From 6a5aa8c7064ce37c4a6368e37f5069725197d617 Mon Sep 17 00:00:00 2001 From: ARS183 <1028762817@qq.com> Date: Tue, 29 Sep 2026 18:23:53 +0200 Subject: [PATCH 1/4] Add the seismic wavefields chapter from the FWch5 lecture notes --- book/4_mechanical_waves/acoustic_waves.md | 105 +++++ .../appendix_acoustic_1d.md | 76 ++++ .../4_mechanical_waves/appendix_strain_p_s.md | 375 ++++++++++++++++++ book/4_mechanical_waves/basic_notions.md | 53 +++ book/4_mechanical_waves/elastic_waves.md | 108 +++++ book/4_mechanical_waves/field_recordings.md | 60 +++ book/4_mechanical_waves/figures/Figure4_1.PNG | Bin 0 -> 2805548 bytes book/4_mechanical_waves/figures/Figure4_2.PNG | Bin 0 -> 1983207 bytes book/4_mechanical_waves/figures/Figure4_3.PNG | Bin 0 -> 2995867 bytes book/4_mechanical_waves/figures/blokjes.png | Bin 0 -> 127950 bytes .../figures/fig_ap_ac_cube.png | Bin 0 -> 8859 bytes book/4_mechanical_waves/figures/huygens.png | Bin 0 -> 28447 bytes book/4_mechanical_waves/figures/pblock.PNG | Bin 0 -> 32548 bytes book/4_mechanical_waves/figures/ptens.PNG | Bin 0 -> 33593 bytes .../4_mechanical_waves/figures/rayelement.png | Bin 0 -> 19415 bytes book/4_mechanical_waves/figures/rays.png | Bin 0 -> 72417 bytes book/4_mechanical_waves/figures/raytimes.png | Bin 0 -> 74229 bytes book/4_mechanical_waves/figures/refltrans.png | Bin 0 -> 14512 bytes book/4_mechanical_waves/figures/sblock.PNG | Bin 0 -> 32996 bytes book/4_mechanical_waves/figures/snell.png | Bin 0 -> 108585 bytes book/4_mechanical_waves/figures/stens.PNG | Bin 0 -> 11847 bytes .../figures/straindispl.PNG | Bin 0 -> 8852 bytes .../figures/stressforce.PNG | Bin 0 -> 10573 bytes book/4_mechanical_waves/figures/tautensor.PNG | Bin 0 -> 30431 bytes book/4_mechanical_waves/fundamental_laws.md | 3 + book/4_mechanical_waves/intro.md | 5 + .../reflection_transmission.md | 77 ++++ book/4_mechanical_waves/snells_law.md | 154 +++++++ book/_toc.yml | 13 + 29 files changed, 1029 insertions(+) create mode 100644 book/4_mechanical_waves/acoustic_waves.md create mode 100644 book/4_mechanical_waves/appendix_acoustic_1d.md create mode 100644 book/4_mechanical_waves/appendix_strain_p_s.md create mode 100644 book/4_mechanical_waves/basic_notions.md create mode 100644 book/4_mechanical_waves/elastic_waves.md create mode 100644 book/4_mechanical_waves/field_recordings.md create mode 100644 book/4_mechanical_waves/figures/Figure4_1.PNG create mode 100644 book/4_mechanical_waves/figures/Figure4_2.PNG create mode 100644 book/4_mechanical_waves/figures/Figure4_3.PNG create mode 100644 book/4_mechanical_waves/figures/blokjes.png create mode 100644 book/4_mechanical_waves/figures/fig_ap_ac_cube.png create mode 100644 book/4_mechanical_waves/figures/huygens.png create mode 100644 book/4_mechanical_waves/figures/pblock.PNG create mode 100644 book/4_mechanical_waves/figures/ptens.PNG create mode 100644 book/4_mechanical_waves/figures/rayelement.png create mode 100644 book/4_mechanical_waves/figures/rays.png create mode 100644 book/4_mechanical_waves/figures/raytimes.png create mode 100644 book/4_mechanical_waves/figures/refltrans.png create mode 100644 book/4_mechanical_waves/figures/sblock.PNG create mode 100644 book/4_mechanical_waves/figures/snell.png create mode 100644 book/4_mechanical_waves/figures/stens.PNG create mode 100644 book/4_mechanical_waves/figures/straindispl.PNG create mode 100644 book/4_mechanical_waves/figures/stressforce.PNG create mode 100644 book/4_mechanical_waves/figures/tautensor.PNG create mode 100644 book/4_mechanical_waves/fundamental_laws.md create mode 100644 book/4_mechanical_waves/reflection_transmission.md create mode 100644 book/4_mechanical_waves/snells_law.md diff --git a/book/4_mechanical_waves/acoustic_waves.md b/book/4_mechanical_waves/acoustic_waves.md new file mode 100644 index 0000000..48a52a0 --- /dev/null +++ b/book/4_mechanical_waves/acoustic_waves.md @@ -0,0 +1,105 @@ +# Wave fields in a fluid: acoustic (P) waves + +With acoustic waves, we assume a fluid in which no shear forces exist, and therefore only P-waves exist. Since waves are physical phenomena, they should have a relation to basic physical laws. The two laws applicable are the conservation of mass and Newton's second law. These two have been used in [the appendix on 1-D acoustic wave motion](./appendix_acoustic_1d.md) to derive the two equations governing the wave motion due to a P-wave. There are some simplifying assumptions in the derivation, one of them being that we consider a 1-dimensional wave. When we denote $p$ as the pressure and $v_x$ as the particle velocity, the conservation of mass leads to: + +$$ +- \frac{1}{K} \frac{\partial p}{\partial t} = \frac{\partial v_x}{\partial x} +$$ (eq:eqdeform) + +in which $K$ is called the bulk modulus. The other relation follows from an application of Newton's law: + +$$ +-\frac{\partial p}{\partial x} = \rho \frac{\partial v_x}{\partial t} +$$ (eq:eqmotion) + +where $\rho$ denotes the mass density. This equation is called the equation of motion. + +We are now going to combine these two equations. Therefore, we let the operator $\partial / \partial x$ work on the equation of motion: + +$$ +- \frac{\partial}{\partial x} \left( \frac{\partial p}{\partial x} \right) + = + \frac{\partial}{\partial x} + \left( \rho \frac{\partial v_x}{\partial t} \right). +$$ (eq:diffeqmotion) + +Now, assuming $\rho$ is constant, it can be taken in front of the $\partial / \partial x$ operator. And differentiations with respect to space $x$ and time $t$ are commutative, so: + +$$ + \frac{\partial}{\partial x} \left( \frac{\partial v_x}{\partial t} \right) + = + \frac{\partial}{\partial t} \left( \frac{\partial v_x}{\partial x} \right). +$$ (eq:commut) + +Now for $\partial v_x / \partial x$, the conservation of mass can be substituted in {eq}`eq:diffeqmotion` to give: + +$$ + -\frac{\partial^2 p}{\partial x^2} = \rho + \frac{\partial}{\partial t} + \left( - \frac{1}{K} \frac{\partial p}{\partial t} \right). +$$ + +Rewriting gives us the 1-Dimensional wave equation: + +$$ +\frac{\partial^2 p}{\partial x^2} + - \frac{1}{c^2} \frac{\partial^2p}{\partial