ڮॢқѪں ۋڌॢ ֨Âًٖ ڙ Ձࣷ Ͽʝτ
лʴܳ
ťս
ڮ३ս
ťġৠ
Three-Dimensional Time-Domain Elastic Wave Modeling using Finite-Difference Method
Dong-Joo Min*, Hyoung-Soo Kim, Hai Soo Yoo and Kwang-Hee Kim
Abstract :We describe a 3-D time-domain, displacement-based, finite-difference elastic wave modeling algorithm that is constructed by defining the material properties within cubes. In the conventional displacement-based finite-difference method, both displacements and material properties are defined at the nodal points, whereas in our finite-difference algorithm, displacements are still assigned to the nodal points but material properties are defined within cubes. In this case, free-surface boundary conditions, which describe stress-free at the free surface, are naturally satisfied by the changes of material properties. Through numerical examples for infinite homogeneous and semi-infinite homogeneous models, we could examine the accuracy of the 3-D finite-difference elastic wave modeling algorithm.
Some numerical examples showed that the 3-D finite-differenc elastic wave modeling algorithm satisfies the reciprocity theorem and successfully generates synthetic seismograms and snapshots.
Key words :3-D, Time-domain, Finite-difference, Elastic wave modeling, Free-surface boundary condition څ أ:֨ÂًٖقԴ Ѻڦχں ۋڌॠي սॱʼə ڙ ѺڦŖԐ ڮॢқ Ձࣷ ϿʝτقԴ ۙڮϸ ąćܓ3 æں ܁ঝ০ ГԐॠş ڦॢ ѓѪڷͿ گϸߕ Ǵҙق ϔݗۆ НՁں ܁ۆॣ ìں ܃؋ॢɰ ϔݗۆ НՁę Ѻڦε. Ͽ˃ üۙ۾ق ܁ۆॠə şܕۆ ѓѪęə ɵν ϔݗۆ НՁں گϸߕ Ǵҙق ܁ۆॣ ąڍ ߸Àۺۍ ۙڮϸ ąć, ܓæں Ԑڌॠݓ ؍Čʪ ϔݗۆ НՁۆ ѺজχڷͿ ۙڮϸقԴ ڿͳۋ Ԑ͆ݕɰə ۙڮϸ ąćܓæں χܔ֨࢈
ս ەɰ Иॢ Œݗ ϔݗę ъИॢ Œݗ ϔݗق ʂॠي ս࠘ۺۍ ३ٮ ३Եۺۍ ३ε Ҽİ॥ڷͿ׆ گϸߕ Ǵҙق.
ϔݗۆ НՁں ܁ۆॠə ڙ ڮॢқ Ձࣷ ϿʝτѪۆ ܁ঝՁں ঝۍॠٕɰ բ֪ڙę սݕşۆ ڦ࠘À3 .
ԴͿ цNJ ąڍق ʂॠي ս࠘ۺڷͿ ĵॢ ३˞ں Ҽİ॥ڷͿ׆ گϸߕ Ǵҙق ϔݗۆ НՁں ܁ۆॠə ڙ3
ڮॢқ Ձࣷ Ͽʝτ ؎Čνˠۋ Ԝъڙν(reciprocity theorem)ε χܔ֨ࢇɰə ìں ؎ ս ەؽɰ ̚ॢ. , սथࠗ ĵܓ ф ɳࠗĵܓق ʂॠي ०ՁՁࣷ ɳϸʪ ф ֟ǹɳϸʪε ՁėۺڷͿ ۚՁॣ ս ەؽɰ.
ܳڅر :3차원 시간영역 변위근사 유한차분 탄성파 모델링 자유면 경계조건, , , , , Vol. 43, No. 1 (2006) pp. 65-75
Դ
Ձࣷ ۙΒۆ ս࠘३Ե şѪۆ ॠǣۍ Ձࣷ Ͽʝ τڹ ɰδ ۙΒ३Ե şѪۍ Ձࣷ ĵܓ҃܁ۋǣ ًԓ قԴ ܛܛ ۋڌʽɰ Ձࣷ Ͽʝτۆ ܁ঝՁ ф মڱՁ. ۋ Ձࣷ ًԓۋǣ ĵܓ҃܁ şѪۆ ܁ঝՁę মڱՁق
ࡾó ٖॳں ܳдͿ ܁ঝॢ Ձࣷ Ͽʝτ ؎Čνˠں
Òьॠə ìۋ ϔڍ ܼڅॠɰ Ձࣷ Ͽʝτۆ ֨ߣə. ڮॢқѪں ۋڌॠي ֨Â ėÂًٖقԴ սॱॠə ì- ۋؽɰ(Kelly et al., 1976).Ŕ Ձࣷ Ͽʝτۆ ܁ঝ, Ձ ф মڱՁں ȭۋş ڦॢ ψڹ ٍĵÀ ەؽڷ϶ ֨, Â ėÂࣷս ًٖ- (pseudospectral method),ܳࣷս ėÂ-
ًٖقԴ ڮॢқѪۋǣ ڮॢڅՙѪں ۋڌॢ Ձࣷ
Ͽʝτ ؎Čνˠۋ Òьʼؽɰ(Kosloff and Baysal, 1982; Fornberg, 1987, 1989; Marfurt, 1984; Jo et al., 1996; Shin and Sohn, 1998; Stekl and Pratt, 1998; Min et al., 2000, 2003, 2004).
