Ray tracing in geophysics

Transcription

1 Ray tracing in geophysics Seminar Author: Klemen Kunstelj Adviser: dr. Andrej Gosar and dr. Jure Bajc

2 Abstract Alp 2002 is an international seismological project. The main issue is to construct the structure of the Lithosphere in the area of Suth-Eastern Alps. The experimental part took place in July 2002 and the analysis of these results is planned to be completed aproximately in two years. In the seminar some basic modelling methods that are used in the analysis of the Alp2002 data are presented. Figure 1: Network of geophones with seismograms in Slovenia during Alp2002

3 Figure 2: Seismograms of an explosion in Vojnik during Alp2002 gathered on the network of seismographs in Slovenia 1

4 CONTENTS 2 Contents 1 Introduction 3 2 Ray theory and seismic waves Wave equations for P- and S-waves Ray equation Ray propagation in media of constant velocity Ray tracing with RAYAMP 6 4 Ray tracing with JIVE3D Model parametrization The forward problem Two-point ray tracing Ray perturbation theory (RPT) Frechet derivatives The inverse problem Conclusion 14

5 1 Introduction 3 1 Introduction The main issue of Alp 2002 is to discover in detail the tectonics and geodynamics at the junction of Europian, Adriatic (fragment of African tectonic plate) and Tisza plates (contact zone between South-Eastern Alps, Dinarides and Panonian basin) by using combined seismic refraction/wide-angle reflection method. For this two techniques the maximal distance between the source (explosion) and the targets (geophone with seismograph) is around few hundred km. The velocity of seismic waves is in general increasing with depth. Waves are refracted at discontinuities and travel in deeper regions almost horizontally (refraction) or they are reflected under blunt angle (wide-angle reflection). Because seismic waves travel long distances, projects as Alp 2002 give us information about the lithosphere. The Earth Lithosphere is in general characterized by three discontinuities: Moho discontinuity between Earth s crust and mantle. It is named after Croatian seismologist Mohorovičič, who discovered it by analysing the earthquake in 1909 in the region of Kolpa. Moho is characterized by an increase of P-wave velocity (compresional waves or prima waves) from km/s in the crust to km/s in the mantle, and an increase of S-wave velocity (shear waves) from km/s to 4.8 km/s.moho is situated km below continents, 5 8 km below the ocean floor, and km below certain mountains regions. Conrad discontinuity between upper part (granite) and lower part (bazalt) of the crust, is located at depth km and is characterized by an increase of P-wave velocity from km/s to 6.5 km/s. It is not observed everywhere. Third discontinuity which is often seen on a seismic data is a junction between the sediments and magmathic base. In the framework of the Alp2002 project a network of 12 profiles of total length 4100 km was spreading over seven countries (Austria, Chech republic, Hungary, Croatia, Slovenia, Italy and Germany) portable seismographs were deployed and 31 strong (300 kg) explosions fired. In Slovenia 127 seismographs were placed along five profiles of total length 575 km and two explosions were fired; one near Vojnik and the other near Gradin. The collected data will allow construction of the three-dimensional model of the lithosphere.