t^2} = 0 +$$ + +in which $c$ can be seen as the velocity of the wave, for which we have: $c = \sqrt{K/ \rho}$. + +This is the basic wave equation for one-dimensional acoustic waves. Equally well, we can derive from the same two equations {eq}`eq:eqdeform` and {eq}`eq:eqmotion` (see also *EXERCISES*) that the particle velocity also satisfies the wave equation: + +$$ +\frac{\partial^2 v_x}{\partial x^2} + - \frac{1}{c^2} \frac{\partial^2 v_x}{\partial t^2} = 0 +$$ (eq:waveqacv) + +where $c$ is the same as in the wave equation of the pressure. + +The solution to the wave equation for the pressure is: + +$$ +p(x,t) = s(t \pm x/c) +$$ (eq:pxt) + +where $s(t)$ is some function. Note the dependency on space and time via $(t \pm x/c)$, which denotes that it is a travelling wave, and behaves as a direct wave as discussed in [](./snells_law.md). The sign in the argument depends on the direction the wave is travelling. + +Often, seismic responses are analyzed in terms of frequencies, i.e., Fourier spectra. The definition used here is: + +$$ +G(\omega) = \int_{-\infty}^{+\infty} g(t) \exp(- i \omega t) \, dt +$$ (eq:fourier1) + +where the radial frequency $\omega$ is used, so $\omega = 2 \pi f$ where $f$ is the (ordinary) frequency. Using this convention, it is easy to show that differentiation with respect to time is equivalent to multiplication with $i\omega$ in the Fourier domain. When we transform the solution of the wave equation ({eq}`eq:pxt`) to the Fourier domain, we obtain: + +$$ +P(x,\omega) = S(\omega) \exp (\pm i\omega x/c). +$$ (eq:Pxw) + +Note here that the time delay $x/c$ (in the time domain) becomes a *linear* phase in the Fourier domain, so the phase $\pm \omega x/c$ is linear as a function of $\omega$. + +In the above, we gave an expression for the pressure, but one can also derive an equivalent expression for the particle velocity $v_x$. For that purpose, we can use the equation of motion as expressed in {eq}`eq:eqmotion`, but then in its Fourier-transformed version, which is: + +$$ +V_x(x,\omega) = -\frac{1}{i\omega \rho} \frac{\partial P(x,\omega)}{\partial x}. +$$ + +When we substitute the solution for the pressure from above ({eq}`eq:Pxw`), we get for the negative sign: + +$$ +\begin{aligned} +V_x(x,\omega) & = -\frac{1}{i \omega \rho} + S(\omega) \frac{- i \omega}{c} \exp(-i\omega x/c) +\\ + & = S(\omega) \frac{1}{\rho c} \exp(-i\omega x/c). +\end{aligned} +$$ + +Notice that the particle velocity is a scaled version of the pressure: + +$$ +V_x(x,\omega) = \frac{P(x,\omega)}{\rho c}. +$$ + +The scaling factor is ($\rho c$), being called the *seismic impedance*. diff --git a/book/4_mechanical_waves/appendix_acoustic_1d.md b/book/4_mechanical_waves/appendix_acoustic_1d.md new file mode 100644 index 0000000..1ad1193 --- /dev/null +++ b/book/4_mechanical_waves/appendix_acoustic_1d.md @@ -0,0 +1,76 @@ +(app-waveq1d)= +# Appendix: derivation of basic equations for 1-D acoustic wave motion + +*In this appendix, the one-dimensional wave motion for an acoustic medium is derived, starting from the deformation law of Hooke and from conservation of momentum (Newton's Second Law). This results in the so-called deformation equation and the equation of motion for acoustic waves.* + +## Derivation + +Here we will derive the basic equations for wave motion in homogeneous media, using the conservation of momentum (Newton's second law) and the deformation law, known as Hooke's Law of elasticity. In this derivation, we consider a single cube of mass when it is subdued to a seismic disturbance (see {numref}`fig-cube`). This cube has a volume $\Delta V$ with sides $\Delta X, \Delta Y$ and $\Delta Z.$ + +```{figure} figures/fig_ap_ac_cube.png +:name: fig-cube +:width: 70% + +A cube of mass, subjected to one-dimensional wave motion. Drawn line: original status. Dotted line: status after deformation. +``` + +We start with an elastic deformation of the cube, using Hooke's Law. This law states that the deformation force that works on a piece of material is linearly related to the extension of that piece. Applying this to our cube of mass, the deformation force gives a relative change in volume: + +$$ +p = -K \frac{dV}{\Delta V} +$$ (eq:hooke) + +where the pressure $p$ is introduced as being the force per unit area, $K$ is called the bulk modulus, and the minus sign expresses that the pressure is opposite to the deformation direction. Now we assume that the volume change is only in one direction (1-dimensional) as shown in {numref}`fig-cube`, so then we get: + +$$ +\begin{aligned} +\frac{dV}{\Delta V} & = + \frac{u_x(x+\Delta X)\Delta Y \Delta Z- u_x(x) \Delta Y \Delta Z} + {\Delta X \Delta Y \Delta Z} \\ +& = \frac{u_x(x+\Delta X) - u_x(x)}{\Delta X} +\end{aligned} +$$ + +where $u_x$ is the displacement in the $x$-direction. Using the situation that $\Delta X$ is rather small, we can approximate $u_x(x+\Delta X)$ by $u_x(x) + ( \partial u_x/\partial x ) \Delta X$; this then gives for Hooke's Law: + +$$ +p = -K \frac{\partial u_x}{\partial x}. +$$ + +Since we are finally interested in particle velocities rather than displacements (since geophones, seismic sensors on land, measure particle velocities), we differentiate both sides of Hooke's Law with respect to time $t$, and introduce $v_x = \partial u_x/\partial t$ to give: + +$$ +\boxed{ +\frac{1}{K} \frac{\partial p}{\partial t} = - \frac{\partial v_x}{\partial x} +} +$$ + +This is the so-called deformation equation. It is one basic relation needed for describing one-dimensional wave motion. + +The other relation is obtained via Newton's Law, applied to the volume $\Delta V$ with mass $M$ in the direction $x$, since we consider 1-dimensional motion: + +$$ +\begin{aligned} +F_x & = M \frac{\partial v_x}{\partial t} \\ + & = \rho \Delta V \frac{\partial v_x}{\partial t} +\end{aligned} +$$ (eq:newton) + +where $F$ is the total force that works on the element $\Delta V$ that induces motion. Consider the total force that is working on the cube in the $x$-direction, through the pressures working on the sides with area $\Delta Y \Delta Z$: + +$$ +\begin{aligned} +F_x & = - [ p(x+\Delta X) - p(x) ] \Delta S_x \\ + & = - \frac{p(x+\Delta X) - p(x)}{\Delta X} \Delta V +\end{aligned} +$$ + +Using $p(x+\Delta X) \simeq p(x) + ( \partial p/\partial x ) \Delta X$ and combining it with Newton's Law as expressed in {eq}`eq:newton` gives: + +$$ +\boxed{ + - \frac{\partial p}{\partial x} = \rho \frac{\partial v_x}{\partial t} +} +$$ + +since $\Delta V$ cancels. This equation is called the equation of motion. It is the other basic relation (next to the deformation equation) needed to describe one-dimensional wave motion. diff --git a/book/4_mechanical_waves/appendix_strain_p_s.md b/book/4_mechanical_waves/appendix_strain_p_s.md new file mode 100644 index 0000000..2fb8610 --- /dev/null +++ b/book/4_mechanical_waves/appendix_strain_p_s.md @@ -0,0 +1,375 @@ +(app-elastic)= +# Appendix: strain, P- and S-waves + +*In this appendix, strain for an elastic solid is explained. Together with the stress (discussed in the main text), it is used to derive the basic equations for P- and S-waves: the deformation equations and the equations of motion.* + +## Strain + +Strain is a geometrical quantity. It is a *relative* displacement under applied forces, i.e., it is a measure of deformation, normalized by its original size. As for stress, strain may be divided into normal strain and shear strain, depending on the type of force applied. For a normal strain in the deformation of a volume, the strain is simply the change of volume divided by its original volume, i.e.,: + +$$ +e_{\text{normal}} = \frac{\Delta V}{V} +$$ + +where $e$ is the strain, $\Delta V$ the change in volume and $V$ the original volume itself. This volume change was also used to derive the acoustic wave equation (see [the appendix on 1-D acoustic wave motion](./appendix_acoustic_1d.md)). + +Let us now look at {numref}`fig-ap-straindispl`, where the displacement $\underline{u}$ is given for a point $\underline{x}$. Notice that the displacement is a function of $\underline{x}$, i.e., $\underline{u} = \underline{u} (\underline{x})$. When a force/stress is applied, we get a difference in displacement, described by: + +$$ +\begin{aligned} +\underline{u} (\underline{x}+ d\underline{x}) & \simeq + \underline{u} (\underline{x}) + + \frac{\partial \underline{u}}{\partial \underline{x} } d\underline{x} \\ + & = \underline{u} (\underline{x}) + d\underline{u} +\end{aligned} +$$ + +where the first term is a translation and the second term is the term associated with deformation, in accordance with the assumption of elasticity. Notice that the term $\partial \underline{u}/ \partial \underline{x}$ is a matrix since both $\underline{u}$ and $\underline{x}$ are vectors: + +$$ + \frac{\partial \underline{u}}{\partial \underline{x} } = + \begin{pmatrix} + \frac{\partial u_x}{\partial x} & \frac{\partial u_x}{\partial y} & + \frac{\partial u_x}{\partial z} \\ + \frac{\partial u_y}{\partial x} & \frac{\partial u_y}{\partial y} & + \frac{\partial u_y}{\partial z} \\ + \frac{\partial u_z}{\partial x} & \frac{\partial u_z}{\partial y} & + \frac{\partial u_z}{\partial z} \\ + \end{pmatrix} +$$ + +The term $\partial \underline{u}/ \partial \underline{x}$ can be written in a symmetric and anti-symmetric part: + +$$ + \frac{\partial \underline{u}}{\partial \underline{x} } = + \underline{\underline{e}} - \frac{1}{2} \underline{\underline{\xi}} +$$ + +in which the elements of $\underline{\underline{e}}$ and $\underline{\underline{\xi}}$ are: + +$$ + e_{ij} = \frac{1}{2} + \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} + \right) +$$ (eq:straindispl) + +$$ + \xi_{ij} = + \frac{\partial u_j}{\partial x_i} - \frac{\partial u_i}{\partial x_j} +$$ + +where $\underline{\underline{e}}$ is the so-called strain tensor, and $\underline{\underline{\xi}}$ represents rigid rotation. + +```{figure} figures/straindispl.PNG +:name: fig-ap-straindispl +:width: 60% + +Displacement field under presence of stress. +``` + +## Tensile stress and strain, and P-waves in a solid + +P-waves are associated with tensile stresses, so therefore, we consider the tensile stress and strain for an infinitesimal rectangular parallelepiped as shown in {numref}`fig-ap-ptens`. The sizes of the parallelepipeds are given by $\Delta X, \Delta Y$ and $\Delta Z$, while the changes therein are given by $\delta U_x, \delta U_y$ and $\delta U_z$. We apply a tensile stress in the $x$-direction only. As was also given in the appendix on acoustic wave motion, Hooke's law states that a force needed to extend or compress a spring is linearly related to the length ($u$) of extension/compression of that spring, so $F = k u.$ Stress is force per unit area, and strain is relative displacement, so Hooke's law applied to our parallelepiped gives: + +$$ +e_{xx} = \frac{\partial u_x}{\partial x} = \frac{\delta U_x}{\Delta X} = + \frac{1}{E} \tau_{xx} +$$ + +where $\tau_{xx}$ is the tensile stress and $e_{xx}$ the tensile strain in the $x$-direction, and $E$ is called Young's modulus. When applying a tensile stress, the size in the perpendicular directions $y$ and $z$ *decreases*, i.e., a lateral contraction occurs. This is quantified by the so-called Poisson's ratio, via: + +$$ +\begin{aligned} +e_{yy} & = \frac{\partial u_y}{\partial y} = \frac{\delta U_y}{\Delta Y} = + - \sigma e_{xx} \\ +e_{zz} & = \frac{\partial u_z}{\partial z} = \frac{\delta U_z}{\Delta Z} = + - \sigma e_{xx} +\end{aligned} +$$ + +We can follow the same procedure for a tensile stress in the $y$- and $z$-directions with tensile strains $e_{yy}$ and $e_{zz}$, giving totally: + +$$ + \begin{pmatrix} + e_{xx} \\ e_{yy} \\ e_{zz} + \end{pmatrix} + = \frac{1}{E} + \begin{pmatrix} + 1 & -\sigma & -\sigma \\ + -\sigma & 1 & -\sigma \\ + -\sigma & -\sigma & 1 + \end{pmatrix} + \begin{pmatrix} + \tau_{xx} \\ \tau_{yy} \\ \tau_{zz} + \end{pmatrix} +$$ (eq:etau) + +```{figure} figures/ptens.PNG +:name: fig-ap-ptens +:width: 60% + +Infinitesimal rectangular parallelepiped deformed by tensile stress. +``` + +So far, we have written down the basic expressions, but they