He Kupu Whakataki ki te Pūmanawa Hurihuri Āhua Ngaru Katoa (FWI)
1. Pendahuluan
I roto i te ao matawhenua o ēnei rā, kei te piki haere te hiahia kia "kite" i te hanganga o raro i te whenua. Ko te umanga pūngao, te whakaiti i ngā aituā, te tūhuratanga o ngā waiariki, tae atu ki te rangahau tektonika e hiahia ana ki ngā tauira matawhenua taipitopito: me pēhea te hurihanga o te tere o ngā ngaru ru i te hohonutanga, kei hea ngā rohe o te paparanga toka, me pēhea te pānga o ngā rerekētanga iti ki te horapa o te ngaru. Ko tētahi o ngā huarahi tino kaha mō tēnei kaupapa ko te Full Waveform Inversion (FWI), he pūnaha hurihanga e whakamahi ana i ngā mōhiohio katoa mai i ngā momo ngaru ru hei whakatau tata i ngā tawhā ā-tinana o raro i te whenua.
E kiia ana te FWI ko te "paerewa koura" o te hurihanga ru whenua nā te mea ka taea e ia te whakaputa whakaahua o raro i te mata he tino kounga teitei, e hipa ana i ngā huarahi tuku iho e whakamahi ana i te wā haere noa, i te kaha whānui rānei. Heoi, he utu tēnei mana: me whai whakatauira ngaru tika te FWI, he rauemi rorohiko nui, me ngā rautaki arotau tupato kia kore ai e taka ki ngā otinga hē.
Ka whakamāramahia e tēnei tuhinga ngā ariā taketake o te FWI, ngā wāhanga matua o tana raupaparorohiko, me ngā wero whānui me ngā rautaki i roto i tōna whakatinanatanga.
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2. He aha te Hurihanga Ngaru Katoa?
I te whakarapopototanga, ko te FWI te tukanga kimi i tētahi tauira o raro i te mata e tino ōrite ana ki ngā raraunga ru whenua kua whakatauirahia ki ngā raraunga ru whenua kua kitea. Ko te "taurite pai" ka tautuhia e tētahi mahi whāinga (kore taurite), pērā i te rerekētanga tapawhā i waenga i ngā raraunga kua kitea me ngā raraunga hangai i ia wā me ia kaiwhiwhi.
Ko ngā rerekētanga matua i waenga i te FWI me ētahi atu tikanga hurihanga ko:
– Ka whakamahia e te FWI te āhua ngaru katoa (wāhanga, kaha, pokanoa, maha, whakarara), ehara i te mea ko te whiriwhiri wā taenga anake.
– Ka whakawhirinaki a FWI ki te whakaoti rapanga o te whārite ngaru (oro, rapa, anerapa rānei) hei whakatauira i ngā raraunga hangai.
– He raruraru arotautanga kore-raina nui te FWI, nā te mea ka taea e ngā tawhā tauira (hei tauira, ngā tere ngaru-P, ngaru-S rānei) te tatau i ngā miriona pūtau i runga i te whatunga 2D/3D.
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3. Ngā Wāhanga Matua i roto i te FWI
3.1 Ngā Raraunga Tirotiro me ngā Raraunga Hanga
Ko ngā mea e hiahiatia ana e te FWI:
– Ngā raraunga tirotiro: ngā rekoata rū whenua o te parae (kohikohi pupuhi) mai i ngā momo pūtake me ngā kaiwhiwhi.
– Raraunga hangai: ngā hua o te whakatauira tau o te horapa ngaru i runga i tētahi tauira rangitahi.
Ka tatauhia ngā raraunga hangai mā te whakaoti i te whārite ngaru (hei tauira, te whārite ngaru oro):
\[
\frac{1}{v^2(\mathbf{x})}\frac{\partial^2 p}{\partal t^2} – \nabla^2 p = s(\mathbf{x},t)
\]
ko \( v(\mathbf{x}) \) te tere o te ngaru, ko \( p \) te pēhanga tauine/ngaru, ā, ko \( s \) te pūtake.