Ձࣷ Ͽʝτۆ ܁ঝՁں ࠑ܁ॠə ॢ ѓѪڷͿ Л܃À ەڷ϶ ۋə ݓशϸ Àūۋق բ֪ڙę
Lamb ,
սݕşÀ ܕۦॠə ąڍق ǣࢍǣə ࢀ ݕफں Íə ۹
ܳࣷۆ ͪێνࣷε ܁ঝ০ ГԐॠəÀق ěॢ Л܃ۋ ɰ. Lamb Л܃ə ۙڮϸقԴ ڿͳۋ Ԑ͆ݕɰə ۙڮ
ț ښ ێ ۿս ț ښ ێ ࢘
2006 1 4 , 2006 2 14
1 ॢĶ३تٍĵڙ ३۹ঞąٍĵԐغɳ
2 ॢĶսۙڙėԐ սۙڙٍĵڙ ݓॠսݓъ ٍĵՙ
*Corresponding Author лʴܳ) E-mail; djmin@kordi.re.kr
Address; Marine geoenvironment research division, Korea Ocean Research & Development Institute, 1270, Sa-2-Dong, Sangrok-Gu, Ansan, Kyunggi 425-170, Korea
ٍĵȦЛ
ϸ ąćܓæں ضυǣ ۞ ŖԐ֨ࢅɗǽق ۆ३ ३Āʽ ɰ Ѻڦχں ۋڌॠə ڮॢқѪں ۋڌॢ Ձࣷ Ͽ. ʝτڹ ۋ͠ॢ ۙڮϸقԴۆ ڿͳܓæں χܔ֨ࢅş ڦ ३ ێйқ Ϳ शইʼə ۙڮϸ ąćܓæں ۋڌॠ
ٕڷǣ ۋəLamb Л܃ق ʂ३ ܁ঝॢ ३ε ܃֨ॣ ս ػڷ϶ Ԝъڙν, (reciprocity theorem)ε χܔ֨ࢅݓ Ї
ॢɰ ۋق ؼüۙε ۋڌॠي ڿͳę ՚ʪε ॥ƍ ܁ۆ. , ॠə ؼüۙѪ(staggered-grid method; Bayliss et al., 1986; Levander, 1988; Virieux, 1986; Fornberg, 1989) ۋ ܃؋ʼؽڷ϶, 3ڙ Ͽʝτ ф ҝő࠙ॢ ݓۆ Ͽ ʝτق ۺڌʼə ˣ Àۤ টь০ ۋڌʼČ ەɰ Ŕ͠ǣ. ۋ͠ॢ ؼüۙѪڹ ڿͳę ՚ʪε ॥ƍ ۋڌॠдͿ Ѻڦ χں ۋڌॠə ѺڦŖԐѪق Ҽ३ মڱۺۋݓ Їॠɰ.
ˣ ę лʴܳٮ ڮ३ս ə ܳࣷս ًٖ
Min (2004) (2003)
ę ֨Â ًٖقԴ ϔݗۆ НՁχڷͿ ۙڮϸ ąćܓæں
܁ঝ০ ГԐॣ ս ەə ԞͿڏ ڙ ѺڦŖԐ ڮॢқ2 Ѫں ܃؋ॠٕɰ ۋ˞ڹ. Lamb Л܃ق ʂॠي ܁ঝॢ
३ε ܃֨ॠə ѺڦŖԐ ڮॢқѪں ڦॠي üۙ۾ق ϔݗۆ НՁں ܁ۆॠʏ şܕ ڮॢқѪęə ɵν Ն Ǵҙق ϔݗۆ НՁں ܁ۆॣ ìں ܃؋ॠٕɰ ۋ ąڍ. Нνۺۍ Ձࣷʴѓ܁֩ߌͤ Ԝъڙνʪ χܔ֨ࢇɰ.
֬܃ ݓॠϔݗقԴ ࣷʴڹ 3ڙۺڷͿ ۻࣷʼдͿ
֬܃ ࣷʴۆ ۻࣷε ܁ঝ০ ГԐॠş ڦ३Դə ڙ 3 Ձࣷ Ͽʝτۋ ज़څॠɰ. 3ڙ Ձࣷ Ͽʝτۆ ąڍ Âɳॢ ֨Â ėÂًٖ Ͽʝτۋ͆ ॣݓ͆ʪ ψڹ ࠻ौ- ࢢ ϭϿνٮ ćԓ֨Âں ज़څͿ ॠдͿ ڿͳę ՚ʪε
॥ƍ ۋڌॠə ؼüۙѪ҃ɰə Ѻڦχں ۋڌॠ϶ ܁ঝ
ॢ ३ε ܃֨ॠə ڮॢқѪۋ ज़څॠɰ. MinęKim ڹ ʆĵܓ ϿقԴۆ
(2005) MASW(Multichannel An- Ѫۆ ۺڌÀɠՁ ٍĵقԴ alysis of Surface Waves) Min ˣ(2004)ۋ ڙ Ձࣷ ϿʝτقԴ ܃؋॰ʏ ѓѪں2 3
ڙ Ձࣷ Ͽʝτق ۺڌॢ ц ەɰ ۋ ٍĵقԴə. ˣ ۋ ڙ Ձࣷ ϿʝτقԴ ܃؋॰ʏ Ն Min (2004) 2
şъۆ ѺڦŖԐ ڮॢқѪۋ ڙڷͿ ر̎ó ঝۤ3 ʼəݓ ۙՃ০ ԕट҃Č Ŕ ܁ঝՁں ۙՃ০ êࢹ३ ҃ Čۙ ॢɰ.
