230912_地震研サマースクール

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November 21, 23

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変分法データ同化を用いた 断層すべり面の摩擦空間特性評価 伊藤 伸一 東京大学地震研究所 c 伊藤伸一 01/21

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Outline 1. 導入:断層すべりと摩擦特性 2. 手法:変分法データ同化 3. 解析:摩擦空間特性の不確実性評価 4. まとめ 地震研 サマースクール c 伊藤伸一 02/21

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Outline 1. 導入:断層すべりと摩擦特性 2. 手法:変分法データ同化 3. 解析:摩擦空間特性の不確実性評価 4. まとめ 地震研 サマースクール c 伊藤伸一 02/21

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断層すべりと摩擦特性 ・すべり運動と摩擦特性は密接に関係 垂直応力 ⌧ = µN <latexit sha1_base64="02qyrxtNtyLb9yvYQd1HSMOPRKI=">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</latexit> 摩擦力 ⌧ = µN <latexit sha1_base64="02qyrxtNtyLb9yvYQd1HSMOPRKI=">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</latexit> 地震研 サマースクール c 伊藤伸一 03/21

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断層すべりと摩擦特性 スティック・スリップの動画 https://www.youtube.com/watch?v=xGUaqd0_XSU 地震研 サマースクール c 伊藤伸一

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断層すべりと摩擦特性 ・すべり運動と摩擦特性は密接に関係 ・クーロンアモントンの法則(古典的) ① 摩擦力は垂直応力に比例する ② 摩擦は見かけの接触面積に依存しない ③ 動摩擦力は静摩擦力より小さく、すべり速度に依存しない ・物質や状況に依存する 地震研 サマースクール c 伊藤伸一 05/21

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断層すべりと摩擦特性 摩擦パラメータ場 ・岩石実験から得られる経験的摩擦則 A(x), B(x), L(x) <latexit sha1_base64="wjl5sSiyJLAw8uGLtQgRFVE9lX8=">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</latexit> 速度-状態依存摩擦則 Dieterich (1979) ⌧t (x) = A(x) log (Vt (x)) + B(x) log (✓t (x)) + ⌧ 0 (x) <latexit sha1_base64="dxt0v5IQrn2jXj1FucQYrVGdwmM=">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</latexit> 摩擦力 Aging 則 すべり速さ Ruina (1983) @✓t (x) =1 @t Vt (x)✓t (x) L(x) 状態変数 ※ 空間一様 かつ 定常 を仮定すると… A B > 0 のとき V % で ⌧ % <latexit 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sha1_base64="FBwjGVsXNuFsiYr++dS5F9jSKqQ=">AAACt3ichVE9S8NQFD2N3/WjVRfBpVgqTuVVBT8m0cWxVduKbSlJfNXQNInJa7WW/gFnwUEUFBzEn+Hi4OrQnyCOCi4O3qQB0aLekLz7zrvn3vNyFEvXHMFYKyB1dff09vUPBAeHhkdC4dGxjGNWbZWnVVM37W1FdriuGTwtNKHzbcvmckXReVYpr7nn2Rq3Hc00tkTd4oWKvGdoJU2VBUE7mbzBZds2D4PFcJTFmReRziThJ1H4kTTDj8hjFyZUVFEBhwFBuQ4ZDj05JMBgEVZAgzCbMs0752giSNwqVXGqkAkt03ePdjkfNWjv9nQ8tkpTdHptYkYQY0/slr2yB3bHntnHr70aXg9XS51Wpc3lVjF0MrH5/i+rQqvA/hfrT80CJSx6WjXSbnmIewu1za8dn71uLm/EGtPsmr2Q/ivWYvd0A6P2pt6k+Mb5H3oU0uL+sdivFQJHNN/0HHDQJCsTP43rTDKz8cRcfDY1H11Z9U3txySmMEPOLWAF60giTXMMnOICl9KSVJRK0n67VAr4nHF8C+ngEwnEnR4=</latexit> <latexit sha1_base64="mb/MmbBEjXUIEosG2FbyDi0feJA=">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</latexit> 速度強化 A <latexit 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sha1_base64="Y77GcsOK5eVQpTsUuk/LgfiHTJE=">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</latexit> B<0 のとき V %で ⌧ & <latexit sha1_base64="FBwjGVsXNuFsiYr++dS5F9jSKqQ=">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</latexit> <latexit 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断層運動モデル 準静的境界要素モデル ・豊後水道 スロースリップ発生域の断層運動のモデル Hirahara et al. (2019) Ground Vlock <latexit sha1_base64="2w2ME6BdKlnLWFQGtTi3j4V/QSA=">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</latexit> Vplate <latexit sha1_base64="Po4sH//aC+eoS63qoc56tPYapl8=">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</latexit> 6.5km 0.5km / yr / yr Obara et al. (2016) 断面図 鳥瞰図 0Z ⌧t (x) = A(x) log (Vt (x)) + B(x) log (✓t@⌧ (x)) + ⌧ (x) t (x) = dx0 g(x, x0 ) (V Vt (x0 )) + K :グリーン関数 plate Z @⌧ (x) Z @t t @⌧t (x) G @VVt (x)) G 0 0 0 0 0 0 = dx g(x, x ) (V V (x )) + K(x) (V :上部プレート lock EOM = dx @tg(x, x ) (Vplate Vt (x ))plate + K(x)t (Vplate Vlock ) plate 2c @t 2c @t からのグリーン Z x) G @Vt (x) 関数 = dx0 g(x, x0 ) (Vplate Vt (x0 )) + K(x) (Vplate Vlock ) G :剛性率 2c @t c :音速 Vt (x)✓t (x) Aging @✓t (x) =1 Law @t L(x) RSDF <latexit 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sha1_base64="PuBoeaGTTzMtIQIP660G3eKSE9Y=">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</latexit> <latexit 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sha1_base64="f6SCGutV9cL+qeOpUsTM6whl4Q0=">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</latexit> <latexit sha1_base64="UoDujNyuXncqJwCG/86TAh40YX8=">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</latexit> 地震研 サマースクール c 伊藤伸一 07/21

