{five}}_{out}$ , and use a similar notation for the wave angles, so the complete energy flux factor is

(xiii.69) $$\begin{gathered} {\left(\frac{{\rho }_{in}{5}_{i\mathrm{due\; north}}}{{\rho }_{out}{\mathrm{five}}_{out}}\right)}^{1/2}{\left(\frac{\cos i_{out}}{\cos i_{in}}\right)}^{\mathrm{ane}/2} \\ \text{or}{\left(\frac{{\rho }_{i\mathrm{north}}{v}_{i\mathrm{due\; north}}}{{\rho }_{out}{v}_{out}}\right)}^{1/2}{\left(\frac{v_{out}{\eta }_{out}}{v_{in}{\eta }_{in}}\right)}^{1/2}. \end{gathered}$$

The foursquare roots arise because the deportation is proportional the square-root of free energy. For a non-mode converted reflection, the free energy-flux cistron is 1, for transmitted and style-converted waves, it is not.

For example, let $u_1(\mathbf{ten},t)$ represent a P-wave displacement pulse incident on a boundary, and let $u_2(\mathbf{x},t)$ represent a transmitted pulse. The two are related by the product of the displacement transmission ratio discussed earlier and the free energy flux factor,

(xiii.lxx) $$\begin{gathered} u_2(\mathbf{x},t)=u_1(\mathbf{ten},t)\times T_{PP} \\ \times {\left(\frac{{\rho }_1{\alpha }_1}{{\rho }_{\mathrm{two}}{\alpha }_2}\right)}^{1/2}{\left(\frac{\cos i_2}{\cos i_1}\right)}^{1/2}. \end{gathered}$$

For a symmetric turning ray (all downwardly and upward transmissions match), the upward leg of a ray volition include the product of the inverses of all the flux factors accumulated on the downward leg so the product of all the factors will exist 1. Consider a P-wave reflection off the cadre mantle boundary (PcP) for a surface source and receiver. Let $u_0$ , correspond the deportation at the source. During propagation, the source aamplitude is modified past the effects of geometric spreading, attenuation, and reflection and transmission using a sequence of multiplications to account for each effect or purlieus interaction. For example, the vertical component of the core reflection PcP amplitude is

(13.71) $$\begin{gathered} u_{PcP}(\mathbf{x},t)\sim u_0(\mathbf{ten},t)\times G_{PcP}(\mathrm{\Delta })\times e^{-\pi f{t^{⁎}}_{PcP}} \\ \times \prod _{i=1}^{\mathrm{northward}}{(T_{PP})}_i\times {R_{PP}}_{core}\times F_{P_{x_{\mathrm{iii}}}}, \end{gathered}$$

where the product of n manual coefficients earlier and after reflection are included in the production operator. Because nosotros have no mode conversions and a symmetric ray, the product of all the free energy flux factors equals ane. A more than complete analysis of the wave aamplitude would include the frequency dependence to include the phase changes associated with the attenuation operator. The point is that a ray-based amplitude analysis is intuitive. We follow the wave from the source to the seismometer and include the aamplitude adjustments for each interaction along the path. For mode conversions, or for deep sources, we must also include the non-symmetric part of the wave'south energy flux factors. We discuss the factors that control $u_0(\mathbf{x},t)$ (the amplitude leaving the source) in a later chapter. Next we investigate wave propagation from a completely different viewpoint – modes, equally we discuss surface waves.

Read total chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128156797000215

Seismology and the Structure of the Earth

J. Virieux , G. Lambaré , in Treatise on Geophysics, 2007

1.04.2.4 Rays and Moving ridge Fronts

The eikonal equation is a nonlinear, fractional differential equation of the first society belonging to the Hamilton–Jacobi variety (Kravtsov and Orlov, 1990), usually solved in terms of characteristics (Courant and Hilbert, 1966). The characteristics are 3D trajectories x  = x(τ) verifying a set of ordinary differential equations (ODEs). If needed, traveltime is integrated through quadratures along these trajectories called rays. For isotropic media, these rays are orthogonal to wavefronts, sometimes called isochrones. A nice tutorial on characteristics has been given by Bleistein (1984, affiliate 1) in relation with differential geometry.

Techniques exist to compute directly wavefronts mainly based either on the Huygens principle (the new wavefront is the envelope of spheres fatigued from an initial wavefront with local velocity) or on ray tracing (short ray elements from an initial wavefront allow the construction of the new wavefront) ( Figure 7 ). Solving straight the eikonal equation for the start-inflow traveltime turns out to be performed quite efficiently: finite divergence (FD) method has been proposed by Vidale (1988, 1990) and numerous more or less precise algorithms have been designed for both isotropic and anisotropic media (Podvin and Lecomte, 1991; Lecomte, 1993; Eaton, 1993). FD techniques such as the fast marching method (Sethian and Popovici, 1999) may also account for large velocity gradients. Techniques for solving these types of Hamilton–Jacobi equations have been developed past Fatemi et al. (1995) and Sethian (1999) for various problems. Combining ray tracing and FD methods may lead to multivalued traveltime estimations as suggested by Benamou (1996) or Abgrall and Benamou (1999). These techniques accept been very attractive in many applications such as commencement-arrival traveltime delay seismic tomography because they always provide a trajectory connecting the source and the receiver and, therefore, a constructed traveltime any is the precision on it. When inverting a huge amount of information, a few miscomputed traveltimes will not touch on the tomographic inversion (Benz et al., 1996; Le Meur et al., 1997). These techniques have besides been thoroughly used for 3-D reflection seismic imaging (Grayness and May, 1994).

Figure 7. Ciphering of wavefronts. On the left panel, the construction is based on the Huygens principle, where the new wavefront is the envelope of spheres fatigued from an initial wavefront with local velocity. The direction is unknown locally. On the right panel, the structure is based on ray tracing equation, where curt ray segments are drawn from an initial wavefront to the new one. The local direction is known by the ray orientation.

The characteristic system of nonlinear first-order partial differential equations (eqn [1]) consists of 7 equations in a 3-D medium connecting the position x, the slowness vector p, and the travel time T, which are very similar to dynamic particle equations of classical mechanics (Goldstein, 1980):

[9] $\begin{array}{l}\frac{\text{d}{x}_i}{\text{d}\tau }=\frac{\partial \mathcal{H}}{\partial p_i}\\ \frac{\text{d}{p}_i}{\text{d}\tau }=\frac{\partial \mathcal{H}}{\partial {\mathrm{ten}}_i},i=1,2,\mathrm{three}\\ \frac{\text{d}T}{\text{d}\tau }=p\mathrm{thousand}\frac{\partial \mathcal{H}}{\partial p_k}\end{array}$

where the variable τ divers as dτ   =   dT/(p _k ∂H/∂p _k ) depends on the selected form of the function H(x, p). The independent variables are position x and slowness vector p, defining a 6-D space, called stage space, on which the abiding Hamiltonian defines an hypersurface, also chosen a Lagrangian manifold (Lambaré et al., 1996).

Read full chapter

URL:

https://world wide web.sciencedirect.com/science/commodity/pii/B9780444527486000043

robertshistoodespil.blogspot.com

Source: https://www.sciencedirect.com/topics/earth-and-planetary-sciences/huygens-principle

Share this post