ڙ ڮॢқ Ձࣷ Ͽʝτ
ڙ ڮॢқٍԓۙ
3
ҼŒݗ ˣѓՁ ϔݗق ʂॢ Ձࣷʴѓ܁֩ڹ ֨Âٖ
ًقԴ ɰڼę Ïۋ शইʽɰ.
(1)
(2)
(3)
يşԴ,
ə нʪ,
ٮ
əLaméԜսۋČ, , ٮ ə ѺڦۋČ, , ٮ ə բ֪ڙۋɰ ֩. (1)قə
,
,
,
,
,
,
,
,
ۆ Òۆ йқ२ۋ ܕۦॢɰ9 . 3ڙ ՁࣷϿʝτں սॱॠş ڦॠي Ϥ۹ Á йқ२ق ʂॢ ڮॢ
қ֩ں ĵՁ३آ ॢɰ. 2ڙ Ձࣷ Ͽʝτق ʂॢ
ۆ ѓѪق şߣॠي ÁÁۆ йқ२ق Min et al. (2004)
ʂॢ ڮॢқ֩ں ĵՁॣ ս ەɰ ۋ ѓѪں ڙق. 3 ۺڌॠş ڦॠيFig. 1ę Ïڹ üۙķں À܁ॢɰ. Fig.
قԴ Ѻڦə üۙ۾ق ܁ۆʼ϶ ϔݗۆ НՁڹ گϸߕ
1 ,
Ǵҙق ܁ۆʽɰ.
ۆ üۙķقԴ
Fig. 1 xѓॳۆ Ѻڦق ʂॢ йқ२2
,
,
ں қজॠϸ ɰڼę Ïɰ.
≈ ∆
∆
(4)
≈ ∆
∆
(5)
≈ ∆
∆
(6)
ঔ०йқ२ۆ ڮॢқ֩ ̚ॢMin et al. (2004)À ܳ
ࣷս ًٖ ڙ Ձࣷ ϿʝτقԴ ۺڌ॰ʏ ѓѪں Ŕ2 ʂͿ ۋڌॠٕڷ϶, xٮ yق ʂॢ йқ२2
ۆ ڮॢқ֩ڹ ɰڼę Ïڹ ѓѪڷͿ ĵՁʽɰ.
∆
(7)يşԴ,
(8)
(9)֩(8)ę (9)ε ֩(7)ق ʂۓॠي ܁νॠϸ,
≈ ∆∆
(10)
ڮԐॢ ѓѪڷͿ
ق ʂॢ ڮॢқ֩ڹ ɰ ڼę Ïۋ शইʽɰ.
≈ ∆∆
(11)
Ïڹ ѓѪڷͿ ɰδ ঔ०йқ२˞ں शইॣ ս ەɰ.
֩(1), (2), (3)ۆ ܟѺق ەə нʪ२ق ʂ३Դə ܳѺ ۆ Òۆ گϸߕۆ нʪε थŒ॥ڷͿ׆ ŖԐ֨ࢇɰ ֩8 .
ۆ ܟѺڹ ɰڼę Ïۋ ŖԐʽɰ
(1) .
≈
∆
(12)
يşԴ ڹ ֨Âۺқɳćε ǣࢍǶɰn .
ۙڮϸ ąćܓæ
ۆ üۙķۋ ۙڮϸں χǣó ʼϸ ڬҙқۋ Fig. 1a
Ԑ͆ݓó ʽɰ(Fig. 1b ܓ ˰͆Դ ֩). , (4)-(12)قԴ1, ق ३ɾॠə НՁں ڷͿ ȮڼڷͿ׆ ۙڮϸ ą
2, 3, 4 0
ćܓæں ГԐॣ ս ەɰ ۋε ֩ڷͿ शইॠϸ ɰڼę. Ïɰ.
≈ ∆
∆
(13)
(a) (b)
Fig. 1. Finite-difference grid sets in a left-handed Cartesian coordinate (a) inside the main body and (b) at the free surface.
≈ ∆
∆
(14)
≈ ∆
∆
(15)
≈ ∆∆
(16)
≈ ∆∆
(17)
≈
∆
(18)
֩(13)-(18)قԴ ؎ ս ەˢۋ گϸߕ Ǵҙق ϔݗۆ Н Ձں ܁ۆॠə ڙ ڮॢқѪۆ ąڍ ۙڮϸقԴۆ3 ڮॢқ֩ڹ ڙ ٚ܃2 (Min et al., 2004)قԴߌͤ ٰ ۻॢ йқ ̚ə йқڷͿ शইʽɰ ˰͆Դ ر1 2 .