9.

沈み込み方向すべり速さの時間発展 5∘ 120km / yr km 15∘ 6.5 m 35 k 100 25km Locked km 沈み込み 方向 Parameters A(x) = 100 [kPa] { B(x) = 135 [kPa] B(x) = 30 [kPa] <latexit sha1_base64="ojr6/pvwkvKumYFJoSJi4bZFi0w=">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</latexit> L(x) = 22 [mm] 実際のパラメータの値やパラメータの不確実性などはよく分からない (地下深く掘って調べるなどは非現実的) → 観測可能な量を使って間接的に調べるアプローチが必要 地震研 サマースクール c 伊藤伸一 08/21

10.

Outline 1. 導入:断層すべりと摩擦特性 2. 手法:変分法データ同化 3. 解析:摩擦空間特性の不確実性評価 4. まとめ 地震研 サマースクール c 伊藤伸一 09/21

11.
[beta]
データ同化
モデル
<latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit>

事後分布

データ

p(xt | D)

D = {D1 , D2 , ...}

予測の高精度化、

<latexit sha1_base64="9Cy5FDqhVcuvIh7OEYFm+wU107A=">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</latexit>

p(xt | D)

逆問題、…
天気予報

出典:Google Earth
地震研 サマースクール

<latexit sha1_base64="uqBeDhpsi6lxGlvwkHHdVQpc2sc=">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</latexit>

実験・観測デザインの
最適化

地震学

Kano et al. (2015)

材料科学

Ito et al. (2017)

c 伊藤伸一

10/21

12.

変分法データ同化 モデルの初期値の事後分布の情報(例えばMAP解など)を効率的に抽出する手法 Dデータと + !t t = h(xt )の関係 Dt = h(xt ) + !t シミュレーションモデル <latexit sha1_base64="yXpqNkT0w6ffWorbjXV9yQ9bbtU=">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</latexit> <latexit 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p(✓):事前分布 <latexit sha1_base64="WGN2PBPUYctwd9zIQsrpAM3Un4c=">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</latexit> ノイズ <latexit 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地震研 サマースクール X log q(Dt h(xt )) t2T obs c 伊藤伸一 11/21

13.

