Singularity Points and Symmetry Planes in the ORT Model
Explore the concept of singularity points and essential symmetry planes in the ORT model, along with the conditions for their existence and exclusion criteria. Learn about the significance of these elements in mathematical analysis and modeling.
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Presentation Transcript
Degenerate Degenerate ORT medium ORT medium Alexey Stovas (NTNU), Yuriy Roganov (Tesseral) & Vyacheslav Roganov (Glushkov Institute)
Outline Outline Singularity points in ORT Singularity point in-between the ORT symmetry planes Degeneracy of solution Conditions Line singularity Group velocity image Conclusions
ORT ORT model model (1 (1- -6 6 singularity singularity points points) ) Conditions (Sylvester s criterion for positive definite matrix) and exclude: 1) abnormal S wave polarization; 2) the singularity points with P wave: ( c ) ( 0, ) c , + , , c , 0, + c c c c c c 11 22 c 33 44 + 55 66 0, 0, c c 23 44 13 55 12 66 2 23 2 13 + 2 12 , , , c 22 33 c c c c + c 11 33 c c c 11 22 c c 2 2 2 2 . 11 23 c c 33 12 c c 23 13 12 c c c 11 22 33 c c c 22 13 Selection of essential symmetry plane (2-3): ( ) ( ) min , max , c c c c c 55 66 44 55 66 Minimum number is 1x4 = 4; Maximum number is 5x4 + 1x8 = 28 (For general ORT, 6x4 + 1x8 = 32). We exclude the singularity points from P&S1 waves
Method Method c c c c c 11 1 2 13 22 23 = = = + + + 0 0 0 d d d c c c c c c c 13 13 55 33 = C 23 23 44 c 44 12 12 66 c 5 5 c 66 Diagonal coefficients are not of importance (fixed) Non-diagonal coefficients are controlling the anellipticity (matter of change)
To To find find the symmetry symmetry planes ( the singularity singularity point planes (Schoenberg Schoenberg and point in in- -between between the and Helbig the , 1997) Helbig, 1997) Second-order minors of the Christoffel matrix are zero U D U D U D = = = 2 1 2 2 2 3 , , , 3 1 2 p p p ( c p ) c + + = 2 1 2 2 2 3 1 0, c f p c p c p ( ( ( )( )( )( ) ) ) c U 11 23 66 55 = = = = + + + , U 12 f f 13 f f ( c p ) 1 35 26 45 64 + + = 2 1 2 2 2 3 1 0, 13 f + p c p 66 22 44 , U ( ) 2 12 34 23 16 45 65 + = 2 1 2 2 2 3 1 0. c p c 12 f p 55 44 33 , U 13 f f 3 24 23 15 + 64 . 65 ( ) D c U c 13 f U 66 1 22 2 44 3 d d d d d d ( ) = = = = , , , 0, 0 12 d 23 13 d 23 12 d 13 13 f 12 f f c c f 23 jk jj kk jk jk 13 12 23
Conditions Conditions for between between the for existance existance of the symmetry symmetry planes of singularity singularity point planes point in in- - 0, 0, 0, U U U 1 2 3 or 0, 0, 0 U U U 1 2 3 = 0, 0, 0: U U U j k l = = = 0, 0 U U U ( ) j k l we arrive to the symmetry plane k l
Degenerate Degenerate ORT ORT = = = = 0, 0, 0 U U U D j k l ( c p ) c + + = 2 1 2 2 2 3 1 0, c f p c p c p 11 23 66 55 Degenerate system of linear equations ( c p ) + + = 2 1 2 2 2 3 1 0, 13 f + p c p 66 22 44 ( ) + = 2 1 2 2 2 3 1 0. c p c 12 f p 55 44 33 This case results not in a singularity point but a singularity line
Degenerate Degenerate ORT ORT = , 45 64 12 f 35 = = 0, 0. 13 f U U 26 1 = + . 2 64 65 f 23 15 13 f 24 One parameter family (f13) of degenerate ORT models
Degenerate Degenerate ORT ORT Eq Eq 1 v ( ) + + = 2 1 p 2 2 p 2 3 = 1, + Eq c p = c 13 f + p c p + + + = 1 2 66 22 44 1 2 f 2 f v ( ) ( f ) + = = 2 1 2 2 2 3 2 3 0, Eq 13 f 12 f p 2 65 24 p 34 p 2 f v v ( ) ( ) 65 = 2 1 2 2 = 0. Eq f p 2 f c 16 23 26 13 45 3 66 1 fv + + = 2 1 2 2 2 3 p p p 2 + c c 55 13 c f = 2 f 2 2 65 66 24 v Phase velocity squared along the singularity line: + f 65 2 4 1 3
Degenerate Degenerate ORT ORT + = + + = + 2 f , 64 65 c 12 f , 64 65 v c 35 64 66 2 f v 12 f 66 35 64 = + 2 f = + , 45 65 v c , 45 65 c 13 f 55 24 65 13 f 2 f v 24 65 55 = 2 f . 45 64 v c = + . 45 64 c f 44 f 23 16 45 2 f v 23 16 45 44 Parameter of the degenerate ORT model is the phase velocity squared along the singularity line
