Transcript
Page 1: Francesco Della Santa - smartdata.polito.it · 30 80 107 number of random transmissivities 0 0.05 0.1 0.15 error (% of real fracture outflow) Mean relative error behavior d=1, B=10,

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d=1, B=10, i=30,80,107d=3, B=10, i=30,80,107d=1, B=30, i=30,80,107d=3, B=30, i=30,80,107d=3, B=10, i=107

Machine Learning for Flux Regression in Discrete Fracture Networks

Francesco Della Santa

NN* Analysis – Predictions

Numerical Experiments - Models and Performance Comparisons

Conclusions and future work

Uncertainty in Discrete Fracture Networks (DFN)

Vector Valued Regression in DFNs

Characterization of flow and transport through fractured media in thesubsurface is a crucial issue in many applications (e.g. in oil and gasextraction or avoidance of drinking water pollution due to industrial waste).Then models such as Discrete Fracture Networks (DFN), that can accuratelysimulate flow through networks of subsurface fractures, are required.

Major issues:• Building mesh on intersecting fractures (non- conforming meshes);• Lack of full deterministic data > DFN stochastically generated using

probabilistic distribution for generating:o Geometrical properties (e.g. fracture density, size, position,

orientation, …);o Hydrogeological properties such as transmissivity (parameter

that characterizes flow facilitation through a given fracture).We suppose distribution of fracture transmissivities __ is:

We study NN regression models on a given DFN with fixed and deterministic geometry consisting of 107 fractures, in a cubic domain (1000 meters wide) and pressure difference between entering and exiting flow surfaces of domain is set to m.

Since QoI’s moments evaluation needs lot of simulations, usage of NeuralNetworks (NN) is considered, to speed up this process (through QoIregression) and then as an alternative model reduction method.

DFNs with fixed and deterministic geometry are taken into account to dealwith smooth QoI. Let be the total number of fractures in the DFN and let _be the total number of fractures with flow exiting the network. Let be thetransmissivity of i-th fracture.In regression problems considered, we want to predict _______ exiting flowsof the given DFN, varying (without loss of generality) transmissivities of first______ fractures (and supposing others fixed to value _________). Moreprecisely we want to build a vector valued regression model that, for eachtransmissivities vector _______________ with corresponding fluxes________________, returns a vector of estimated fluxes:

We built vector valued NNs for flux regression problems in DFNs and studied their performances varying parameters, with particular attention to NN* network. NN* has good performances (very few serious prediction mistakes) especially considering distribution and moments of QoI considered.Future works will improve NN architectures and tune parameters for training. A new dataset creation criterion, using homogeneous quality simulation (adaptive methods), will be considered.

Collaborators logos – delete box if not required

Joint work with:Stefano Berrone, Sandra Pieraccini (supervisor), Francesco Vaccarino (supervisor)

Numerical solution:• Non-conforming meshes generated independently for each fracture;• Strongly parallel approach (fracture-local problems solved within a

gradient-based approach);• Different choices available for space discretization (FEM, XFEM, VEM).

Uncertainty and quantities of interest (QoI):Given a probability space and an outcome , let be solution of our mathematical problem and ______________ be the application’s QoI (e.g. the overall flux flowing through the network).

Main target: evaluates moments of via non-intrusive methods:

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�̂ = ['̂1, . . . , '̂m]> s.t. �̂ ⇡ �<latexit sha1_base64="r0i5holaRe27kQyz9IHWYngyCxA=">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</latexit><latexit sha1_base64="r0i5holaRe27kQyz9IHWYngyCxA=">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</latexit><latexit sha1_base64="r0i5holaRe27kQyz9IHWYngyCxA=">AAACe3icdVFNT9tAEF2bfkCgJYUjVFo1QqqqyLKrIrggoXLhCFIDSFkTjdcDWbH2bnfHqMjKub+xh/4EfgISdpIDkPZd9um9+dqZzGrlKY7/BOHSq9dv3i6vdFbX3r1f737YOPOmchIH0mjjLjLwqFWJA1Kk8cI6hCLTeJ7dHLX++S06r0z5g+4spgVcl+pKSaBGGnV/izFQLTKjc39XNE8t7FhNJvyAD2fWLbhWGSV9oXNDnvef60V6KchY8bOCnPuIohn7T10B1jrza8EYdXtxFE/BF0kyJz02x8mo+1fkRlYFliQ1eD9MYktpDY6U1DjpiMqjBXkD1zhsaAkF+rSeLmzCdyoPZLhFx5XmUxGfZtRQ+Ha6JrIAGvuXXiv+yxtWdLWf1qq0FWEp20akNE4beelUcwnkuXJIBO3kyFXJJTggQqc4SNmIVXOaTrOP5OXvF8nZ1yiJo+T0W+/w+3wzy2yLfWKfWcL22CE7ZidswCS7DzaD7eBj8BD2wi9hfxYaBvOcTfYM4e4jgRPFJA==</latexit><latexit sha1_base64="r0i5holaRe27kQyz9IHWYngyCxA=">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</latexit>