変分法データ同化による初期値更新 x0 = ✓ C(✓) = <latexit sha1_base64="3tnKdZhOroykKE9AkYQLgeiHt+E=">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</latexit> X t2T obs <latexit sha1_base64="ZS3Z0hDusosxrFqeLGFiyaBbqyI=">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</latexit> :データ ✓guess <latexit sha1_base64="YnLegehdSrKrSmzbu3/KDfP7jUU=">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</latexit> Time 適当に選んだ x0 <latexit sha1_base64="AqZRKhAp+ail3X3ypq29XDB6qtc=">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</latexit> 地震研 サマースクール = ✓guess ではデータと全く合わないが,,, c 伊藤伸一 12/21

14.

変分法データ同化による初期値更新 x0 = ✓ C(✓) = <latexit sha1_base64="3tnKdZhOroykKE9AkYQLgeiHt+E=">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</latexit> X t2T obs <latexit sha1_base64="ZS3Z0hDusosxrFqeLGFiyaBbqyI=">AAALqnicbdbbbtowGAfwtDt1ZWztdrkbNKqpldYKumnrzaSez+cCPZkhJzjgNk7S2NCUKHuCPc1utxfZ2ywE1C/kS26wv5//DkTEtu5aXKpS6d/Y+JOnz56/mHg5mXuVf/1mavptTTodz2BVw7Ec70KnklncZlXFlcUuXI9RoVvsXL9d6/t5l3mSO3ZFPbisLmjL5iY3qIpKjamPa7NEtZmic4XvBSI7ohEowu1C5UdAFPNV4OgyDMPGVLG0UIqvAm6Uh42iNryOG9OTFmk6RkcwWxkWlfK6XHJVPaCe4obFwjzpSOZS45a22HXUtKlgsh7EPygsJDWgQsoHoYfpoqCqnZpHmUv1gNtuRzHbGA0o33RsJcN8ntjs3nCEoHYzINRrCeqHAenP5rgB8UQhqv3sF4nFBe9HUILbGYmo+JjIkyYzSZcZs8XyXDRUd/yA6I7V7IcKM8XyTPg4iGKnfR6ojlUHNbAaoE2sTVCGlYGaWE3QFtYWaBtrG5Rj5aA3WG9Ab7HeglpYLVCBVYDaWG1QB6sD6mJ1Qe+w3oF6WD1QiVWCKqwKtIO1A9rF2gW9x3oP6mP1QR+wPoD2sPYSL8MK5hUIr2JdBV3Duga6jnUddAPrBugm1k3QLaxboNtYt0F3sO6A7mLdBd3Duge6j3Uf9ADrAegh1kPQI6xHoMdYj0FPsJ6AnmI9BT3DegZawVoBrWKtgtaw1kDPsZ6DXmC9AL3Eegl6hfUq8TL0mJex8JQSy5JgrYztg8T1xCrCo00uY1hcTywn/XNAxrC4nlhlo9NFM2vcABKvevZ0vZHZ/IztgPiJDSF7kpE5RMZSR0RisWtRkfkA4nriVq7klpOxE5ChJP5t8bOKxpgFErfhXxo/7IHE7TB95vCEY7Ow/zlL2jI6oLBgobzERMhHu3NhOqfunVSuFOf4aD8j2PbY8JaPY+eH4XQfp83otJkd7ib6XzPvbPJu+rfGI7ujXRyU3B/NzaeCie+bj86n5fRpFDdqiwvlzwuLJ1+Ky6vDk+qE9l77oM1qZe2btqxta8daVTO0X9pv7Y/2N/cpd5q7zF0Pho6PDTPvtJEr1/wPhiw4kg==</latexit> :データ ✓ˆ <latexit sha1_base64="ehxCx+d3pIESsKl7ZfcPjJ0tcZ8=">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</latexit> ✓guess <latexit sha1_base64="YnLegehdSrKrSmzbu3/KDfP7jUU=">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</latexit> Time 勾配法による最適化により データに適合する初期値 X C(✓) の微分(勾配) = <latexit sha1_base64="ZS3Z0hDusosxrFqeLGFiyaBbqyI=">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</latexit> 地震研 サマースクール x0 = ✓ˆ を探す <latexit sha1_base64="Lo0LEFFzTOuOLCRfu+MtakxyflQ=">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</latexit> の計算が必要 t2T obs c 伊藤伸一 13/21

15.