are not in the desired form: we want to have a form in which the partial derivatives of the particle displacements are given (and not strains), and we want to have the stress on the left-hand side of the equation and the strains/particle-displacements on the right-hand side. To address the first issue, we have: + +$$ + \begin{pmatrix} + e_{xx} \\ e_{yy} \\ e_{zz} + \end{pmatrix} + = + \begin{pmatrix} + \frac{\partial u_x}{\partial x} \\ + \frac{\partial u_y}{\partial y} \\ + \frac{\partial u_z}{\partial z} + \end{pmatrix} +$$ + +To address the second issue, we have to invert the matrix in {eq}`eq:etau`. Since the inverted matrix has terms that are not very simple, the so-called Lamé parameters are introduced: + +$$ +\begin{aligned} + \lambda & = \frac{\sigma E}{ (1-2\sigma)(1+\sigma)} \\ + \mu & = \frac{E}{2 (1+\sigma)} +\end{aligned} +$$ + +where $\mu$ is known as the shear modulus. Using these, we then get: + +$$ + \begin{pmatrix} + \tau_{xx} \\ \tau_{yy} \\ \tau_{zz} + \end{pmatrix} + = + \begin{pmatrix} + \lambda + 2\mu & \lambda & \lambda \\ + \lambda & \lambda + 2\mu & \lambda \\ + \lambda & \lambda & \lambda + 2\mu + \end{pmatrix} + \begin{pmatrix} + \frac{\partial u_x}{\partial x} \\ + \frac{\partial u_y}{\partial y} \\ + \frac{\partial u_z}{\partial z} + \end{pmatrix} +$$ + +This is the desired expression that we will use next. + +Let us now derive the equations necessary to describe 1-D wave motion of a P-wave, travelling in the $x$-direction. To that end, consider {numref}`fig-ap-pblock`, in which again an infinitesimal block with size $\Delta X$, $\Delta Y$ and $\Delta Z$ is given. The mass $\Delta M$ of the little block is given by $\rho \Delta X \Delta Y \Delta Z$. We initiate a plane displacement field with displacements in the $x$-direction only (and therefore there is no contraction $\sigma$ for this 1-D case): + +$$ + \underline{u} (x,y,z,t) = + \begin{pmatrix} + u_x(x,t) \\ 0 \\ 0 + \end{pmatrix} +$$ + +In that case, Hooke's law gives $\tau_{xx} = (\lambda + 2\mu)\partial u_x/\partial x$. As for the acoustic case, we are finally interested in particle velocities rather than displacements (since geophones measure particle velocities), we differentiate both sides of Hooke's law with respect to time $t$, and introduce $v_x = \partial u_x/\partial t$ to give: + +$$ +\boxed{ + \frac{ \partial \tau_{xx} }{\partial t} = (\lambda + 2 \mu) + \frac{ \partial v_x }{\partial x} +} +$$ + +Note that this equation has the same shape as the equation for acoustic waves, i.e., + +$$ + - \frac{ \partial p }{\partial t} = K \frac{ \partial v_x }{\partial x} +$$ + +```{figure} figures/pblock.PNG +:name: fig-ap-pblock +:width: 60% + +Infinitesimal rectangular block under plane stress, used for deriving basic equations for P-wave motion. +``` + +This equation alone does not describe wave motion; to that end we need to apply Newton's law as well. Referring to {numref}`fig-ap-pblock`, the force is, in terms of stress: + +$$ + F_x = \Delta Y \Delta Z [ \tau_{xx}(x+\Delta X) - \tau_{xx}(x) ] +$$ + +and the force is given as mass times acceleration, i.e.: + +$$ + F_x = \Delta M \frac{\partial^2 u_x}{\partial t^2}. +$$ + +Equalizing these forces gives: + +$$ + \frac{ \tau_{xx}(x+\Delta X) - \tau_{xx}(x) }{\Delta X} + = + \frac{\Delta M}{\Delta X \Delta Y \Delta Z} + \frac{\partial^2 u_x}{\partial t^2} +$$ + +or, since $\tau_{xx}(x+\Delta X) \simeq \tau_{xx}(x) + ( \partial \tau_{xx}/\partial x ) \Delta X$: + +$$ + \frac{ \partial \tau_{xx} }{\partial x} + = + \rho \frac{\partial^2 u_x}{\partial t^2}. +$$ + +Again, we need to introduce the particle velocity for the displacement, giving: + +$$ +\boxed{ + \frac{ \partial \tau_{xx} }{\partial x} = + \rho \frac{\partial v_x}{\partial t}. +} +$$ + +This is the other desired equation, the equation of motion for P-waves. As expected, this equation has the same shape as the one for acoustic waves: + +$$ + - \frac{ \partial p }{\partial x} + = + \rho \frac{\partial v_x}{\partial t}. +$$ + +## Shear stress and strain, and S-waves in a solid + +S-waves are associated with shear stresses. We will follow the same line of derivation as for the P-waves but then with shear stresses: consider the shear stress and strain for an infinitesimal rectangle, deformed into a parallelogram (2D) as shown in {numref}`fig-ap-stens`, with sizes $\Delta X$ and $\Delta Z$. Let us first consider the left-hand figure (a), with $\tau_{xz}$. The shear modulus $\mu$ is defined as the constant that relates the stress and displacement, in this case as: + +$$ + \tau_{xz} = \mu \frac{\delta U_x}{\Delta Z} +$$ + +which is the shear equivalent of Hooke's law for elastic deformation. Since the associated strain $e_{xz}$ was defined as $(1/2) ( \partial u_z/\partial x + \partial u_x/\partial z)$ (see {eq}`eq:straindispl`), which for the parallelogram becomes: + +$$ + e_{xz} = \frac{1}{2} \frac{\delta U_x}{\Delta Z}, +$$ + +the stress-strain relationship for shear deformation becomes: + +$$ + \tau_{xz} = 2 \mu e_{xz}. +$$ + +```{figure} figures/stens.PNG +:name: fig-ap-stens +:width: 70% + +Infinitesimal rectangle deformed to parallelogram by shear stress. +``` + +This was the analysis for shear deformation in the horizontal direction, but deformation can also take place in the vertical direction, as shown in {numref}`fig-ap-stens`(b). In that case we have for the stress and strain: + +$$ +\begin{aligned} + \tau_{zx} & = \mu \frac{\delta U_z}{\Delta X} \\ + e_{zx} & = \frac{1}{2} \frac{\delta U_z}{\Delta X} \\ + \tau_{zx} & = 2 \mu e_{zx}. +\end{aligned} +$$ + +The deformations as shown in {numref}`fig-ap-stens`(a) and (b) are actually deformations with a rigid-body rotation where $e_{xz} \neq e_{zx}$. We are interested in the deformation which show no rigid-body rotation, as shown in {numref}`fig-ap-stens`(c). For no rigid-body rotation $\xi_{xz}=0$, so: + +$$ + \frac{\partial u_x}{\partial z} = \frac{\partial u_z}{\partial x}, + \qquad \text{or} \qquad + \frac{\delta U_x}{\Delta Z} = \frac{\delta U_z}{\Delta X}, +$$ + +We then have: + +$$ + e_{xz} = e_{zx}, +$$ + +and