̅ ѻॢ ۙڮϸ ąćܓæۋ ػۋ ۙڮϸ ڬҙқقԴ ۆ ϔݗۆ НՁۋ ۋ͆ə ܓæق ۆॠ0 ي ۙڮϸقԴ ڿ ͳۋ Ԑ͆ݕɰə ۙڮϸ ąćܓæۋ ۹ۼͿ χܔʾ ս ەɰ.
܁ঝʪ қԵ
ڙ ڮॢқٍԓۙÀ ĵՁʼϸ ۋ˞ ڮॢқٍ
3 ,
ԓۙÀ ܁ঝ০ ĵՁʼؽəÀε ԕट҃ş ڦॠي ڙ3
Ձࣷ Ͽʝτۆ ܁ঝʪ қԵں սॱॠيآ ॢɰ ҃ࣀ. ڹ ܁ঝʪ қԵق ؘԴ ս࠘қԓ३Եں սॱॠي ܳرݕ ١ۆ ॢć ǴقԴۆ ࣷۤ ɾ üۙսε Ā܁ॠČ ؋܁
ܓæ қԵں ࣀॠي ֨ÂۺқÂüں Ā܁ॠó ʽɰ л( ʴܳٮ ڮ३ս, 2002).Ŕ͠ǣ ڙ ࣷʴѓ܁֩ۆ ąڍ3 ս֩ۋ ҄ۡॠдͿ қԓܓæ ф ؋܁ܓæں Ā܁ॠşÀ رͷɰ ˰͆Դ ࣷۤ ɾ üۙս ф ؋܁ܓæڹ ąॹۺۍ. ѓѪں ࣀॠي Ā܁ॠٕɰ ܁ঝʪ қԵڹ ИॢŒݗϔݗ.
ę ъИॢ Œݗϔݗق ʂॢ ս࠘ۺۍ ३ε ३Եۺۍ ३ ٮ ҼİॠČ Ԝъڙνε χܔ֨ࢅəÀε êݒ॥ڷͿ׆, սॱʽɰ.
ИॢŒݗϔݗ
Ձࣷʴѓ܁֩ق ʂॢ ڮॢқ֩ۋ ĵՁʼϸ Ϥ۹ Àۤ ş҆ۺۍ Иॢ Œݗ ϔݗق ʂॠي ս࠘ۺڷͿ ĵ
ॢ ३ε ३ԵۺڷͿ ĵॢ ३ٮ Ҽİ॥ڷͿ׆ ܁ঝʪε қԵॢɰ ИॢŒݗϔݗق ʂॢ ३Եۺۍ ३ə. Pilant À ܃֨ॢ ֩ں ս࠘ۺڷͿ ۺқ॥ڷͿ׆ ĵॠٕ
(1979)
ڷ϶ ɰڼę Ïۋ शইʽɰ, .
(19)
(20)
يşԴ ڹ R(
)ѓॳ ѺڦۋČ,ə ѓॳ Ѻڦۋ϶,
ۋČ,
ۋɰ.
ə բ֪ڙۆ ܳࣷսًٖڷͿۆ ऻνق Ѻঞʽ É ۋɰ.ɰڼۆ ěć֩ں ۋڌॠϸ, x ѓॳ(u), y ѓॳ(v), z ѓ ॳ(w) Ѻڦε ĵॣ ս ەɰ.
(21)
(22)
(23)
ИॢŒݗ ϔݗق ʂॢ ܁ঝʪ қԵں ڦॠي ࣷۆ ՚P ʪ, Sࣷۆ ՚ʪ нʪÀ ÁÁ, 2000 m/s, 1000 m/s, 2.0 g/cm3ۍ ϔݗں À܁ॠČ ս࠘ۺۍ ३ٮ ३Եۺۍ ३ε ĵॠٕɰ բ֪ڙڷͿə ߯ʂܳࣷսÀ. 20 Hz (ܳ ܳࣷ
սÀ5 Hz)ۍ йқۆ Àڍ֟ ॥սε ۋڌॠٕɰ1 ս࠘ۺۍ ३ۆ ćԓقԴ ࠻ौࢢۆ şزڌ͟ۆ (Fig. 2).