6 2 Ray theory and seismic waves 4 2 Ray theory and seismic waves 2.1 Wave equations for P- and S-waves Displacements u(x, t) in Navier s equation ρ 2 u = (λ + µ) ( u) + µ 2 u t 2 form a vector field in an elastic medium. We use Helmholtz theorem in order to represent u in terms of a curl-free scalar potential φ and divergence-free vector potential ψ u = φ + ψ. (1) The Navier equation may than be re-written as [ (λ + 2µ) 2 φ ρ 2 φ t 2 ] + [ µ 2 ψ ρ 2 ψ t 2 ] = 0, (2) where ρ, λ and µ are taken to be costant troughout the medium and the change in the gravitational field strength are ignored. Equation (2) is setisfied, if 2 φ 1 2 φ λ + 2µ α 2 t = 0, α = (3) 2 ρ and 2 ψ 1 2 ψ µ β 2 t = 0, β = 2 ρ. (4) Equations (3) and (4) are two independent wave equations that describe compressional waves traveling at the P -wave velocity α and shear waves which travel at the S-wave velocity β. They represent a good aproximation to the equation of motion for heterogeneous media provided that the elastic propertis of the medium do not change significantly over a single wavelength. If the elastic properties do change rapidly, both wave equations become coupled and P - and S-waves may no longer be considered to propagate independently. In the layer-interface formalism both wave equations are normally treated as independent within each layer, but converted waves may be produced at intefaces, where there is a sudden change in the elastic properties of the medium. 2.2 Ray equation The procedure of execution the ray equation is similar in geophysics and optics. The solution of (3) for harmonic waves of frequency ω with constant amplitude and zero initial phase can be written (for P -waves) φ(x, t) = φ 0 (x)e iωt (5)

7 2.2 Ray equation 5 without the loss of generality. If φ 0 (x) is defined as φ 0 (x) = A(x)e ik 0S(x), (6) where k 0 = ω/α 0 and α 0 is a reference wave speed, equation (5) will form a solution to the wave equation provided that certain constraints on A(x) and S(x) (called the eikonal) describe the spatial variation of amplitude and phase within a general wave field. Let us now introduce the concept of a ray, which describes the path of a wave packet through the isotropic medium, being at all times perpendicular to the wavefront. It is clear that the unit vector ˆp = α α 0 S (7) is always perpendicular to the wavefront in the direction of the increasing phase. Like the travel time T (x), which is defined as a time taken for wavefront to travel from a reference point x 0 to the arbitrary point x, the eikonal is defined relative to the phase at the reference point S(x) = α 0 T (x) and the eikonal equation can be re-written in terms of the travel time and the wave speed v(x), ( S) 2 = α2 0 = ( T ) 2 = 1 α 2 v(x). (8) 2 The ray path may be described by the function x(s) where s is the curvilinear distance from the reference point along the ray path. Let us also define a second function p(s) for which p = T. This is called the slowness vector, because its magnitude at a point x(s) is equal to the reciprocal of the velocity at that point. Applying the condition that the ray path is orthogonal to the wavefronts yields dx = v T = vp. (9) ds By differentiating the eikonal equation (8) with respect to s and combining the result with (9) we obtain the ray equation ( ) d 1 dx = ds v ds ( 1 v ). (10) Once the ray trajectory has been obtained, travel times may be calculated by integrating ( ) 1 T = T ds = ds. (11) raypath raypath v(x(s)) So the travel time from A to B is equal to the travel time from B to A. This principle of reciprocity may be used to improve the efficiency of ray-tracing for certain problem types.

8 2.3 Ray propagation in media of constant velocity Ray propagation in media of constant velocity In this case we will take a layer of thickness H and velocity v over a halfspace of velocity v. This situation allows the study of ray trajectories and travel times due to the presence of a surface of contact between the layer and a half-space. The thickness of the layer is the parameter that scales a spatial dimension of the model. For this reason, the application of ray theory is valid, if the wave lengths are much smaller than the thickness of the layer. This example represents, in a simplified way, the situation of the Earth s crust over the upper mantle for small distances, for which the flat-earth approximation is valid. We consider the case v < v and the focus F at the Figure 3: Seismic waves surface (h = 0)(figure 3). There are three types of rays travelling from F to P, which is a distance x from the focus F: (a) direct rays; (b) rays reflected on the surface between the layer and the half-space (FCP); and (c) criticaly refracted rays or head waves, i.e. rays with a critical angle of incidence i c, propagate a certain distance horizontally through the half-space and come back to the free surface with the same angle of incidence i c (FBDP). In the next sections of this seminar I will present some possible approaches of modelling the seismic data. 3 Ray tracing with RAYAMP Rayamp is a ray tracing modelling program that is used to obtain 2D models without inversion stage. To define the velocity structure within a 2D model, there are two types of boudaries, model and divider boundaries. A model boundary is a straight line of arbitrary dip. It has assigned a costant velocity along its length and a nonzero velocity gradient normal to its length. A divider boundary is assigned a velocity zero and it separates two regions

