Practical Migration, deMigration and Velocity Modeling The Partial Differential Wave Equations Bee Bednar Panorama Technologies, Inc. 14811 St Marys Lane, Suite 150 Houston TX 77079 July 11, 2013 Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 1 / 28
Outline Full Two-Way Wave Equations 1 Newton and Hooke The Coupled Elastic System The Stress Tensor and the C Matrix 2D Isotropic Elastic Wave Equation Example First Order Elastic Systems First Order Elastic System Solution Second Order Equations Summary Wavefield Characteristics 2 Frequencies and Wavenumbers One-Way Wave Equations 3 XT, FX, TK, and FK Various Domains Summary Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 2 / 28
Full Two-Way Wave Equations Outline Full Two-Way Wave Equations 1 Newton and Hooke The Coupled Elastic System The Stress Tensor and the C Matrix 2D Isotropic Elastic Wave Equation Example First Order Elastic Systems First Order Elastic System Solution Second Order Equations Summary Wavefield Characteristics 2 Frequencies and Wavenumbers One-Way Wave Equations 3 XT, FX, TK, and FK Various Domains Summary Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 3 / 28
Full Two-Way Wave Equations Newton and Hooke 3D Stress Equation Newton in 3D 3 ∂ 2 u i ∂ t 2 = 1 ∂σ ij � . ∂ x j ρ j = 1 In 3D,the forces that can affect a point are in-line compressional and orthogonal shear. Looking at a small cube each of the nine faces of the cube can move both inward and outward as compressional as well as shear along vertical and horizontal planes. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 4 / 28
Full Two-Way Wave Equations Newton and Hooke 3D Stress Equation Newton in 3D 3 ∂ 2 u i ∂ t 2 = 1 ∂σ ij � . ∂ x j ρ j = 1 The stresses σ ij can generated up to three wavefields, u i . The existence of a wavefield and its strength is completely determined by the properties of the rocks governing propagation. Newton’s law relates acceleration to the nine possible forces per unit area (stresses) through the equation above. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 5 / 28
Full Two-Way Wave Equations Newton and Hooke 3D Hooke For a linear 3D medium, Hooke’s law can be rephrased as A CHANGE in FORCE per unit volume is equal to the bulk modulus times the increase in volume divided by the original volume. The 3D stress equation has nine stress factors, σ ij , one for each of the three dimensions and three coupled wavefields, u i . Hooke’s law says that each component of stress σ ij is linearly proportional to every component of strain E mn so that 1 � ∂ u m + ∂ u n � � � σ ij = c ijmn E mn = c ijmn 2 ∂ x n ∂ x m m , n m , n In this case the c ijmn are elements of what is called the stress tensor. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 6 / 28
Full Two-Way Wave Equations The Coupled Elastic System Coupled Full Elastic Equations The two equations 3 ∂ 2 u i ∂ t 2 = 1 ∂σ ij � ρ ∂ x j j = 1 1 � ∂ u m + ∂ u n � � σ ij = c ijmn 2 ∂ x n ∂ x m m , n form a coupled system for full elastic wave propagation. Note that superficially there are 81 elements in the stress tensor defined by the c ijmn . Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 7 / 28
Full Two-Way Wave Equations The Stress Tensor and the C Matrix The C = [ c ij ] Matrix vs the c ijmn Tensor Notice that c ijmn = c mnij , c ijmn = c ijnm , c ijmn = c jimn and c ijmn = c mnij , so that after applying the indexing scheme (Voigt scheme) index ij = 11 22 33 23 13 12 map ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ index k , l = 1 2 3 4 5 6 one gets c 11 c 12 c 13 c 14 c 15 c 16 c 12 c 22 c 23 c 24 c 25 c 26 c 13 c 23 c 33 c 34 c 35 c 36 , c 14 c 24 c 34 c 44 c 45 c 46 c 15 c 25 c 34 c 45 c 55 c 56 c 16 c 26 c 36 c 46 c 56 c 66 which is the C = [ c ij ] matrix shown earlier. The symmetry reduces the number of c ij to 21 volumes. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 8 / 28