ॢć ф ćԓ֨Â ɳ߹ں ڦॠي ࣷۤ ɾ Òۆ üۙս5 ε ۋڌॠٕɰ ۋə शܵ ڙ Ձࣷ Ͽʝτ ؎Čν. 2 ˠۋ1%ۆ ١ε ڮݓॠş ڦॠي ࣷۤ ɾ 33.3Òۆ üۙսε ज़څͿ ॰ʏ ìں ČͲॠϸ ϔڍ ۺڹ üۙս,
ε ۋڌॢ ìڷͿ أÂۆ ս࠘қԓ ١ε ڮьॣ ìۋ, ɰ ս࠘қԓق ۆ३ ьԦॠə ١ə ࣷۤ ɾ üۙսε. ɚͲܹڷͿ׆ ३Āॣ ս ەڷǣ ࣷۤ ɾ üۙսε ݒÀ,
֨࢈ ąڍ ज़څॢ ߪ üۙսÀ ɚرǣдͿ ψڹ ࠻ौࢢ şزڌ͟ę ćԓ֨Âں ज़څͿ ॢɰ ˰͆Դ. PC-cluster ε ۋڌॠäǣ ֙ऎ࠻ौࢢε ۋڌ३آ ॢɰ ИॢŒݗϔ. ݗͿ À܁॰ݓχ ֬܃ ϿʝτقԴə1000 m × 1000 m
ࡾşۆ Ͽں À܁ॠٕČ ėÂüۙ Âü
× 1000 m (∆
, :ϔݗ ՚ʪۆ ߯ՙÉ, : ࣷۤɾ üۙս, : ߯ʂܳࣷս ڹ) 10 mε ۋڌॠٕɰ ˰͆. Դ ߪ ۋڌʽ üۙսə101 × 101 × 101ۋɰ.
ę ə բ֪ڙۋ Ͽۆ ܁ ܼؖق ڦ࠘ॣ ˺ Figs. 3 4
ڙ ڮॢқѪں ۋڌॠي ĵॢ ս࠘ۺۍ ३ٮ ३Ե 3
ۺۍ ३ε ҃يܳČ ەɰ. Fig. 3ڹ բ֪ڙں ܼ֮ڷͿ x߹ڷͿ 0 m, y߹ڷͿ200 m, z߹ڷͿ200 m ʼə Ė قԴ ćԓॢ ३ε ҃يܳČ ەڷ϶ ۋ ąڍ, x ѓॳ Ѻڦ ə ܕۦॠݓ ؍ڷдͿ ʪ֨ॠݓ ؍ؕɰ. Fig. 4ə բ֪ڙ ں ܼ֮ڷͿx߹, y߹ ߹ڷͿ ÁÁ, z 200 mʼə ĖقԴ ćԓॢ ३ε ҃يܳČ ەɰ. Fig. 4قԴ ҃ϸ, x߹ ѓॳ Ѻڦٮy߹ ѓॳ ѺڦÀ ԴͿ ێ࠘॥ں ؎ ս ەɰ. Figs.
ę قԴ ս࠘ۺۍ ३ٮ ३Եۺۍ ३ Ԑۋۆ ١ə ս 3 4
࠘ۺۍ ѓѪڷͿ ३ε ĵॣ ˺ ࣷۤ ɾ üۙսε ߿қ০
ܳݓ ؍؉Դ Ԧşə ս࠘қԓق ۆॢ ١ٮ ३Եۺۍ ३ε ĵॠə ę܁قԴ ऻνق Ѻঞ ę܁ ˣقԴ ьԦॢ
١ۋɰ ̚ॢ ܟڍۆ ąćϸقԴۆ ъԐࣷε ܃äॠݓ.
؍ؕڷдͿ ۻࣷ֨Âۋ ţرݗ ąڍ ܟڍ ąćϸقԴ ъ Ԑʽ ࣷ˞ۋ ٖॳں й࠘ó ʽɰ.
ъИॢ Œݗϔݗ
Ձࣷ Ͽʝτ ؎Čνˠۆ ܁ঝʪε êݒॠş ڦॠي Л܃ق ʂॠي ܁ঝॢ ३ε ܃֨ॠəÀε ԕट҇
Lamb
ज़څÀ ەɰ(e.g., Ewing et al., 1957; Levander, 1988;
Л܃ə ъИ Min et al., 2003; Min et al., 2004). Lamb
ॢ Œݗ ϔݗقԴ ݓशϸۋǣ ݓशϸ Ŗߌق բ֪ڙۋǣ սݕşÀ ܕۦॠə ąڍ, Pࣷ ф शϸࣷۍ ͪێνࣷق ʂॠي ܁ঝॢ ३ε ܃֨ॠəÀε êݒॠə ìۋɰ.
Л܃ق ěॢ ३Եۺۍ ३ə ۋ
Lamb Ewing et al.(1957)
܃֨ॢ ३ε ۋڌॠٕڷ϶ ɰڼę Ïۋ शইʽɰ, .
∞ ′ (24)Fig. 2. The first derivative of Gauss function whose maximum frequency is 20 Hz (major frequency is 5 Hz).
Fig. 3. Analytic solutions (solid line) and numerical solutions (plus symbol) obtained by the 3-D finite-difference method for (a) y- and (b) z-displacements at the distance of (0 m, 200 m, 200 m) from the source. Gs indicates the number of grid points per wavelength.