9 3 Ray tracing with RAYAMP 7 Figure 4: Profil from Rijeka to Vojnik with different velocity and velocity gradient. Blocks may thus be defined, in which the velocity, magnitude and direction of velocity gradient are arbitrary. The ray path within a given block is a circular arc (because of constant velocity gradient), for which the travel time and the distance traveled may be calculated using very simple analytical expresions. The source may be located along any model boundary, targets are usually located on the surface of the model. If the incidence to a boundary is at the Figure 5: Profil from Vojnik to Ivanič grad critical angle, the head waves may occur. Beyond the point of intersection of

10 4 Ray tracing with JIVE3D 8 the critical ray with the boundary, the head waves are simulated by shooting critically refracted rays off the boundary at regular intervals along its length. If the incident angle is more than the critical, then the ray reflects from the boundary. For the precritical and multiple reflections, only the boundaries at which reflection is studied need to be specified. At all other boundaries the behaviour of a ray is controlled by the angle of incidence at the boundary. Thus, if no precritical or multiple reflections are studied, then a single specification of the range of take-off angles gives all wide-angle reflections, turning rays and head waves. The corresponding travel-time curve is divided into branches such that the distance along each branch increases or decreases monotonically with distance. The family of rays associated with each traveltime branch is labeled with a unique identification number, which is used in the synthetic seismogram routine for purposes of interpolation within a given ray family. 4 Ray tracing with JIVE3D Modelling with Jive3D can be divided into forward modelling and inversion stjpg. At each iteration of the algorithm, a set of synthetic travel-time data are produced from a working velocity model, and the Frechet derivatives (subsection 4.2.3), which link small changes in model parameters to small changes in travel-time data are calculated. These synthetic data and Frechet derivatives are then passed to the inversion stage, which compares the synthetic data with the given real data and calculates a new model based on a set of linear aproximations until the model converges to a point that optimizes the specified norm for smoothness and best fit. 4.1 Model parametrization An important feature of any tomographic inversion method is its approach to model parametrization. We require a parametrization that is able to describe models in the conventional layer-interface formalism. Each layer and interface is then parametrized by a grid of velocity and depth nodes, allowing the inclusion of reflected and refracted travel times. To define a 3D field of seismic velocity and an interface in a 3D velocity model, we will use qudratic B-splines in three-dimensions B2 3D (x, x ) = β 2 (x 1, x 1)β 2 (x 2, x 2)β 2 (x 3, x 3) and cubic B-splines in two-dimensions B3 2D (x, x ) = β 3 (x 1, x 1)β 3 (x 2, x 2). The β functions are basis functions for interpolation, also called B-spline functions.

11 4.2 The forward problem 9 Velocity field is then given by v(x) = 27 i=1 B 3D 2 (x, x i )v i, (12) where v i are the velocity B-spline coefficients that have influence at position x; x i are the node positions corresponding to those coefficients. Similarly, an interface in a 3D velocity model (modelled as a 2D surface), constructed using cubic B-splines is given by z(x) = 16 i=1 B 2D 3 (x, x i )z i, (13) where z i are depth B-spline coefficients that have influence at position x; x i are equivalent node positions. The interfaces represent discontinuities in seismic velocity at which reflections and refractions may occur. 4.2 The forward problem Two-point ray tracing In the section 2.2 it was stated that the ray equation may be used to determine the trajectory of a ray through a velocity model given the appropriate boundary conditions. These conditons normally take one of the following forms: 1. Source position and direction of propagation specified - find the position at which the ray returns to the surface (if at all) and the travel time; 2. Source position and direction specified - find the position within the model at which a given travel time is specified; 3. Start and end points specified - find all rays which join the two together and their travel times. In order to ensure that all two-point solutions may be found for any possible ray phase (including reflections and refractions through multiple layers), the shooting method is used in JIVE3D. This approach is relatively easy to implement for the models using the ray perturbation theory (RPT). It may be applied in a consistent manner to wide-angle reflection, wide-angle refraction and normal incidence reflection phases. All possible ray paths from source to receiver are being explored.