Full Two-Way Wave Equations 2D Isotropic Elastic Wave Equation Example 2D Isotropic Elastic Wave Equation As an example, the 2D Isotropic Elastic Wave Equation is � � ∂σ 1 , 1 λ + 2 µ ∂σ 1 , 1 ∂ x 1 + ∂σ 1 , 3 ∂ x 1 + λ ∂ v 1 ∂ v 3 ∂ v 1 1 = = ∂ t ρ ρ ∂ x 3 ∂ t ρ ∂ x 3 � � ∂σ 1 , 3 µ ∂ x 1 + ∂ v 1 ∂ v 3 � � ∂σ 1 , 3 ∂ x 1 + ∂σ 3 , 3 = ∂ v 3 1 = ∂ t ∂ x 3 ρ ∂ t ρ ∂ x 3 ∂σ 3 , 3 λ + 2 µ ∂ v 3 ∂ v 1 ∂ x 3 + λ = ∂ t ρ ρ ∂ x 1 where, in the usual geophysical notation, x 1 = x , and x 3 = z . Thus, v 1 represents particle velocity in the horizontal and v 3 is particle velocity in the vertical direction. In this case the C matrix is defined by λ + 2 µ and µ . Note that these are actually 2D numeric fields. That is, they are 2D functions of x and z . Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 9 / 28
Full Two-Way Wave Equations First Order Elastic Systems First Order System Although the algebra is quite tedious, for any given C matrix, the coupled system in the previous slide can be written as the first order vector system 2 3 v 1 v 2 6 7 6 v 3 7 6 7 6 7 ∂ v ∂ v ∂ v ∂ v σ 1 , 1 6 7 ∂ t = X 1 + X 2 + X 3 6 7 v = σ 1 , 2 ∂ x 1 ∂ x 2 ∂ x 3 6 7 6 7 σ 1 , 3 6 7 6 7 σ 2 , 2 6 7 6 7 σ 2 , 3 4 5 σ 3 , 3 where the elements of the X 1 , X 2 , and X 3 matrices are determined by the c ij volumes in the C matrix. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 10 / 28
Full Two-Way Wave Equations First Order Elastic System Solution First Order System Solution This latter equation is appealing because it’s a one-dimensional-time-domain differential system whose solution is easily expressed as t � v ( t ) = exp [ t H ] v ( 0 ) + exp [ ξ H ] S ( t − ξ ) d ξ 0 where v ( 0 ) represents the initial conditions, S ( t ) is the source term and H is the operator ∂ ∂ ∂ H = X 1 + X 2 + X 3 ∂ x 1 ∂ x 2 ∂ x 3 Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 11 / 28
Full Two-Way Wave Equations Second Order Equations Second Order Full Elastic Equation Substitution of � ∂ u m � 1 + ∂ u n � σ ij = c ijmn 2 ∂ x n ∂ x m m , n into 3 ∂ 2 u i ∂ t 2 = 1 ∂σ ij � ρ ∂ x j j = 1 yields the second order version of the full elastic system ∂ 2 u i ∂ 2 u m c ijmn � ∂ t 2 = ρ ∂ x n ∂ x j m , n , j Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 12 / 28
Full Two-Way Wave Equations Second Order Equations Second Order Isotropic Elastic Equation When the C matrix represents a isotropic elastic system, the two shear or transverse waves are identical, so, after considerable algebraic manipulation, one can write ∂ 2 u ∂ t 2 = ( λ + 2 µ ) ∇ ( ∇ · u ) − µ ρ ∇ × ∇ × u ρ where the first component of u = ( u 1 , u 3 ) is the compressional wave and the third component is the transverse or shear wave. From a physical viewpoint, the dot product annihilates the compressional component, while the cross product annihilates the shear component. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 13 / 28
Full Two-Way Wave Equations Second Order Equations Second Order Scalar Wave Equation In a purely acoustic media, the shear parameters are zero, so there is no propagation of shear waves. The 3D elastic equation reduces to the scalar form ∂ 2 u � ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u � ∂ t 2 = λ ∂ z 2 ρ � λ Setting v = ρ produces the traditional scalar wave equation. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 14 / 28
Full Two-Way Wave Equations Summary Two-Way Wave Equation Summary In the interest of clarity, the previous derivations were performed under some overly simplistic assumptions. Most notably was the assumption that the density, ρ , was constant as a function of position. Had this not been the case, the full scalar wave equation would have taken the form � ∂ ∂ 2 p � 1 ∂ p ∂ x + ∂ 1 ∂ y + ∂ ∂ p 1 ∂ p ∂ t 2 = ρ v 2 . ∂ x ∂ y ∂ z ∂ z ρ ρ ρ and the fully elastic wave equation would have been a bit more complex. Fortunately, this assumption will not significantly impair out ability to understand the computational aspects of digital wave propagation, so the discussion is continued with the equations as previously derived. The anisotropic models of interest are VTI , TTI , ORT , and TORT , all of which are incorporated within the fully elastic wave equation. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling July 11, 2013 15 / 28
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