3.2 Tauira Tuatahi
He tino tairongo te FWI ki te tauira tuatahi. Mena he tawhiti rawa te tauira tuatahi i ngā āhuatanga tūturu, ka pāngia pea te pūnaha e te pekepeke huringa, arā, ko te rerekētanga ā-wāhanga i waenga i ngā raraunga hangai me ngā raraunga i tirohia he nui atu i te haurua o te wā, ka meinga te rōnaki kia pana i te otinga ki te huarahi hē.
Ko te tauira tuatahi ka riro mai i:
– te hurihanga o te wā haere (tomography),
– ngā tauira tere whānui mai i te tātaritanga matawhenua,
– he huarahi maha-tauine rānei (tīmata mai i ngā auau iti).
3.3 Mahi Whāinga (Kāore i te ōrite)
Ngā mahi whāinga noa:
\[
J(m)=\frac{1}{2}\sum_{s}\sum_{r}\int (d_{\text{syn}}(t; m)-d_{\text{obs}}(t))^2 \, dt
\]
ko \( m \) he tawhā tauira (hei tauira, tere), ko \( s \) te taupū pūtake, ā, ko \( r \) te taupū kaiwhiwhi.
Haunga te L2 misfit matarohia, kei reira anō ētahi atu misfit hei whakaiti i te pekepeke pahikara, hei tauira:
– te kore taunga e hangai ana ki te kōpaki,
– te kore taunga o te wāhanga anake,
- te kawe pai (Wasserstein),
– ngā tātari ōrite rānei.
3.4 Tātaitanga Rōrahi: Tikanga Āhua-Tāpiri
Ko tētahi o ngā āhuatanga matua o te FWI ko te huarahi ki te tatau pai i ngā rōnaki. Nā te nui o ngā tawhā tauira, kāore e taea te tango tau tika. Ko te otinga ko te tikanga āhua-piri.
Te pikitia mārama:
1. Whakatauira whakamua: tatauhia te mara ngaru whakamua mai i te pūtake o te tauira o nāianei.
2. Tātaihia te toenga raraunga \( \Delta d = d_{\text{syn}} – d_{\text{obs}} \).
3. Whakatauira tāpiri: werohia te toenga hei "pūtake whakahoki" mai i te tūranga kaiwhiwhi ka horahia whakamuri i te wā.
4. Ka puta he rōnaki ki ngā tawhā tauira i te hononga i waenga i ngā mara ngaru whakamua me ngā mara ngaru hono.
Mā tēnei tikanga, ko te utu tatau o te rōnaki he rite tonu ki te rua o te whakatauira ngaru mō ia pūtake (whakamua + tāpiri), nō reira he nui tonu engari ka taea te whakamahi i te HPC/GPU.
3.5 Kaupapa Whakahōu Tauira (Whakaarotautanga)
Kia whiwhihia te rōnaki \( \nabla J \), ka whakahoutia te tauira mā te whakamahi i tētahi tikanga arotau, hei tauira:
– Te Hekenga Tino Paripari (te mea māmā rawa),
– Te Rōnaki Honohono
– L-BFGS (e whakamahia whānuitia ana nā te mea he whai hua mō ngā raruraru nui),
– te tikanga Newton/Quasi-Newton rānei.
Ngā whakahōutanga taketake:
\[
m_{k+1} = m_k – \alpha_k \, H_k^{-1}\nabla J(m_k)
\]
ko \( \alpha_k \) te roa o ngā hikoinga , ā, ko \( H_k^{-1} \) te whakatata Hessian whakamuri (hei tauira, i roto i te L-BFGS).
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4. Rerenga Mahi o te Pūmanawa FWI (Poto)
I te nuinga o te wā, ka rere auau te FWI:
1. Tīpakohia te tauira tuatahi \( m_0 \).
2. Mō ia pūtake:
– te whakatauira whakamua → raraunga hangai,
– tatauhia ngā toenga ki ngā raraunga tirotiro,
– te mahi tauira tāpiri,
– te kohikohinga piki me te heke.
3. Whakamahia te whakatikatika i mua i te whakamahinga (hei tauira, te whakatikatika i te rama, te whakangawari rānei).
4. Whakahōuhia te tauira mā roto i ngā tikanga arotau.
5. Whakahokia kia tae rā anō ki te whakatakotoranga, ki te rohenga tāruarua rānei.
I te nuinga o te wā ka whakahaerehia te FWI i ngā tauine maha, ka tīmata i ngā auau iti (te whakatika i ngā wāhanga tauira tauine nui) kātahi ka neke ake ki ngā auau teitei ake (te tāpiri taipitopito).
