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Probing into the Upper Mantle Using Surface Waves:

Beyond the Geometrical Ray Theory

Kazunori YDH=>O6L6

Division of Earth and Planetary Sciences, Graduate School of Science, Hokkaido University,North 10 West 8, Kita-ku, Sapporo 060�0810, Japan

(Received January 6, 2004; Accepted July 21, 2004)

A variety of methods of surface wave inversion, which enables us to investigate detailed imagesof the upper mantle on a regional scale, are reviewed. The study of surface wave tomographybeginning in the 1980’s has brought us with a significant jump in our understanding of the Earth’sinterior, particularly the upper mantle. Most of the studies of surface wave tomography have beenbased upon a geometrical ray theory, working with either dispersion curves or waveforms of surfacewaves. Such a simple representation of surface wave propagation has allowed us to treat a greaternumber of data sets, which are indispensable for obtaining high resolution tomography models.However, the ray theory, which is relying upon the high-frequency approximation, is no longer validwhen the scale-length of heterogeneity is comparable to the wavelength of waves to be considered.The e#ects of finite frequency are particularly important for the higher-frequency surface waves,which mainly sample the crust and uppermost mantle where very strong lateral heterogeneity islikely to exist. Recent development of the 3-D sensitivity kernels allows us to treat the e#ects of finitefrequency in the tomographic inversions. The use of such finite frequency theory will furtheradvance the methods of surface wave tomography.

Key words : Surface waves, Upper mantle, Ray theory, Finite frequency, Tomography

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Fig. 3. An example of a synthetic test ofwaveform inversion for retrieving a 1-Dshear wave speed profile based on a lin-earized inversion scheme of Nolet (1986).(a) 1-D shear wave speed models. A start-ing model (black solid line) has 3� fastershear wave speed in the mantle than thereference model (dashed line). (b) Initialwaveform fit and (c) final fit for a retrievedmodel [modified from Hiyoshi (2001)].

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Fig. 4. An example of fully non-linear waveform inversion using the Neighbourhood Algorithm ofSambridge (1999) for an event in Sumba, Indonesia, recorded at CAN station. (a) Density plots ofall the models created by the NA, ranked in order of increasing misfit. The initial model is shownin a dashed line and the best-fit model is in solid black line. (b) Initial waveform fit and (c) final fit.(d) Phase speed perturbations for the first three modes, estimated from the best-fit 1-D shearvelocity profile. A gray solid line (top panel) is a dispersion curve for the fundamental modeRayleigh wave which is measured directly from the observed waveform [after Yoshizawa andKennett (2002a)].

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and Dahlen (2000)]� P! ¦)"7�@#��I��U-4 ��������§�N�+-L� [Laske

and Masters (1996), Yoshizawa et al. (1999), Spetzler

et al. (2001), Yoshizawa and Kennett (2002b)]����4 "�����-MI� 2���/01�230��U�/-�� 01���V��tu-O)�vY�¨�� ©��ª«��=-ZI��V��tu-O)�4 Marquering et al. (1998, 1999), Dahlen

et al. (2000), Hung et al. (2000), Zhao et al. (2000)LK�¬­R�!)�

4.1 ����������P§ ����-A�)!"���=�/01�230-O)�vY�¨&,� STR��01�"����dy�"������ dc��Z®4 �¯�&,L��-56!�°��0R�� [Woodhouse and Wong

(1986)]�

dy(w)±²k(w)� dc(s�w)c

ds (4)

��� s4��-56!³2��I� � �����-&�"�´�;µ¶�4 ��&,L�.Z®-A�)�)�������4 Fig. 5(a) �&,- !�Y���$�3-¦)"7�)�#���!���N W�45���#6·�L)VQ ���.-4��"7����47¸R�L)� (� �VL���NO'��4 ��p(���)� l(��-4 µ�¹�º�0�0£R��m -��I�"7�89�����N���)�(Fig. 5(b))� �!#6� q-STR����=4 1

��-56!89�8L��7�4L' W�p()

Fig. 5. A schematic view of (a) a geometrical ray and (b) a finite-width ray. Actual surface waves willhave some sensitivity to a region around the path.

01�/01�230-&��5h0´�45�¼& 399

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���� K2Dc (q, f�w)$� ��������)���

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4.2 ����������� ��

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(2002b)]�k[�� 2������������)��������:�l� ���������b���m;� �n� �����5�� ��)!o��"!F� #p��:$q �Ars%t��AQ�:�bAu�;� ��vw:&C ��: x<,y4z,{|2,} A;��~ �h�$� ��Z������������)����!��'��m��� ����:(�y�34|�,�)�������5�A:�;C [Yoshizawa & Kennett (2002b)]��<,y4z,{|2,:��;C 2��+,-./0./1234 P�� 80*� Rayleigh�S : Fig.