therefore: + +$$ + \tau_{xz} = \tau_{zx}. +$$ + +Similarly, for no rigid-body rotation in 3-D: + +$$ +\begin{aligned} + e_{xy} & = e_{yx} \\ + e_{yz} & = e_{zy} \\ + \tau_{xy} & = \tau_{yx} \\ + \tau_{yz} & = \tau_{zy} +\end{aligned} +$$ + +Let us now derive the equations necessary to describe 1-D wave motion of an S-wave, travelling in the $x$-direction (as for the P-wave). To that end, consider {numref}`fig-ap-sblock` in which again an infinitesimal block with size $\Delta X$, $\Delta Y$ and $\Delta Z$ and mass $\Delta M = \rho \Delta X \Delta Y \Delta Z$ is given. We initiate a plane displacement field, giving displacements in the $z$-direction only (for P-waves they were in the $x$-direction only): + +$$ + \underline{u} (x,y,z,t) = + \begin{pmatrix} + 0 \\ 0 \\ u_z(x,t) + \end{pmatrix} +$$ + +This means that there are no tensile stresses (since $u_z = u_z(x,t)$ only), and the other shear stresses $\tau_{xy}$ and $\tau_{yz}$ are zero. The shear equivalent of Hooke's law gives: + +$$ + \tau_{zx} = 2\mu e_{zx} = 2\mu \cdot \frac{1}{2} \left( + \frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z} \right) + = \mu \frac{\partial u_z}{\partial x} +$$ + +As for P-waves, in order to introduce the particle velocity $v_z$, we differentiate both sides with respect to time $t$ and take the particle velocity as $v_z = \partial u_z/\partial t$ to give: + +$$ +\boxed{ + \frac{\partial \tau_{zx}}{\partial t} = \mu \frac{\partial v_z}{\partial x}. +} +$$ + +This is the desired expression for the deformation. + +```{figure} figures/sblock.PNG +:name: fig-ap-sblock +:width: 60% + +Infinitesimal rectangular block under plane stress, used for deriving basic equations for S-wave motion. +``` + +As before, this equation alone does not describe wave motion since we need to apply Newton's law as well. Referring to {numref}`fig-ap-sblock`, the force is, in terms of stress: + +$$ + F_z = \Delta Y \Delta Z [ \tau_{zx}(x+\Delta X) - \tau_{zx}(x) ] +$$ + +and the force is given as mass times acceleration, i.e.: + +$$ + F_z = \Delta M \frac{\partial^2 u_z}{\partial t^2}. +$$ + +Equalizing these forces gives: + +$$ + \frac{ \tau_{zx}(x+\Delta X) - \tau_{zx}(x) }{\Delta X} + = + \frac{\Delta M}{\Delta X \Delta Y \Delta Z} + \frac{\partial^2 u_z}{\partial t^2} +$$ + +or, since $\tau_{zx}(x+\Delta X) \simeq \tau_{zx}(x) + ( \partial \tau_{zx}/\partial x ) \Delta X$: + +$$ + \frac{ \partial \tau_{zx} }{\partial x} + = + \rho \frac{\partial^2 u_z}{\partial t^2}. +$$ + +Again, we need to introduce the particle velocity via the displacement, giving: + +$$ +\boxed{ + \frac{ \partial \tau_{zx} }{\partial x} = + \rho \frac{\partial v_z}{\partial t}. +} +$$ + +This is the other desired equation, the equation of motion for S-waves. diff --git a/book/4_mechanical_waves/basic_notions.md b/book/4_mechanical_waves/basic_notions.md new file mode 100644 index 0000000..6f92f15 --- /dev/null +++ b/book/4_mechanical_waves/basic_notions.md @@ -0,0 +1,53 @@ +# Basic physical notion of waves + +The seismic method makes use of the properties of the velocity of seismic waves. This velocity is different for different rocks and it is this difference which is exploited in the seismic method. When we create a wave field at or near the surface of the earth, some energy will be reflected back (bounced back). They can be viewed as echoes. From these echoes we can determine the velocities of the rocks, as well as the depths where the echoes came from. In this chapter we will discuss the basic principles behind the behaviour of seismic wave fields in solid materials. When we use the seismic method, we usually discuss two types of seismic methods, depending on whether the distance from the sound source to the detector (the "ear") is large or small with respect to the depth of interest: the first is known as refraction seismics, the other as reflection seismics. Of course, there are overlaps and differences between those two types which will be discussed in this chapter. + +Everybody knows what waves are when we are talking about waves at sea. Seismic wave fields in materials have the same kind of behaviour as these waves, only they travel much faster than the waves we see at sea. Waves can occur in several ways. We will consider two types of them, namely the longitudinal and the shear wave. Longitudinal waves behave like waves in a large spring. When we push from one side of a spring, we will observe a wave going through the spring which characterizes itself by a thickening of the wires running through the spring in time (see also {numref}`fig-pands`). A property of this type of wave is that the motion of a piece of the wire is in the same direction as the wave moves. These waves are also called Push-waves, abbreviated to P-waves, or compressional waves. Another type of wave is the shear wave. A shear wave can be compared with a chord. When we push a chord upward from one side, a wave will run along the chord to the other side. The movement of the chord itself is only up- and downward: characteristic of this wave is that a piece of the chord is moving perpendicular to the direction of that of the wave (see also {numref}`fig-pands`). These types of waves are referred to as S-waves, also called shear waves. Characteristic of this wave is that a piece of the chord is pulling its "neighbour" upward, and this can only occur when the material can support shear stress. In fluids, one can imagine that a "neighbour" cannot be pulled upward simply because it is a fluid. Therefore, in fluids only P-waves exist, while in a solid both P- and S-waves exist. + +```{figure} figures/blokjes.png +:name: fig-pands +:width: 80% + +Particle motion of P and S waves, where the particle motion for SV is in the vertical direction only and for SH in the horizontal direction only. +``` + +In the real world, we deal with three spatial dimensions, so a wave will spread in three directions. In a homogeneous medium (so the properties of the material are everywhere constant and the same), the wave will spread out like a sphere. The outer shell of this sphere is