One One- -parameter parameter family of of degenerate degenerate ORT family ORT models models ( ) v ( ) ( ) v c 2 f 2 f c c c v 11 12 13 2 f c c 22 23 = C 33 c 44 c 55 c 66 Number of parameters: 9 3 + 1 =7
Singularity Singularity line line HTI-type 1 v ( ) + + = 2 1 2 2 2 3 = 2 2 2 p p p sphere 1.8 fv km s 2 f 2 3 2 1 2 2 p p p ( ) + + = 0 cone 2 f 2 f 2 f v c v c v c 44 55 66 2 f , c v c 55 44 2 f . c v c ( ) 44 66 + + = 2 1 2 2 2 3 1 n n n sphere 2 3 2 1 2 2 n n n ( ) VTI-type + + = 0 cone 2 f 2 f 2 f v c v c v c 44 55 66 = 2 2 2 2.05 fv km s 2 f ( ) c v c hyperbola 55 44 2 2 2 64 1 n n c ( ) + = 2 f ( ) c v c ellipse 1 65 ellipse or hyperbola 44 66 2 f 2 f v c v 44 55
HTI-type Physically Physically realizable realizable medium medium VTI-type 2 fcr v No singularity points VTI type v One singularity point HTI type No singularity points 2 fcr c c c c c c c c c 66 55 66 66 44 55 44 55 44 + + + 2 f 45 64 64 65 c v c VTI type range 44 66 16 45 35 64 The range for vf^2 is more narrow comparing with the standard one.
Singularity Singularity lines ( lines (horizontal horizontal symmetry symmetry plane) plane) = 2 f 2 2 Points A, B, C are singularity points in symmetry planes 1.8 c v km s c = 2 f 2 2 2.05 c v km s c 55 44 44 66 2 f 2 v c c c v ( ) f cr 55 0, A HTI-type 66 55 2 f 2 f c c v c v c c c 44 55 , B 44 55 44 55 The phase velocity for both S1 and S2 waves is vf=const along this line VTI-type 2 f v c c c 44 ,0 C 2 v ( ) f cr 66 44 The phase velocity for both S1 and S2 waves is vf=const along this line Points A, B, C are points with the wedge-type singularity (the transversal semi-axis of the group velocity ellipse is zero)
Singularity Singularity line (HTI line (HTI- -type of S1 type of degeneracy degeneracy) ) S2 = 2 2 2 1.8 fv km s According to Crampin (1991), there is no singularity line in ORT. He was wrong.
Singularity Singularity line (VTI line (VTI- -type of S1 type of degeneracy degeneracy) ) S2 = 2 2 2 2.05 fv km s
From From phase phase to to group group domain domain 1 2 ( ) p ( ) ( ) ( ) ( ) ( )( ) T T = + + + 2 p p p p p p p p p ... F F F F v v ( ) 1 ( ) 2 = = S S S S S S V V , 1 2 T T v p v p 1 2 S S 1 2 ( ) ( )( ) ( ) ( ) = = = + 2 T T p 0 A p v v v v F F = T T T p A p v p v p 1 2 2 1 S S 1 2 Degenerate matrix
Group Group velocity velocity surfaces surfaces trough lacuna S2 S1 cone lacuna
( ) v 2 f c c c 11 12 13 Alternatives Alternatives ( ) v 2 f c c 22 23 ( ) v 2 f c = C 33 c 44 c 55 c 66 ( )( ) : Forbidden combinations c c c c c 2 f 2 f v c c v 13 12 d d d 55 66 = + + 2 f , c v , , , , , , ; ( ) 11 11 12 c c 13 c c 2 f v c 23 44 ; . 22 12 23 ( )( ) 2 f 2 f v c c v d d 44 66 33 23 13 = + + 2 f , 12 d 23 c v ( ) 22 2 f v c 13 Total number = 6 3 55 : 3 17 ( )( ) 2 f 2 f v c v c d d 55 44 = + + 2 f . 13 d 23 c v ( ) 33 2 f c v 5 2 1 2 12 66 = = 5 Number of different c M jk
Two Two alternative alternative degenerate degenerate models models = 2 2 2 2.05 fv km s 9 4.511 3.028 3.376 9.84 5.94 = C 2 1.6 2.18 8.297 3.6 8.673 2.25 2.4 5.15 = C 2 1.6 2.18
Conclusions Conclusions We define the one-parameter (vf^2) family of degenerate ORT models with line singularity for S1&S2 waves. There are two types of singularity lines (VTI-type and HTI-type) depending on the value for vf^2. The phase velocity of both S waves computed along these lines equal to vf. In group domain, the singularity line results in two lines. For vf^2= vf^2cr, the additional singularity point in symmetry plane moves to the singularity line, two lines in group domain converge, and ellipse collapses.
Acknowledgements Acknowledgements GAMES project at NTNU