∞ (25) ′ (26)
يşԴ
(27)
′ (28)
ۋɰ ֩. (24)قԴ ə սथѓॳ Ѻڦѯࢢq (q = iu + jv) ۋČ ֩(25)قԴwəzѓॳ Ѻڦۋɰ ֩. (24)ٮ(25) قԴ Ѻڦε ćԓॠş ڦॠي ܳࣷս ًٖقԴ ق ʂॢk ۺқÉں ćԓॢ ऻνق ًѺঞ֨ࢅə ѓѪں ۋڌॠ
ٕɰ ėÂࣷս ق ʂॢ ۺқڹ ս࠘ۺۍ ćԓѓѪں. k ۋڌॠٕڷ϶ ۋ ˺, dkÉۋ ۚںս ܁ঝॢ ३ε ص əɰ ۋ ٍĵقԴə. LambЛ܃ۆ ३Եۺۍ ३ε ĵॠ
ş ڦॠي dk = 0.001ں ۋڌ॰ɰ.
ۋ ٍĵقԴ Č؋ʽ ڙ ڮॢқѪ3 ں ۋڌॢ Ͽʝ τ ؎ČνˠۋLambЛ܃ق ʂॠي ܁ঝॢ ३ε ܃֨ॠ əÀε ԕट҃ş ڦॠي ИॢŒݗϔݗقԴٮ Ïۋ ࣷP ۆ ՚ʪ, Sࣷۆ ՚ʪ нʪε ÁÁ, 2000 m/s, 1000 m/s, 2.0 g/cm3 ۍ ϔݗں À܁ॠٕČ բ֪ڙڹ ߯ʂܳࣷս, À20 Hz(ܳ ܳࣷսə5 Hz)ۍ йқۆ Àڍ֟1
॥սͿ ݓशϸۆ ܼؖق ڦ࠘ॠČ ەɰ ИॢŒݗϔݗق. Դٮə ɵν ս࠘ۺۍ ३ۆ ćԓں ڦॠي ࣷۤ ɾ üۙ
սε20Òε ۋڌॠٕڷдͿ1000 m × 1000 m × 1000
ࡾşۆ Ͽں Ͽʝτॠş ڦॠي
m 401 × 401 × 401
Òۆ üۙսε ۋڌॠٕɰ ܟڍ ąćϸڷͿҙࢢ ԦՁʼ. ə ъԐࣷۆ ܃äε ڦॠيShin(1995)ۋ ܃؋ॢ ֟ऐݓ ąćܓæں ۋڌॠٕɰ. Fig. 5ə բ֪ڙڷͿҙࢢ x߹
ѓॳڷͿ100 m, 200 m, 300 m̆رݕ ݓशϸقԴ ĵ
ॢ ߹ ѓॳ Ѻڦۆ ս࠘ۺۍ ३ٮ ३Եۺۍ ३ε ҃يz
ܳČ ەɰ. Fig. 5قԴ ҃ϸ ३Եۺۍ ३ٮ ս࠘ۺۍ ३, Fig. 4. Analytic solutions (solid line) and numerical solutions (plus symbols) obtained by the 3-D finite-difference method for (a) x-, (b) y- and (c) z-displacements at the distance of (200 m, 200 m, 200 m) from the source. Gs indicates the number of grid points per wavelength.
À ϔڍ ۞ ێ࠘ॠČ ەڼں ؎ ս ەɰ.
Ԝъڙν(Reciprocity)
گϸߕǴق ϔݗۆ НՁں ܁ۆॠə ڙ ڮॢқ3 Ѫۋ Ԝъڙνε χܔ֨ࢅəÀε ԕट҇ ज़څÀ ەɰ.
ۋε ڦॠيFig. 6ę Ïڹ Ͽں À܁ॠČ(1000 m,
ٮ ق ÁÁ բ֪ڙ
800 m, 0 m) (1000 m, 1200 m, 0 m)
ę սݕşε ڦ࠘֨ࢅČ ĵॢ ս࠘ۺۍ ३(forward)ٮ բ֪ڙę սݕşۆ ڦ࠘ε ԴͿ цƾԴ ĵॢ ս࠘ۺۍ ३(reverse)ε Ҽİ३ ҃ؕɰ. Fig. 7ڹzѓॳ Ѻڦق ʂ ॠي ս࠘ۺڷͿ ĵॢ ३˞ں ҃يܳČ ەɰ. Fig. 7قԴ
҃ϸ բ֪ڙę սݕşۆ ڦ࠘ε ԴͿ цƴرԴ ĵॢ ३ À ԴͿ ێ࠘॥ں ؎ ս ەڷ϶ ۋͿ׆ گϸߕ Ǵҙق, ϔݗۆ НՁں ܁ۆॠə ڙ ڮॢқѪڹ Нνۺۍ3
Ձࣷʴѓ܁֩ę υÀݓͿ Ԝъڙνε χܔ֨ࢇɰə
ìں ؎ ս ەɰ.
ս࠘Ͽʝτ Āę
Иॢ Œݗ ϔݗę ъИॢ Œݗ ϔݗق ʂॢ ܁ঝʪÀ êݒʼČ ǣϸ ڮॢқѪں ۋڌॢ ڙ ֨Âًٖ 3 Ձࣷ Ͽʝτ ؎Čνˠۋ ݓݗॡۺ Ͽق ʂॠي ܁ঝॢ
Ձࣷ ɳϸʪε ۚՁॣ ս ەəÀε ԕट҇ ज़څÀ ە ɰ ۋε ڦॠي սथࠗĵܓ Ͽق ʂॠي ०ՁՁࣷ. ɳϸʪε ۚՁॠČ ɳࠗĵܓ Ͽق ʂॠي ֟ǹ ɳϸ, ʪε ۚՁॠٕɰ.