k = [1, . . . ,n]>

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Input Layer

Hidden Layer 1

Hidden Layer 2

HiddenLayer 3.2

HiddenLayer 3.1

HiddenLayer 4.1

HiddenLayer 4.2

OutputLayer 2

OutputLayer 1

Example of NN for vector valued regression for flux prediction ____________________ .(n = 3 , m = 2 , d = 2)

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NN architecture:• Input layer of _ softplus units;• shared layers of _ softplus

units;• task-specific layers of _

softplus units for each one of the _ outputs;

• Output layer of _ units (linked to its last task-specific layer).

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-2 -1 0 1 2-2

-1

0

1

2Softplus function

Error measures:

• Absolute Error:

• Relative Error:

• Weighted Absolute Error:

NN parameters:• fixed (fractures {8,

33, 62, 105}; );• ;• ;Training parameters:• E = 1000 (maximum training

epochs);• B = 10, 30 (mini-batch size

for training);• P = 150 (Early stopping

patience parameter).

ea = [|'̂1 � '1|, . . . , |'̂m � 'm|]><latexit sha1_base64="pQYWwe1jtItcoMQUUbh0JHbxbqE=">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</latexit><latexit sha1_base64="pQYWwe1jtItcoMQUUbh0JHbxbqE=">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</latexit><latexit sha1_base64="pQYWwe1jtItcoMQUUbh0JHbxbqE=">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</latexit><latexit sha1_base64="pQYWwe1jtItcoMQUUbh0JHbxbqE=">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</latexit>