6�� ������Y$������'Y���;� �+�,��rR��� �C� 1-������o.�+,-./0./:�b�������*;� 7�1234�$� 6����KL/0G����+,-./0./:�b<,y4z,{|2,��1$� \��� ��!o��"!F� #p����[C�"!Ars%���5� A�u���2�J��d� ��3������4��[�� ;CF[�� k5�����y/2���;�'C�����:���<,=2>?,:@ �AF�'����y/2����A6 ;�¡A� <,y4z,{|2,��~F¢7����5�AF��Yoshizawa and Kennett (2002b)�k[�� £������y/2�M4����A�`¤:�l�¡� �+���������y/2���YJ!o���$) 3,000 km� �,��$) 40*�5 ��A'��<¥2����#p0G��<,y4z,{|2,��$) 200 km�� %8� 2-���F9¦ !"� \�:b�<,y4z,{|2,F¦§;C���*¨��$) 300 kmA�d� �J$�� 40*����;C���������y/2�M4�©!����7ªA���;

Fig. 6�� �<,y4z,{|2,1234$� ��«2{��¥¬�<� 3�����=��­�YJ� \� �®F�YJ�� [Yoshizawa (2002),

Yoshizawa and Kennett (2004a)]<,y4z,{|2,:�� ����/$� \�U¯8>�5 e��°?3��VW�C+,-./0./1234 P� 5�±@ S �U¯�$�A��U¯':B CD� �²C� PC³�{N�� �´�DEU¯µ���F¶FA� ;�;� <,y4z,{|2,$� Z·���G¸�¹¡"Iº�¡�U¯�

Fig. 6. The influence zone kernel for a Ray-leigh wave at 80 s for a path between anevent near Irian Jaya and NWAO stationin south-eastern Australia [after Yoshizawaand Kennett (2004a)].

¼ H I J400

����� ������ ������� 10��������������� ��� !"#$%�&'()*'+,-'�./01�23456�#789:�;<4 ��=� >?#@AB4�� CDE'F)GHIJD���� !�KLMNGO� 30PQR�STUVW�X<Y�=� � 5�Z[O�;<4 3\]^_�`O�a'bcdecdf-g)�h#i�����

4.3 ���������� ����&'()*'+,-'j���&'k-lm'#�Fnop(d-nq)�rst�u�;<4KLvw��0� xY�Fnop(d-�y9:��z�{;<� Yoshizawa and Kennett (2004a)#|-+Fp}~H��VW�Fnop(d-nq)�`���������f-�-�c+F��� Fig. 7 ��O�

���f-�-�c+F=� Fnop(d-nq)�rst������h9:�#� ������=� ��q-�����&'k-lm'�B�� (4)��;<���������������#h9:�� �0�� ��;<4��q-��=� ���4������� �j��� ¡¢x£¤4rst¥��B=��¦�§����r¨B©'ª'o�KLz����B�4��«¬1=� ;­¢x£rstV¢O��®� ¯�nq) (Fig. 7(b))�`O���q-�� ��°±B&'()*'+,-'�� (5)��;<4²³´r�;��µ®�� �������KL¶·���q-�h��rstc+F���� ��Y� h����r¨= Fig. 7(a)����¸����������� (4)��;��¹]9:�nq)

Fig. 7. Realistic resolution tests using the ray theory and influence-zone theory. (a) Ray paths, (b)input checker board model with 5-degree cells, (c) retrieved model based on ray theory and (d)retrieved model which is updated from (c) working with both ray tracing and the influence zone[after Yoshizawa and Kennett (2004a)].

VW�&'k-lm'�;�CDE'F)JD�»¼ 401

� Fig. 7(c)���� ��� ������ (Fig. 7

(c))����� �������������������� ����������� (5) �!"��#���� � Fig. 7(d)����

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and Kennett (2004a)]� !^�� ��*�Z[���^a�� �@��ST�Y �+�-� '()e�]1f�56894g� !Rhi4D�jE�k��� �#'l*�4+�

� 5. ����������� �������� 4-.� ��� ��������^m��no+�,����� ,-��@��ST����p���� �3�� q��567'.r���� ������s/3bq�t0�'u1�4+vw�.� no.x^04+� ��� _�y2z{-.40� �834������ �@��ST&|R�2���*�Z[���^�'-]2}� !R5~4,-�67ST�Y ��D�jE�k��� �8+�'-]+� ��>�.� !R�49 ����� 3�#)�� ':��4+� -.��� Yoshizawa

(2002)f Yoshizawa and Kennett (2004b)���]�1�;�t0�������� _�'(<�"�+ 2

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(1990), Tromp and Dahlen (1992)]�uR�rU(r)cR�k

�1R V(r)�1cR: Rayleigh wave (6)

uL�k�1L W(r)(�r��1)cL: Love wave (7)

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(1992b, 1993)]�Bornno-.� ,-������ ¬=�¶·­���H¥+'� �`�._�f<�y2'��12+�'��� vw� Rytovno���+�N�-L+ [Yomogida and Aki (1987)]. Rytovno-.,-������ "Q�|R� F�ln c�ln A�iy��+�-� <��_���2�2`��¸��O-]+� �]� Bornnovw�P�� dF.��!^4­�C 8-H¥b2+�

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¼ Q R S402

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x�, xs, w)�KBornc (x, x�, xs, w)/c0(x, w�xs)� ������

�������� � 30!��" Rayleigh# $%& 100'�� 50'( )*+,-./01/02345Im�K Rytov

F �� Fig. 867�89 (11)�) Im�K RytovF �

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����@� [Marquering et al. (1999), Hung et al.