ݔۿࣷ ъԐࣷ ےćĹۼࣷ Ͽ˚ۻঞࣷ ˣۋ ܃ʂͿ, , ,
ՁʼəÀε ԕट҃ş ڦॠي Âɳॢ սथ ࠗ ĵܓق2 ʂॠي ०ՁՁࣷ ɳϸʪε ۚՁॠٕɰ սथ ࠗ ĵܓ. 2 ۆ ࡾşə1000 m × 565 m × 200 mۋ϶ Ԝҙࠗۆ ˃ Fig. 5. Analytic solutions (solid line) and numerical solutions (plus symbols) obtained by the 3-D finite-difference method for vertical displacements of Lamb's problem at the distance of (a) 100 m, (b) 200 m, and (c) 300 m from source.
ƍə20 mۋɰ Ԝࠗę ॠࠗۆ НՁڹ. Table 1ق ǣࢍǣ ەɰ բ֪ڙڷͿə ߯ʂܳࣷսÀ. 100 Hz (ܼ֮ܳࣷս À25 Hz)ۍ Àڍ֟॥սۆ йқε ۋڌॠٕɰ1 . բ֪ڙڹ ݓशϸۆ ܼ֮ق ȮيەɰČ À܁ॠٕČ սݕ, şə शϸقԴx߹ں ˰͆ ۻÒʼر ەə ìڷͿ À܁ॠ
ٕɰ. Fig. 8ڹ սݔѓॳ Ѻڦۆ ०ՁՁࣷ ɳϸʪε ҃ يܵɰ. Fig. 8قԴ ҃ϸ ݔۿࣷ ъԐࣷ ےćĹۼࣷٮ, ,
ࢀ ݕफں Íə ۹ܳࣷսۆ ͪێνࣷÀ ǣࢍǣČ ەɰ.
ɳࠗĵܓͿəy-zɳϸʪÀFig. 9ۆ Ͽę Ïڷ϶x
߹ ѓॳڷͿ НՁۆ ѺজÀ ػə Ͽں À܁ॠٕɰ բ.
֪ڙڹ(750 m, 750 m, 0 m)Ϳ ݓशق ڦ࠘३ ەڷ϶, սݕşə(750 m, 300 m, 0 m)قԴ(750 m, 1200 m, 0 m)ūݓy߹ں ˰͆ қपॠČ ەɰ բ֪ڙڷͿə ߯. ʂܳࣷսÀ60 Hzۍ йқ Àڍ֟čԸں ۋڌॠٕ1 ɰ. Fig. 10ڹ ۻࣷ֨Âۋ ÁÁ0.1333 s, 0.2 s, 0.2666 ێ ˺ ɳࠗϿق ʂॠي صڹ ֟ǹɳϸʪε s, 0.3333 s
҃يܵɰ. Fig. 10قԴ ݔۿࣷ ъԐࣷ Ĺۼࣷ ͪێνࣷ, , , ˣۋ ۻࣷ३ Àə Ͽ֥ں ěॣ ս ەɰ.
Ā
ϔݗۆ НՁں üۙ۾ق ܁ۆॠə şܕۆ ڮॢқѪ ęə ɵν ϔݗۆ НՁں گϸߕ Ǵҙق ܁ۆॠə ڮॢ
қѪں ۋڌॠə ڙ Ձࣷ ϿʝτѪں ۙՃ০ ՙ3 ÒॠČ Ŕ ܁ঝʪε қԵॠي ҃ؕɰ ϔݗۆ НՁں گ. ϸߕ Ǵҙق ܁ۆॣ ąڍ üۙ۾ق ϔݗۆ НՁں ܁ۆ ॠʏ şܕۆ ڮॢқѪęə ɵν ۙڮϸقԴ ڮॢқ
֩ڹ ٰۻॢ йқ ̚ə йқۆ ε ̿ó ʼ϶1 2 , Нνۺۍ Ձࣷʴѓ܁֩ߌͤ Ԝъڙνε ŷ̲νݓ ؍ əɰ ˰͆Դ ڮॢڅՙѪں ۋڌॢ Ձࣷ ϿʝτقԴߌ. Fig. 6. A y-z cross section of the 3-D model for
reciprocity check.
Fig. 7. Numerical examples for reciprocity theorem:
vertical displacements for the forward and reverse problems.
Fig. 8. Synthetic seismograms for the horizontally homogeneous layered model where one layer is overlain on homogeneous media.
P-velocity S-velocity Density
1 892 m/s 477 m/s 2.0 g/cm3
2 2000 m/s 1000 m/s 2.2 g/cm3
Table 1. Material properties for the layered model where one layer is overlain on a homogeneous medium
ͤ ѻॢ ۙڮϸ ąćܓæ ػۋʪ ϔݗۆ НՁχڷͿ
ۙڮϸقԴ ڿͳۋ ܕۦॠݓ ؍əɰə ۙڮϸ ąćܓæ ں ܁ঝ০ ГԐॣ ս ەɰ.