er =

|'̂1 � '1|

|'1|, . . . ,

|'̂m � 'm||'m|

�>

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ew =eaPm̂i=1 'i

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= 10�5<latexit sha1_base64="eS2xemGj6vEUVt44+/cUbJU9bDM=">AAACBHicbVA9SwNBEN2LXzF+nVraLAbBxnAnijZC0MYygvmAJIa5zSQu2btbducC4Ujrr7DVyk5s/R8W/hcvMYVGX/V4b4Z58wKtpCXP+3ByC4tLyyv51cLa+sbmlru9U7NxYgRWRaxi0wjAopIRVkmSwoY2CGGgsB4MriZ+fYjGyji6pZHGdgj9SPakAMqkjuu2BqA18Avue3fp0em44xa9kjcF/0v8GSmyGSod97PVjUUSYkRCgbVN39PUTsGQFArHhVZiUYMYQB+bGY0gRNtOp8nH/CCxQDHXaLhUfCriz40UQmtHYZBNhkD3dt6biP95zYR65+1URjohjMTkEEmF00NWGJlVgrwrDRLBJDlyGXEBBojQSA5CZGKSdVTI+vDnv/9Lascl3yv5NyfF8uWsmTzbY/vskPnsjJXZNauwKhNsyB7ZE3t2HpwX59V5+x7NObOdXfYLzvsXYP+Wzw==</latexit><latexit sha1_base64="eS2xemGj6vEUVt44+/cUbJU9bDM=">AAACBHicbVA9SwNBEN2LXzF+nVraLAbBxnAnijZC0MYygvmAJIa5zSQu2btbducC4Ujrr7DVyk5s/R8W/hcvMYVGX/V4b4Z58wKtpCXP+3ByC4tLyyv51cLa+sbmlru9U7NxYgRWRaxi0wjAopIRVkmSwoY2CGGgsB4MriZ+fYjGyji6pZHGdgj9SPakAMqkjuu2BqA18Avue3fp0em44xa9kjcF/0v8GSmyGSod97PVjUUSYkRCgbVN39PUTsGQFArHhVZiUYMYQB+bGY0gRNtOp8nH/CCxQDHXaLhUfCriz40UQmtHYZBNhkD3dt6biP95zYR65+1URjohjMTkEEmF00NWGJlVgrwrDRLBJDlyGXEBBojQSA5CZGKSdVTI+vDnv/9Lascl3yv5NyfF8uWsmTzbY/vskPnsjJXZNauwKhNsyB7ZE3t2HpwX59V5+x7NObOdXfYLzvsXYP+Wzw==</latexit><latexit sha1_base64="eS2xemGj6vEUVt44+/cUbJU9bDM=">AAACBHicbVA9SwNBEN2LXzF+nVraLAbBxnAnijZC0MYygvmAJIa5zSQu2btbducC4Ujrr7DVyk5s/R8W/hcvMYVGX/V4b4Z58wKtpCXP+3ByC4tLyyv51cLa+sbmlru9U7NxYgRWRaxi0wjAopIRVkmSwoY2CGGgsB4MriZ+fYjGyji6pZHGdgj9SPakAMqkjuu2BqA18Avue3fp0em44xa9kjcF/0v8GSmyGSod97PVjUUSYkRCgbVN39PUTsGQFArHhVZiUYMYQB+bGY0gRNtOp8nH/CCxQDHXaLhUfCriz40UQmtHYZBNhkD3dt6biP95zYR65+1URjohjMTkEEmF00NWGJlVgrwrDRLBJDlyGXEBBojQSA5CZGKSdVTI+vDnv/9Lascl3yv5NyfF8uWsmTzbY/vskPnsjJXZNauwKhNsyB7ZE3t2HpwX59V5+x7NObOdXfYLzvsXYP+Wzw==</latexit><latexit sha1_base64="eS2xemGj6vEUVt44+/cUbJU9bDM=">AAACBHicbVA9SwNBEN2LXzF+nVraLAbBxnAnijZC0MYygvmAJIa5zSQu2btbducC4Ujrr7DVyk5s/R8W/hcvMYVGX/V4b4Z58wKtpCXP+3ByC4tLyyv51cLa+sbmlru9U7NxYgRWRaxi0wjAopIRVkmSwoY2CGGgsB4MriZ+fYjGyji6pZHGdgj9SPakAMqkjuu2BqA18Avue3fp0em44xa9kjcF/0v8GSmyGSod97PVjUUSYkRCgbVN39PUTsGQFArHhVZiUYMYQB+bGY0gRNtOp8nH/CCxQDHXaLhUfCriz40UQmtHYZBNhkD3dt6biP95zYR65+1URjohjMTkEEmF00NWGJlVgrwrDRLBJDlyGXEBBojQSA5CZGKSdVTI+vDnv/9Lascl3yv5NyfF8uWsmTzbY/vskPnsjJXZNauwKhNsyB7ZE3t2HpwX59V5+x7NObOdXfYLzvsXYP+Wzw==</latexit>