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5. Ngā Wero Matua i roto i te FWI
5.1 Te Peke Pahikara
Koinei te raruraru tino mōhiotia. Ina kore ngā raraunga hangai me ngā raraunga i tirohia e "i roto i te wāhanga", ka taea e te kore taunga o L2 te arahi i te arotautanga ki te iti rawa o te rohe rūpahu. Ko te otinga whānui:
– tīmata mai i ngā auau tino iti,
– whakapai ake i te tauira tuatahi (tomography),
– te whakamahi i ngā momo rerekē,
– te whakamahi i ngā rautaki tātaritanga wā me te whiriwhiri raraunga.
5.2 Utu Tātai
Ka taea e te FWI 3D me ngā pūtake maha te hiahia ki ngā mano ki ngā miriona o ngā whakatauira ngaru. E hiahiatia ana tēnei:
– rorohiko whakarara (rōpū, GPU),
– te tiaki mahara (te tirotiro hei tiaki i ngā mara ngaru),
– me ngā rautaki whiriwhiri pūtake iti (te whakakōwae pūtake, te ako mīhini iti rānei).
5.3 Te Āhuatanga Ā-tinana me te Haruru Kāore i te Taurite
Kei roto i ngā raraunga mara te haruru, ngā pānga o te taputapu, te anisotropy, te whakaitiiti (Q), te āhua whenua uaua, me te koretake o te pūtake. Mēnā he tino māmā rawa te whakatauira whakamua (hei tauira, te oro ina he ngāwari te reo), ka taea te whakapēhi i ngā hua hurihanga.
5.4 Ngā Tawhā Tauira
Ka pā te whiriwhiri i ngā tawhā hei huri (hei tauira, \( v_p \), te mātotoru, \( v_s \), te anisotropy, Q) ki te aro me te pumau. Ka taea e te nui rawa o ngā tawhā te whakaputa taupatupatu, ā, ka kino ake te huihuinga.
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6. Tono FWI
E whakamahia whānuitia ana te FWI mō:
– te tūhuratanga hinu me te hau: te whakapai ake i te whakataunga o ngā tauira tere, te āwhina i te hekenga me te whakamārama i ngā hanganga,
– ngā waiariki: te mahere i ngā rohe pakaru me ngā huringa āhuarangi,
– te ru whenua ā-ao/ā-rohe: te whakaahua i te koroka me te kirinuku,
– te hangarau whenua me te hangarau tata-mata: te mahere papaku mō ngā turanga, ngā kauhanga, me te whakaiti i te kaha o te whenua.
E kitea ana ngā painga o te FWI ina he uaua te reo, ā, he nui ngā mōhiohio ngaru kei roto i ngā raraunga: ko ngā rerekētanga iti, ngā ara maha, me ngā tau maha e kiia ana he pokanoa, ka taea tonu te whakamahi hei puna mōhiohio.
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7. Whakamutunga
Ko te Hurihanga Āhua Ngaru Katoa he pūnaha hurihanga whenua e whakamahi ana i ngā mōhiohio whai kiko kei roto i ngā āhua ngaru hei hanga tauira o raro i te mata whenua taumira teitei. Ko tōna kī ki te angitu kei roto i te whakatauira whārite ngaru tika, ngā tataunga rōnaki whai hua mā te whakamahi i te tikanga āhua-piri, me ngā rautaki arotau e karo ana i te iti rawa o te rohe pēnei i te pekepeke huringa. Ahakoa te nui o ngā tono rorohiko me te hoahoa reremahi tupato, kua whakamātauhia te FWI hei huarahi kaha i roto i te whakaahua whenua me te tūhuratanga rauemi.
Ki te hiahia koe, ka taea e au te haere tonu me ētahi atu tuhinga hangarau whai muri—hei tauira, te matapaki i te takenga mai o te pikinga ā-taumata tāpiri i runga i te āhua ōkawa, ngā tauira waehere-pūāhua FWI, ngā rautaki maha-tauine e hangai ana ki te band-pass me te matapihi mō ngā raraunga tūturu.