called the wave front. Another way of describing this wave front is in terms of the normal to the wavefront: the ray. We are used to rays in optics and we can use the same notion in the seismic method. When we were explaining the behaviour of P- and S-waves, we already used the term "neighbour". This was an important feature; otherwise the wave would not move forward. A fundamental notion included in this, is Huygens' principle. When a wave front arrives at a certain point, that point will behave also as a source for the wave, and so will all its neighbours. The new wavefront is then the envelope of all the waves that were generated by these points. This is illustrated in {numref}`fig-huygens`. Again, the ray can then be defined as the normal to that envelope which is also given in the figure. + +```{figure} figures/huygens.png +:name: fig-huygens +:width: 80% + +Using Huygens' principle to locate new wavefronts. +``` + +So far, we only discussed the way in which the wave moves forward, but there is also another property of the wave we have not discussed yet, namely the amplitude: how does the amplitude behave as the wave moves forward? We have already mentioned spherical spreading when the material is everywhere the same. The total energy will be spread over the area over the sphere. This type of energy loss is called spherical divergence. It simply means that if we put our "ear" at a larger distance, the sound will be less loud. There is also another type of energy loss, that is due to losses within the material, which mainly consists of internal friction losses. This means that the amplitude of a wave will be extra damped because of this property. S-waves usually show higher friction losses than P-waves. Finally, we give a table of common rocks and their seismic wave velocities in {numref}`tab-wavevel`. + +```{list-table} Seismic wave velocities for some common media, soils and rocks. +:name: tab-wavevel +:header-rows: 1 + +* - Material + - Wave velocity (m/s) + - Material + - Wave velocity (m/s) +* - Air + - 330 + - Sandstone + - 2000-4500 +* - Water + - 1500 + - Shales + - 3900-5500 +* - Soil + - 20-300 + - Limestone + - 3400-7000 +* - Sands + - 600-1850 + - Granite + - 4800-6000 +* - Clays + - 1100-2500 + - Ultra-mafic rocks + - 7500-8500 +``` diff --git a/book/4_mechanical_waves/elastic_waves.md b/book/4_mechanical_waves/elastic_waves.md new file mode 100644 index 0000000..d6a60de --- /dev/null +++ b/book/4_mechanical_waves/elastic_waves.md @@ -0,0 +1,108 @@ +# Wave fields in a solid: elastic (P and S) waves + +## Stress + +Here we define *stress*. Stress can be defined as the force per unit area exerted on a body across an imaginary surface (see also {numref}`fig-stressforce`). Since the dimension of stress is of a pressure, i.e., force per unit area, the quantities of stress are also given in the dimension of pressure, i.e., Pa. Therefore, the stress can be seen as the equivalent of pressure in an (elastic) solid. One can discern normal stress as the stress perpendicular to a surface, and shear stress as the one parallel to a surface. The normal stress is also called compressional or tensile stress. Since a stress can be in any direction to a surface, the stress can be decomposed into components perpendicular and parallel to the surface. This decomposition matrix is also called the stress tensor. In formula form, this can all be written as: + +$$ + \frac{\underline{F}}{\Delta S} = + \underline{\underline{\tau}} \, \underline{n} +$$ + +where $F$ is the force, $\Delta S$ is the surface, $\underline{\underline{\tau}}$ is the stress tensor and $\underline{n}$ is the vector normal to the surface $\Delta S$. When writing out the matrix of the stress, i.e., the stress tensor, in terms of the coordinates $x$, $y$ and $z$, we get: + +$$ + \underline{\underline{\tau}} = + \begin{pmatrix} + \tau_{xx} & \tau_{xy} & \tau_{xz} \\ + \tau_{yx} & \tau_{yy} & \tau_{yz} \\ + \tau_{zx} & \tau_{zy} & \tau_{zz} + \end{pmatrix} +$$ + +Some of these factors are visualized in {numref}`fig-tautensor`. + +```{figure} figures/stressforce.PNG +:name: fig-stressforce +:width: 60% + +Definition of stress for a force working on a surface $\Delta S$. +``` + +In order to see the equivalence with the acoustic case, consider the case where no shear stresses exist. Then the diagonal elements of the matrix are the only non-zero ones, and the stress matrix can be written as: + +$$ + \underline{\underline{\tau}}_{\text{no-shear}} = + \begin{pmatrix} + -p & 0 & 0 \\ + 0 & -p & 0 \\ + 0 & 0 & -p + \end{pmatrix} +$$ + +where $p$ is the pressure. Notice that there is a minus sign since commonly the positive sign for pressure is for a compressional force, while for stress it is for a tensile force. + +```{figure} figures/tautensor.PNG +:name: fig-tautensor +:width: 60% + +Definition of stress components for stresses working on an infinitesimal parallelepiped. +``` + +## P-waves in an elastic solid + +For 1-D wave motion in the $x$-direction, Hooke's law gives for the plane tensile stress $\tau_{xx}$ (see [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ + \frac{\partial \tau_{xx}}{\partial t} = (\lambda + 2\mu) + \frac{\partial v_x}{\partial x} +$$ + +where $\lambda$ and $\mu$ are the so-called Lamé parameters. $\lambda$ is the first Lamé parameter, and $\mu$ is the shear modulus; in 3-D, $\lambda$ is related to the bulk modulus $K$ through $K = \lambda + (2/3)\mu$. This equation is called the deformation equation for a P-wave in an elastic solid. Note that this equation has the same form as the deformation equation for acoustic waves (see {eq}`eq:eqdeform`). + +Then applying Newton's law gives an equation nearly equivalent to the one for the acoustic case, but now having tensile stress instead of pressure (see also [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ +\frac{\partial \tau_{xx}}{\partial x} = \rho \frac{\partial v_x}{\partial t} +$$ + +This is the equation of motion for a P-wave. Combining the deformation equation and the equation of motion gives, as for the acoustic case, the wave equation