Adjoint法による勾配計算 勾配法によって最適化したいが, C(✓) = log p(✓) <latexit 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Adjoint法 L=C+ <latexit 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Z T X logq(D q(Dtt h(x h(xtt)) )) log t2T obs ✓ で直接微分ができない ✓ ◆ d T :終端時刻 > dt t f (xt ) xt t:Adjoint変数 dt <latexit 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<latexit sha1_base64="jRRHIl4aQaDCLy6WjaPD1Ye14kE=">AAACs3ichVFNLwNRFD0d39/FRmLTaCpWzWsRYiVsLLW0JG0jM9NXns5XZl4bNP6AlZ1gRWIhfoaNha1Ff4JYVmJj4c50EkFwJzPvvvPuufe8OZpjCE8y1owoHZ1d3T29ff0Dg0PDI9HRsbxn11yd53TbsN1tTfW4ISyek0IafNtxuWpqBt/Sqqv++Vadu56wrU156PCSqe5aoiJ0VRKUL8o9LtWdaJwlWRCxn0kqTOIIY92OPqKIMmzoqMEEhwVJuQEVHj0FpMDgEFZCgzCXMhGccxyjn7g1quJUoRJape8u7QohatHe7+kFbJ2mGPS6xIwhwZ7YLWuxB3bHntn7r70aQQ9fyyGtWpvLnZ2Rk4mNt39ZJq0Se5+sPzVLVLAYaBWk3QkQ/xZ6m18/OmttLGUTjWl2zV5I/xVrsnu6gVV/1W8yPHv5hx6NtPh/LPFrhcQBzbcDBzwck5Wp78b9TPLpZGo2mc7MxZdXQlN7MYkpzJBzC1jGGtaRozn7OMU5LpR5paBoSrldqkRCzji+hGJ+AHhQm64=</latexit> <latexit 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0 <latexit 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(ラグランジュの未定乗数) 変分をとって係数比較 Adjoint モデル d dt t = (r x f )> t + X t0 2T obs <latexit 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(t t0 )r xt0 C <latexit 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T =0 直接微分が取れなくても、adjointモデルを解くことで微分が得られる ※ t=Tから時間後ろ向きに解く必要があることに注意! 地震研 サマースクール c 伊藤伸一 14/21

16.
[beta]
時間後ろ向きに計算する
Adjoint モデル

d
dt

t

= (r x f )>

t +

X

t0 2T obs

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(t

t0 )r xt0 C
<latexit sha1_base64="h5FlaEbyb5cnhAtKyCqna4vs1GU=">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</latexit>

T

=0

t
<latexit sha1_base64="RWXuDuJHXJvlN8xJDO9R8AqtrUs=">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</latexit>
<latexit

Input

✓
<latexit sha1_base64="jRRHIl4aQaDCLy6WjaPD1Ye14kE=">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</latexit>

Output

<latexit sha1_base64="pbpm7LokrN6XiD4C/gJ6Clb8cQ0=">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</latexit>

dC
d✓

t
<latexit sha1_base64="RWXuDuJHXJvlN8xJDO9R8AqtrUs=">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</latexit>
<latexit

<latexit sha1_base64="ryuXTQ0mJCzzWoku/BcVPGMW6NI=">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</latexit>

地震研 サマースクール

T obs = {0, t1 , t2 , . . . , tn = T }

c 伊藤伸一

15/21

17.

変分法データ同化を使ってできること 直接微分のとれないコスト関数の(高階も含め)微分が計算できる 微分がわかるといろんなことがわかる ・最適化によるパラメータやモデルの初期値のMAP解推定 ・感度解析(パラメータに対するモデルの敏感さ) ・不確実性評価(MAP解周りの標準偏差) <latexit sha1_base64="aKB9kaQCOyKzH6yU8shDDgcB/1o=">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</latexit> p(✓ | D) ・・・ ・分布の偏り、鋭さ 地震研 サマースクール ✓ˆ ✓ <latexit sha1_base64="6L/sfM20+IbPGS/8f9cvkF/lmnE=">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</latexit> <latexit sha1_base64="ehxCx+d3pIESsKl7ZfcPjJ0tcZ8=">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</latexit> c 伊藤伸一 16/21

18.