12 4.2 The forward problem Ray perturbation theory (RPT) RPT is a powerful set of tehniques allowing ray paths, travel times, amplitudes, paraxial rays and even waveforms to be approximated rapidly by applying perturbations to models, for which a previous solution has been calculated. Two mathematical formulations of RPT have emerged: Hamiltonian formulation and Lagrangian formulation. In both, a similar procedure is adopted. A ray is traced through a reference medium cell by cell for which an analytical solution exists and the quantities sought by the procedure (travel time, amplitude, final position and slowness vector) are calculated for reference ray (figure 6). One or more perturbations to the system are then applied and their effect on the ray is estimsted to first order, producing the perturbed ray. When ray-tracing through isotropic media, the Hamiltonian may be defined as follows H(x, p, τ) = 1 2 (p2 u 2 (x)), (14) where u(x) is slowness function, or the reciprocal of the seismic velocity; x and p are the position and the slowness vector perspectively, and the parameter τ is a measure of the extent of propagation along the ray, and may be defined by its differential relation to the travel time dt = u 2 dτ. Hamilton s equations yield a representation of the ray equations, dx dτ = ph, dp dτ = xh. (15) In order to apply RPT, the function u 2 is written in the form u 2 = u 2 ref + u 2 where u 2 ref is a reference function for which rays may be traced analytically and u 2 is perturbation that will be applied. This perturbation of u 2 results in a perturbation of the Hamiltonian H = 1 2 u2. The perturbations to the position and slowness now read x(τ) = x(τ 0 ) + (τ τ 0 ) p(τ 0 ) τ τ 0 (τ τ ) x ( u 2 (x))dτ, (16) p(τ) = p(τ 0 ) + 1 x ( u 2 (x))dτ. (17) 2 τ 0 The travel time as a function of τ is given by T (τ) = τ τ τ 0 u 2 (x)dτ (18) and the perturbation in travel time is Taylor expansion up to first order T (τ) = τ τ 0 u 2 (x(τ ))dτ + τ τ 0 Γ 0 x(τ )dτ, (19)

13 4.2 The forward problem 11 where Γ 0 is the gradient of the slowness squared function (Γ 0 = u 2 (x 0 )). These results are all expressed as polynomials in τ therefore the integrals in (16), (17),and (19) are straightforward. The equations above provide a mechanism for tracing a ray through the medium described in subsection 4.1 using quadratic B-splines Frechet derivatives Once the final position, final slowness and travel time have been determined for a ray within a single cell in the grid, the Frechet derivatives relating the travel time to each of the model parameters defining the velocity within this cell must be obtained (see figure 6). Frechet derivatives for ray solutions to Figure 6: A represantation of cell ray-tracing the two-point problem describe a change in ray path as well as travel time in response to small changes in the velocity or slowness squared parameter (see figure 7). Frechet derivative of the travel time with respect to the j-th model is given by T = s2 u(x) u(x)ds = ds (20) m j m j x 0 (s) s 1 m j where (s 1, s 2 ) is the range along the ray over which the parameter m j influences the travel time. Making use of the identities udτ = ds and d(u 2 ) = 2udu, we obtain T = 1 τ2 u 2 (x) dτ (21) m j 2 τ 1 m j