ڙ Ձࣷ Ͽʝτۆ ąڍ қԓěć֩ۋ ϔڍ ҄ۡ
3
ॢ ε ̿дͿ қԓܓæ ф ؋܁ܓæں ݔۿ ĵॠş À ֖ݓ ؍ɰ ˰͆Դ қԓܓæ ф ؋܁ܓæں ֬ॹۺۍ. ѓѪڷͿ Ā܁ॠٕɰ ИॢŒݗϔݗ ф ъИॢ Œݗ ϔ. ݗق ʂॠي ս࠘ۺڷͿ ĵॢ ३ε ३ԵۺڷͿ ĵॢ ३ ٮ Ҽİ॥ڷͿ׆ ܁ঝʪ қԵں սॱॢ Āę گϸߕ Ǵ ق ϔݗۆ НՁں ܁ۆॠə ڙ 3 Ձࣷ Ͽʝτۋ ३Ե ۺۍ ३ق ϔڍ ŖԐॢ ս࠘ۺۍ ३ε ܃֨ॢɰə ìں
؎ ս ەؽɰ ̚ॢ բ֪ڙę սݕşۆ ڦ࠘ε ԴͿ цƾ. À϶ ս࠘ۺڷͿ ३ε ĵॠČ Ҽİ३ ҆ Āę Ԝъڙν ʪ χܔ֨ࢇɰə ìں ؎ ս ەؽɰ ̚ॢ սथࠗ ĵܓ. , ق ʂॠي ०ՁՁࣷ ɳϸں ۚՁ॥ڷͿ׆ ݔۿࣷ ъ, , Ԑࣷ ےćĹۼࣷ ͪێνࣷÀ ܁ঝ০ ГԐʽɰə ìں, ,
؎ ս ەؽڷ϶ ɳࠗĵܓ Ͽق ʂॠي ֟ǹɳϸʪε,
ۚՁ॥ڷͿ׆ ࣷۆ ۻࣷتԜۋ ܁ঝ০ ГԐʽɰə ìں Fig. 9. A y-z cross section of the 3-D vertical step
model.
Fig. 10. Snapshots obtained by the 3-D finite-difference method for the vertical step model: at (a) 0.1333 s, (b) 0.2 s, (c) 0.2666 s, and (d) 0.3333 s.
ঝۍॣ ս ەؽɰ ۋ͠ॢ Āę˞ں цڷͿ Ѻڦχں. ۋڌॠə ڙ Ձࣷ Ͽʝτڹ ݓݗॡۺ őϿۆ Ͽ3 ۆ ϿʝτقԴ মęۺڷͿ ۋڌʾ ս ەں ìڷͿ şʂ ʼ϶ ॳ ڙ ًԓۋǣ ĵܓ҃܁ ˣقԴ মڱۺڷͿ, 3 টڌʾ ìڷͿ şʂʽɰ.
Ԑ Ԑ
ۋ ٍĵə ॢĶ३تٍĵڙPE93300, PM31600ę ॢ Ķսۙڙ ėԐ KIWE-DRC-04-14ۆ ۦ܁ۺ ݓڙڷͿ սॱʽ ìڷͿ ěćşěق ÇԐ˚ςɦɰ.
ČЛॶ
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л ʴ ܳ ť ս
1993țԴڐʂॡİ ݓĵęॡİگę ۋ ॡԐ
ț Դڐʂॡİ ęॡİگę ݓĵН 1995
νۻė ԵԐ
ț Դڐʂॡİ ęॡİگę ݓĵН 1999
νۻė чԐ
ț ԴڐʂॡİݓݗęॡęۋॡԐ 1986
ț ԴڐʂॡİݓݗęॡęݓĵН 1988
νۻėԵԐ
ț ԴڐʂॡİݓݗęॡęݓĵН 1994
νۻėчԐ
ইۦ ॢĶ३تٍĵڙ Ըےٍĵڙ (E-mail; djmin@kordi.re.kr)
ইۦ ॢĶսۙڙėԐ սۙڙٍĵڙ սԵٍĵڙ (E-mail; hskim@kowaco.or.kr)
ڮ ३ ս ť ġ ৠ
ț ॢتʂॡİ ۙڙėॡę ėॡԐ 1982
ț ॢتʂॡİ ۙڙėॡę ݓĵН 1984
νۻė ԵԐ
ț ॢتʂॡİ ۙڙėॡজ ݓĵН 1996
νۻė чԐ
ț ॢتʂॡİ ݓĵ३تęॡę
1996 ,
ۋॡԐ
1999țUniversity of Memphis, Department of Geological Science, ݓĵНνۻė ԵԐ
2003țUniversity of Memphis, Department of Earth Sciences, ݓĵНνۻė чԐ
ইۦ ॢĶ३تٍĵڙ ےٍĵڙ (E-mail; hsyoo@kordi.re.kr)
ইۦ ॢĶ३تٍĵڙ Ըےٍĵڙ (E-mail; kwanghee@kordi.re.kr)