for the stress, assuming the density being constant: + +$$ +\frac{\partial^2 \tau_{xx}}{\partial x^2} + - \frac{1}{c^2_P} \frac{\partial^2 \tau_{xx}}{\partial t^2} = 0 +$$ + +where $c^2_P = (\lambda + 2\mu)/\rho$ is the squared P-wave velocity. This wave equation describes the wave propagation for P-waves in an elastic solid. + +## S-waves in an elastic solid + +Let us now consider shear stresses which, as we shall see, will result in a wave equation for S-waves. To that end, we consider 1-D displacement in the $z$-direction for a wave propagating in the $x$-direction for the 1-D shear motion. The equivalent of Hooke's law for shear stress results in (see [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ + \frac{\partial \tau_{zx}}{\partial t} = \mu \frac{\partial v_z}{\partial x} +$$ (eq:deformS) + +where $\tau_{zx}$ is the shear stress. This is the deformation equation for an S-wave. + +Then applying Newton's law for a shear stress gives (see also [the appendix on strain, P- and S-waves](./appendix_strain_p_s.md)): + +$$ + \frac{\partial \tau_{zx}}{\partial x} = \rho \frac{\partial v_z}{\partial t} +$$ (eq:eqmotionS) + +which is the equation of motion for S-waves. Combining the deformation equation and the equation of motion gives the wave equation for the shear stress, assuming the density being constant: + +$$ +\frac{\partial^2 \tau_{zx}}{\partial x^2} + - \frac{1}{c^2_S} \frac{\partial^2 \tau_{zx}}{\partial t^2} = 0 +$$ (eq:waveqStau) + +where $c^2_S = \mu / \rho$ is the squared S-wave velocity. This wave equation describes the wave propagation for S-waves in an elastic solid. + +With respect to the (1-D) reflection coefficients for P- and S-waves at the boundary of two elastic solids: the derivations go similar as for the acoustic case, as worked out in the previous section: + +- For P-waves, we require the continuity of the tensile stress $\tau_{xx}$ and the particle velocity $v_x$ at the interface, and +- For S-waves we require the continuity of the shear stress $\tau_{zx}$ and the particle velocity $v_z$ at the interface. + +The resulting expressions for the reflection and transmission coefficient are the same as for the acoustic case, only now the velocities are for the elastic solid. (See *EXERCISES*) diff --git a/book/4_mechanical_waves/field_recordings.md b/book/4_mechanical_waves/field_recordings.md new file mode 100644 index 0000000..3175135 --- /dev/null +++ b/book/4_mechanical_waves/field_recordings.md @@ -0,0 +1,60 @@ +# Seismic wavefield recordings from the field: physical interpretation using kinematic concepts + +In this section, we will look at the physical interpretation of seismic wavefield data as recorded in the field, two recordings on land (one with rocky near-surface and one with loose-soil near-surface) and one recording at sea. + +## Seismic wavefield recording on land: rocky near-surface + +Let us first consider a seismic wave field recorded on land. In {numref}`fig-raw-shot-land` this wavefield record is shown. The first event we are considering is the "first arrival". It shows itself by giving a pulse after a quiet period. This arrival can be a *direct* arrival or a *refraction*, both of which we discussed earlier. When looking closely at the record, it can be seen that there is a slight change of dip, occurring at around an offset of 800 m. Therefore, the arrival until this distance is a direct arrival, while beyond that distance, it is the refraction. By drawing a straight line through the direct arrival, the velocity with which it propagates along the surface is: 2400 meters in 670 ms, so approximately 3600 m/s. This is a relatively high velocity, so there is probably some very hard rock at the surface. The velocity of the refracted arrival can also be determined by a straight line through that arrival; it amounts to 2400 meters in around 500 ms (note that the arrival does not go through the origin, so it is the difference between the time at $x=0$ m and at $x=2400$ m), so a velocity of 4800 m/s. Below the record in {numref}`fig-raw-shot-land`, a simple model is shown that explains this first arrival; the synthetic record belonging to this model is given on the top right of the figure. + +The next events we consider are the strong events that cross 2400 meter at some 1.3 seconds. This event is interpreted as ground-roll or surface waves (they propagate along the surface). Calculating the velocity, we come to 1850 m/s. Again, the simple model below the record explains this ground roll; the synthetic record on the top right of the figure also shows this arrival. + +```{figure} figures/Figure4_1.PNG +:name: fig-raw-shot-land +:width: 100% + +Field seismic wavefield recording from land survey (top left), synthetic seismogram (top right) using model of near surface (middle) and model at larger depths (bottom). +``` + +Also in this field record, a "high-frequency" event can be observed, which goes through the 4-second mark at about 1300 meters distance. Calculating the velocity from this, we come to some 325 m/s. It may be clear that this is a wave that goes through the air. This event is also synthesized in the top right figure, using the model as given below it. + +Last but not least, are "high-frequency" events which are slightly curved, e.g. the ones at 0.9 and 1.6 seconds. These events are interpreted as reflections from layer boundaries in the deep subsurface. Those are usually the events we are interested in when we want to obtain an image of the subsurface. Using a simple model as given at the bottom of the figure (which explains the deeper part of the earth), the synthetic record for these events is also shown in the top right of the figure. + +In the above, we have interpreted five types of events, which can be captured in one combined model and are shown in one combined synthetic seismogram. These synthetics explain the most important events in the raw seismic record. Still, when looking at the resulting synthetic seismogram, we see that we are very over-simplifying the situation, since the synthetic and field record are