14 4.2 The forward problem 12 Figure 7: Ray geometry for Frechet derivatives calculated under (a) initial value boundary condition and (b) two-point boundary conditions. Shaded area is part of velocity field that is perturbed. The new ray path is shown as a dotted line. which can be rewritten, by substituting x(τ) = x ref (τ)+ x(τ), as a reference and a perturbation term, where and ( ) Tref m j ( T m j = 1 2 ) ( ) ( ) T Tref T = +, (22) m j m j m j τref τ 0 = 1 2 ( u 2 (x 0 ) + Γ ) 0 (x ref x 0 ) dτ (23) m j m j τfinal τ 0 ( ) Γ0 x + u2 (x 0 ) dτ. (24) m j m j The 3D quadratic B-spline used to produce the slowness squared field in a cell is u 2 (x) = Since Γ 0 = u 2 (x 0 ), we have 27 j=1 B 3D 2 (x, x j )m j. (25) u 2 (x 0 ) m j = B 3D 2 (x, x j ), Γ 0 m j = B 3D 2 (x, x j ). (26)

15 4.3 The inverse problem 13 So far, these expressions have been derived for the case, in which the model parameters describing the seismic velocity field contain B-spline coefficients in units of u 2 (represented as m u 2). When solving the inverse problem using regularized inversion (subsection 4.3), the model parameters must be converted to units of velocity using the relation m v = 1/ m u 2 and the Frechet derivatives T = 2 T m v (m v ) 2 m u 2 (27) Beginning with the differential expression for the change in travel time δt = pδx along a short segment of a ray δx, for which the slowness vector is p, a simple expression for the change in travel time for a ray crossing an interface due to perturbation in that interface may be obtained from Snell law; δt = δp 3 δz, where δp 3 is the change in the vertical component of the slowness vector at the reflection/refraction and δz is the change in the interface depth. The interface is defined by equation (13), so an expression for Frechet derivative may be obtained as T z = δp 3 = δp 3 B3 2D (x, x j ) (28) m j m j The results derived above allow a ray to be traced through a single cell in a model parametrized as described in subsection 4.1, producing final position and slowness vectors, travel times, and Frechet derivatives. 4.3 The inverse problem In the linearized inversion approach, the inverse problem may be formulated as: we have a model, described by the vector m, the components of which are the interface depths and velocity parameters included in the inversion. From this model, the forward modelling code provides a set of synthetic travel time data, contained in the vector t and a matrix A of Frechet derivatives, which measure the sensitivity of the model and synthetic travel times. We also have the set of real travel-time data, t real, the uncertainties in those data, σ, and information on the geometric arrangement of model parameters. In order to measure the fit of the current model to the data, we define the traveltime residual vector r = t real t, which is a subject of the optimization by least-squares formulation. For example I would like to present the evolution of salt dome inversion from the starting to the final model. We can notice from pictures bellow that χ 2 is decreasing step by step, so we are getting more and more realistic two dimensional model of investigating area.

16 5 Conclusion 14 5 Conclusion Figure 8: From starting to the final model We conclude by comparing both modelling techiques. Rayamp is only used for 2D forward modelling, which means, that it can not produce inversions to the starting model. If more realistic models are to be obtained with Rayamp, one would have to include some sort of alghoritm for automatic search of the best set of parameters, which are defining the model, such as; different model boundaries, velocity field and smoothing criteria. The procedure of Jive3D is more practical, it also includes inversion and is therefore able to constuct the final model as an evolution of the starting model.

17 REFERENCES 15 References [1] Agustin Udias: Principles of Seismology (Cambridge University Press 1999); [2] James William Douglas Hobro: Three-dimensional tomographic inversion of combined reflection and refraction seismic travel-time data (Chuchill College Cambridge 1999); [3] Andrej Gosar: Exploration of the South-Eastern Alps lithosphere with 3D refraction seismic (Project Alp2002) data acqusition in Slovenia (Geologija); [4] G. D. Spence, K. P. Whittall, R. M. Clowes: Practical Synthetic Seismograms for latterally varying media calculated by asymptotic ray theory (BSSA 1984; );