Outline 1. 導入:断層すべりと摩擦特性 2. 手法:変分法データ同化 3. 解析:摩擦空間特性の不確実性評価 4. まとめ 地震研 サマースクール c 伊藤伸一 17/21

19.
[beta]
摩擦空間特性の不確実性評価(私の研究紹介)
豊後水道 LSSE model
Hirahara and Nishikiori (2019)

{
<latexit sha1_base64="H6hgW2UPuyTeXlurJ5jRNw1vdhc=">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</latexit>

速度ー状態依存摩擦則
<latexit sha1_base64="0o8QTSYemkNYuwrVKhd5a9dH1Ks=">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</latexit>

⌧ (x, t) = A(x) log v(x, t) + B(x) log ✓(x, t)

Notation

x = (X, Y )>
<latexit sha1_base64="5dacfQWuafX1KYITsussZk7uFPA=">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</latexit>

Aging 則

v : Slip velocity
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✓ : State variable
K, k : Green’s functions

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釣り合い方程式
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⌧˙ (x, t) =

⌘v(x, t) +

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<latexit sha1_base64="Wk0EDy5M1U6wmcRbR5QZdrqDnOo=">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</latexit>

⌘
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: Damping parameter

vpl , vlock : Speeds of plates
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Z

dx0 [K(x, x0 ) {v(x0 , t)

vpl } + k(x0 ) (vlock

vpl )]

Slip velocity

興味のあるパラメータ場

A(x), B(x), L(x)

既存研究では、パッチ的なパラメータ場に制約していた。
→ 原理的にパッチ的な不確実性しか得られない。

<latexit sha1_base64="RiOxOi+Aigg/w08Ttms07/zVG1g=">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</latexit>

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v
<latexit sha1_base64="gP7u9a7XWLEtdQaoNfEGPENAAWw=">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</latexit>

Y (km)

˙ t) = 1
✓(x,

v(x, t)✓(x, t)
L(x)

<latexit sha1_base64="ZPVqY7MykWZuyoUvp/K5MAQEwGI=">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</latexit>

X(km)

→ すべり運動の詳細を調べるには制約のない高解像の不確実性評価の方法が必要不可欠

地震研 サマースクール

c 伊藤伸一

18/21

20.

摩擦特性分布不確実性の高解像評価 得られた不確実性の場 問題設定 仮定した摩擦パラメータの場: 100km 35km (a) 120km (b) データ: 時間窓 (a)—(d) における すべり速度 のスナップショット v <latexit sha1_base64="gP7u9a7XWLEtdQaoNfEGPENAAWw=">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</latexit> (c) 上の動画を ———— で切った断面図 (d) ・単純形状の摩擦特性にも関わらず、 複雑な不確実性の場が得られる. ・活発な地震活動を含むデータは、 不確実性を大幅に小さくする. → 効率的な高精度化への指針 地震研 サマースクール c 伊藤伸一 19/21

21.

摩擦特性分布不確実性の高解像評価 得られた不確実性の場 問題設定 仮定した摩擦パラメータの場: 100km 35km (a) 120km (b) データ: 時間窓 (a)—(d) における すべり速度 のスナップショット v <latexit sha1_base64="gP7u9a7XWLEtdQaoNfEGPENAAWw=">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</latexit> (c) 上の動画を ———— で切った断面図 (d) ・長時間の同化は不確実性を小さくする ことができる. (計算量は増える) ・活発な地震活動データが取り込まれる と不確実性は大幅に下がる. 地震研 サマースクール c 伊藤伸一 20/21

22.

まとめ ・断層すべり運動と摩擦について ・大規模モデルのための変分データ同化について ・変分法データ同化による摩擦空間特性評価の研究紹介 ・午後の演習では変分データ同化を簡単なモデルへ適用した デモプログラムを通じて変分データ同化にふれてみる 地震研 サマースクール c 伊藤伸一 21/21

23.

Adjoint法による勾配計算:解ける非線形方程式を例として ・厳密に解ける方程式でadjointモデルを解いてみる ・ロジスティック方程式 <latexit 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{ <latexit 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生物の個体数変化などを表す数理モデル ⇣ xt ⌘ d 厳密解 xt = rxt 1 Kert dt K xt = rt e + K/✓ x0 = ✓ r, K :既知の定数とする <latexit 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<latexit 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<latexit 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1 Kert xt = rt e + K/✓ r, K <latexit 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・問題設定 時刻 t = T の時に個体数 y が観測できたとして、 時刻 t = 0 での個体数 ✓ を推定する問題を考える <latexit 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<latexit 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1 C = (xT 2 y <latexit 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コスト関数 y) 2 1 dC f (xT ) @C 2 = を求めてみよう C の最小化に必要な勾配 = (y xT ) d✓ f (✓) @xT 2 <latexit 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✓ <latexit 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t=0 <latexit 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<latexit 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<latexit 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T 午後のデモでは pythonプログラムでこれを解く実験を予定 地震研 サマースクール c 伊藤伸一 1

24.