only resembling in the arrival times of the most important events. When looking at the general characteristics, they are very different indeed. + +## Seismic wavefield recording on land: loose-soil near-surface + +The field record we discussed so far was recorded on some hard rocks where the velocities are relatively high. However, when shooting data on land with loose top soil, in this case sand on a beach, the characteristics are much different. In {numref}`fig-raw-shot-wassenaar`, a field record of this situation is given. Again, we can determine the main events in this record. Let us first consider the "first arrival", i.e. the arrival that is coming in first after a quiet period. As usual, this is interpreted as a refraction, as shown in the figure below the record. The velocity can be determined: we come to some 1600 m/s. This velocity is very close to the velocity of water, so this refraction may be due to the water table. In the figure on the right, the synthetic shows this arrival. + +The next event is the most prominent one, namely the event that goes through the 1-second mark at some 180 m, so its velocity is around 180 m/s. This arrival is interpreted as "ground-roll"/surface waves, which travel along the surface. The model which explains this arrival is given again below the record, and its synthetic is shown on the top right. + +The most important events for this wavefield recording are the "high-frequency" events that are the slightly curved arrivals, which can all be interpreted as reflections from deep layers. The number of reflections are too many; only a few are synthesized in the record on the top right, using the model as given at the bottom of the figure. + +Again, when comparing the synthetic to the field seismogram, it is obvious that we have very over-simplified the earth; the positive side is that we have probably been able to understand most of the events in the field recording. + +```{figure} figures/Figure4_2.PNG +:name: fig-raw-shot-wassenaar +:width: 100% + +Field seismic wavefield recording from land survey with loose top soil (top left), synthetic seismogram (top right) using model of near surface (middle) and model at larger depths (bottom). +``` + +## Seismic wavefield recording at sea + +{numref}`fig-raw-shot-marine` shows a seismic wavefield recording, made at sea. This record is much "cleaner" than the land record, as we measure in a water layer, which is a very homogeneous layer. + +Let us analyze some separate events again. The first event in the marine record is the faint one, going nearly through the origin. It crosses the 500 meter at some 340 ms; this means a velocity of some 1470 m/s. It may be clear that this is the direct arrival from the source to the receivers through the water, as explained in the model below the record. This direct arrival is thus a body wave, since it travels with the velocity of water. + +The next event is the first arrival at farther offsets; this arrival is interpreted as a refractive event. When analyzing the distance travelled over time, i.e. the apparent velocity, a velocity of roughly 2000 m/s is obtained. Using the equation for a refraction ({eq}`eq:trefract`), a depth of 300 meter is obtained. This is quantified in the model below the figure, and its associated synthetic seismogram in the figure on the top right. + +The third event we analyze is the first strong event that looks hyperbolic: starting at some 0.4 seconds and bending down to some 2.2 seconds at 3200 m offset. Clearly, because of its hyperbolic behaviour, it is interpreted as a reflection. When looking at later times, we have some more strong hyperbolic events, such as at 0.8 seconds (bending downward toward some 2.3 seconds), and at 1.2 seconds (bending downward toward 2.4 seconds), and even more. These events are interpreted as so-called multiply reflected waves, i.e., waves that bounce up and down in the water layer. In fact almost all events we see below 0.8 seconds are due to multiply reflected waves, or short-hand: multiples. The times at which the multiply reflected waves arrive seem to be periodic; this is indeed the case. + +Combining the interpretation of the refraction and the reflection, it must be noted that the refracted arrival does not converge to the first reflection but to a later reflection. This means that the refraction occurs at a deeper layer; the shallower layers are probably loosely consolidated, so that the velocity of sound has not changed much compared to the one from water. + +A simple model explaining all these events is shown in the figure below the record, with a water layer of 300 meter. The refracted layer is estimated at a depth of 550 meter, where we assumed that the velocity was a constant of 1500 m/s above. The resulting synthetic seismogram is shown in the top right of the figure. It may be clear that the multiply reflected waves come from the same reflective boundary in the subsurface, namely, the sea bottom (and the sea surface, of course), and are therefore superfluous. They are considered as noise; the only one being "signal" is the one at around 0.4 seconds. + +```{figure} figures/Figure4_3.PNG +:name: fig-raw-shot-marine +:width: 100% + +Seismic wavefield recording from marine survey (top left), synthetic seismogram (top right) using model of water layer and sea-water bottom, where only the path of one multiple reflection is drawn (bottom). +``` diff --git a/book/4_mechanical_waves/figures/Figure4_1.PNG b/book/4_mechanical_waves/figures/Figure4_1.PNG new file mode 100644 index 0000000000000000000000000000000000000000..dcc2ab1e6d8afad9505854d09a700e2bb7c418e8 GIT binary patch literal 2805548 zcmZsD2|Sf;*L{jek|Y%pk|arzB$<6gd_<`I?0rf%po&LLgr+iL&!`r z^Iu2L^M1qsJHLL<`*!4<`@XJg@4fa~Yu~;~3Kw@!GEi>Xv}wmB87bvWn<&q3+O%tp zVl&=3?0ijQ6UnAcm!!m19NWhVjjXg<2tVD0x@*s}8Qt5oZ2$X~mgt=j)mBQ(x_kkF=_cHH-=41xCO6}mE}T@;3!*@Gmv48O8j zb9daHos#Eac*=8Nc{z&qM4n}j)kN-m#E@`Zm27+?sW=U(c;L~emF%ourG1_C*|ff- zVKs&sZH}9EZTt*B{^!qtbl(qFG5m$uS~)?5D}>FG*TnpCi0rK^SFTX+5(-y{ 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