(2000)]� �#)��: 2������9SM� (11)

�)2345�: p/4)*+)����R�� ~�#}�),-./01/0����:@N@� [Spetzler et

al. (2002)]5.2 3��������uv),-./01/02345:� 2���@*+�!���H9K)���� ��� ¡IJ 3��S#�!���H9�#,-./01/02345�LM����� [Yoshizawa and Kennett (2004

b)] ¢£�X)¤¥�¦ 9�� R� Bornq§�¨N�S�#©/-.ª5)«¬ (9)���I� (7)��Q�##­)«¬ du�LM ���: Rayleigh

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Fig. 8. Two-dimensional sensitivity kernels for Rayleigh waves at (a) 100 s and (b) 50 s, derived fromthe Rytov approximation. Representation of imaginary parts of the sensitivity kernels (top), andthe cross-path profile of both real and imaginary parts of the kernels at the center of the path(bottom).

�#w-�3·¸-��6�¹-º5»�)¼P 403

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�� QR� 30S100��O�� QRT0�U�O�CB� Fig. 8�V��9:;WHFX�Y�Z[('/\]^?�@_�� `C������ Pa���b�c�d��@!e� �<�$�fg�$h;#i�;j@_�� `C� Pa!9klm�=��n��'()*+,*+�����

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Fig. 9. Representation of time-dependent 3-D sensitivity kernels for a vertical component of Rayleighwaveform in a frequency range between 0.01 and 0.03 Hz. Instantaneous sensitivity kernels at (a)850 s, (b) 870 s, (c) 890 s and (d) 950 s. Corresponding waveforms for a fundamental-mode Rayleighwave (top), horizontal slices of the kernel at 90 km depth (middle panels), and cross-path slices atthe center of the path (bottom panels).

¼ B C =404

���� �������������� ���������������� ��� ��� ���������� ����� ������� �!��"�#$%� �#��&�� �����' ( ���)�*+���� �%�$!� �,�-$!�.,���������/0�� /�1�2��34�� ����5��*+����6�,!��.�7� �*���7%������� �89�� ��-:������( �*+;�<"� �����������=�7���> �� ��6?"�����7%������� � ����6����� �@ABC�DE�0��F7 G)!�HIJKA�7"LM��#��NO ��� �����P LM$ 40%QR� ST��P �LM$ 20

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(1990)]� �%�$!� -�.g��h/��i01j��%���� �,.�HIk l2�%���346���Bm� �*�&nol2�%�����������/0�pZ����,�-:��$!X% [Yoshi-

zawa (2002), Ritzwoller et al. (2002), Friederich

(2003), Yoshizawa and Kennett (2004a)]� �#�� �,.�89��7!#�,!7��*����� YZ�mq�-$!5rs�,!��� �t��.�����M6�,��� 5 uv�% 3TjwExy�zy�a�{0 �� 1T^+�HIl2�!��� ����WK �|�}7�~7�"LM�����#���6��� ��� �#������� � k Bornmq���8�9�� �#.� �7� ST��P�#��NO��� F7 G)�-$!:����P`abcdE��HI�l2��34��� [Kennett and Nolet

(1990), Marquering and Snieder (1995)]� �%�$!���HIk l2�%a�{0�#��;� ���������<��� -�[����=N�����k >��%�*�a�{07!���E����E�:���� �?������' @ �A�!� BC���D346��� �#�� mE���F(����SFs6��( ��G�-�� 3Tja�{0#�%������� ��H6��I�� �:� �7-:��4,��

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������� �k m'K.,�#� 67:���� �I ���'(k J��,%6?$!-7W¡:� �,�� �����' ( ���6 80EKQ¢������������-�CX��o ��� �#�� LM 30%QR�"LM�������� �����-�O���WK �£�)* X�7� k%"LM����O���� �@���¤�� 3Tji0�3¥6��� �� ¦�%�*�&a�{07%=�7������� �� ������-�§L� ��������m'K�%��� CX�§¨�����6M6 X��

� ���©M�ª«N"! W�$%O¬§P­�®¯�k�� �� uv�%����°Y�E����E �89±²��%�� Brian Kennett, Eric Debayle,

Malcolm Sambridge, Sergei Lebedev, Tony Dahlen,

Barbara Romanowicz, Jeannot Trampert� QR S�I³T´� °­#.�µ�¶¨?UXk�%� �89�§��� V�·�¸·�89WX¨¹ ]��Yº15740266e �-�»¨JKk�%�

� �

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����E����E�-�C�DE�0���±< 405

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