Adjoint法による勾配計算:解ける非線形方程式を例として 厳密解を代入したコスト関数 1 C = (xT 2✓ <latexit 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y) 2 <latexit 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1 J= 2 を直接 x0 = <latexit 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y ✓ rT erT Ke + K/✓ 1 ◆2 で微分したもの ✓ @C f (xT ) @C K/✓ dC = = rT + K/✓ @✓ d✓ f e(✓) @xT <latexit 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<latexit 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<latexit 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とAdjointモデルの厳密解 地震研 サマースクール ◆2 erT (xT y) 1 ✓ K/✓ rT = (y xT )e 0 が一致するか計算してみる。 erT + K/✓ c 伊藤伸一 1 ◆2

25.

Adjoint法による勾配計算:解ける非線形方程式を例として Adjointモデル { ✓ <latexit 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<latexit 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d dt <latexit 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T t 2xt K =r 1 dC = xT = dxT ◆ t y Forwardモデル <latexit sha1_base64="bnMJwvYjLtMTGOa+nSaakkrHmUY=">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</latexit> Adjointモデルの厳密解 <latexit 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d log dt <latexit 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✓ ◆ 2xt t =r 1 K  Z t ✓ dt r 1 t = T exp T 地震研 サマースクール d xt = f (xt ) dt x0 = ✓ <latexit 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Adjointモデル <latexit sha1_base64="xtxQWpbQzIsxyhdenZHKkKtlsY0=">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</latexit> 2xt K ◆ d dt t <latexit sha1_base64="H0lB3rz9YK4l4/TbewLEkvCBPXw=">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</latexit> T = (r x f )> = r xT C c 伊藤伸一 t

26.

Adjoint法による勾配計算:解ける非線形方程式を例として <latexit sha1_base64="7s/aBs0w5v0+H3dMvVSPpP86YSc=">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</latexit> 0 = T exp  Z <latexit sha1_base64="c9AV6W9ub0CUaDofgdh0ZhC7qxs=">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</latexit> 0 = Te rT exp <latexit sha1_base64="duZCQpEmOOH96Qph8pCw6TEt3jk=">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</latexit> 0 = (xT y)e 0 ✓ dt r 1 T " rT 2r K ✓ Z T 2xt K # ◆ dt xt 0 erT K/✓ + K/✓ 1 ◆2 * <latexit 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✓ @C K/✓ dC f (xT ) @C = = rT + @✓ d✓ f e(✓) @xK/✓ T <latexit 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1 ◆2 モデルに依存した補正 地震研 サマースクール <latexit 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dt xt = <latexit 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T 勾配の厳密解 <latexit 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Z erT (xT K log ert + K/✓ r = xT y y) と一致することが確かめられた 観測とのズレ c 伊藤伸一 1

27.

Adjoint法による勾配計算:解ける非線形方程式を例として 腕だめし 問題1 <latexit sha1_base64="qq4ASKbRCnqtH0AUwK5D5aIgeuk=">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</latexit> erT ✓ erT K/✓ + K/✓ 1 ◆2 <latexit sha1_base64="b1DPnFeej0eKMvjUwmgaz1pBj7I=">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</latexit> f (xT ) rxT (1 = は始点と終点での右辺の比 f (✓) r✓(1 xT /K) ✓/K) に等しい このことを確かめてみよう。 問題2 先の例題のような1変数自励系(時間tを陽に含まない方程式)について、 観測が終点の1点のみの場合には <latexit sha1_base64="PyKu8uN2My5PhfDj6yo2lu7r4qo=">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</latexit> dC f (xT ) dC = d✓ f (✓) dxT ← 勾配が始点と終点の情報のみで決まる が常に成立する。なぜか考えてみよう。 地震研 サマースクール c 伊藤伸一