�H = 10<latexit sha1_base64="z33SSzCBZZ8bWSQCnrbSaXAUF0k=">AAAB/3icbVC7SgNBFJ2NrxhfUUubwSBYhV0RtBGCWqSMYB6QLOHu5CYOmX04c1cISwq/wlYrO7H1Uyz8F3fXFJp4qsM593LPPV6kpCHb/rQKS8srq2vF9dLG5tb2Tnl3r2XCWAtsilCFuuOBQSUDbJIkhZ1II/iewrY3vsr89gNqI8PgliYRuj6MAjmUAiiV3N41KgJe5xfcsfvlil21c/BF4sxIhc3Q6Je/eoNQxD4GJBQY03XsiNwENEmhcFrqxQYjEGMYYTelAfho3CQPPeVHsQEKeYSaS8VzEX9vJOAbM/G9dNIHujPzXib+53VjGp67iQyimDAQ2SGSCvNDRmiZtoF8IDUSQZYcuQy4AA1EqCUHIVIxTusppX04898vktZJ1bGrzs1ppXY5a6bIDtghO2YOO2M1VmcN1mSC3bMn9sxerEfr1Xqz3n9GC9ZsZ5/9gfXxDSfnlQ0=</latexit><latexit sha1_base64="z33SSzCBZZ8bWSQCnrbSaXAUF0k=">AAAB/3icbVC7SgNBFJ2NrxhfUUubwSBYhV0RtBGCWqSMYB6QLOHu5CYOmX04c1cISwq/wlYrO7H1Uyz8F3fXFJp4qsM593LPPV6kpCHb/rQKS8srq2vF9dLG5tb2Tnl3r2XCWAtsilCFuuOBQSUDbJIkhZ1II/iewrY3vsr89gNqI8PgliYRuj6MAjmUAiiV3N41KgJe5xfcsfvlil21c/BF4sxIhc3Q6Je/eoNQxD4GJBQY03XsiNwENEmhcFrqxQYjEGMYYTelAfho3CQPPeVHsQEKeYSaS8VzEX9vJOAbM/G9dNIHujPzXib+53VjGp67iQyimDAQ2SGSCvNDRmiZtoF8IDUSQZYcuQy4AA1EqCUHIVIxTusppX04898vktZJ1bGrzs1ppXY5a6bIDtghO2YOO2M1VmcN1mSC3bMn9sxerEfr1Xqz3n9GC9ZsZ5/9gfXxDSfnlQ0=</latexit><latexit sha1_base64="z33SSzCBZZ8bWSQCnrbSaXAUF0k=">AAAB/3icbVC7SgNBFJ2NrxhfUUubwSBYhV0RtBGCWqSMYB6QLOHu5CYOmX04c1cISwq/wlYrO7H1Uyz8F3fXFJp4qsM593LPPV6kpCHb/rQKS8srq2vF9dLG5tb2Tnl3r2XCWAtsilCFuuOBQSUDbJIkhZ1II/iewrY3vsr89gNqI8PgliYRuj6MAjmUAiiV3N41KgJe5xfcsfvlil21c/BF4sxIhc3Q6Je/eoNQxD4GJBQY03XsiNwENEmhcFrqxQYjEGMYYTelAfho3CQPPeVHsQEKeYSaS8VzEX9vJOAbM/G9dNIHujPzXib+53VjGp67iQyimDAQ2SGSCvNDRmiZtoF8IDUSQZYcuQy4AA1EqCUHIVIxTusppX04898vktZJ1bGrzs1ppXY5a6bIDtghO2YOO2M1VmcN1mSC3bMn9sxerEfr1Xqz3n9GC9ZsZ5/9gfXxDSfnlQ0=</latexit><latexit sha1_base64="z33SSzCBZZ8bWSQCnrbSaXAUF0k=">AAAB/3icbVC7SgNBFJ2NrxhfUUubwSBYhV0RtBGCWqSMYB6QLOHu5CYOmX04c1cISwq/wlYrO7H1Uyz8F3fXFJp4qsM593LPPV6kpCHb/rQKS8srq2vF9dLG5tb2Tnl3r2XCWAtsilCFuuOBQSUDbJIkhZ1II/iewrY3vsr89gNqI8PgliYRuj6MAjmUAiiV3N41KgJe5xfcsfvlil21c/BF4sxIhc3Q6Je/eoNQxD4GJBQY03XsiNwENEmhcFrqxQYjEGMYYTelAfho3CQPPeVHsQEKeYSaS8VzEX9vJOAbM/G9dNIHujPzXib+53VjGp67iQyimDAQ2SGSCvNDRmiZtoF8IDUSQZYcuQy4AA1EqCUHIVIxTusppX04898vktZJ1bGrzs1ppXY5a6bIDtghO2YOO2M1VmcN1mSC3bMn9sxerEfr1Xqz3n9GC9ZsZ5/9gfXxDSfnlQ0=</latexit>

(NN*)

m = 4<latexit sha1_base64="82gm1PoIiHh8Y0IRmoT67kovAmg=">AAAB93icbVBNS8NAFNzUr1q/qh69LBbBU0mkoBeh6MVjBdMW2lA229e6dHcTdl+EEvobvOrJm3j153jwv5jEHLQ6p2HmPd68CWMpLLruh1NZWV1b36hu1ra2d3b36vsHXRslhoPPIxmZfsgsSKHBR4ES+rEBpkIJvXB2nfu9BzBWRPoO5zEEik21mAjOMJN8RS9pa1RvuE23AP1LvJI0SInOqP45HEc8UaCRS2btwHNjDFJmUHAJi9owsRAzPmNTGGRUMwU2SIuwC3qSWIYRjcFQIWkhws+NlClr5yrMJhXDe7vs5eJ/3iDByUWQCh0nCJrnh1BIKA5ZbkTWAtCxMIDI8uRAhaacGYYIRlDGeSYmWS21rA9v+fu/pHvW9Nymd9tqtK/KZqrkiByTU+KRc9ImN6RDfMKJII/kiTw7c+fFeXXevkcrTrlzSH7Bef8CH9eSTw==</latexit><latexit sha1_base64="82gm1PoIiHh8Y0IRmoT67kovAmg=">AAAB93icbVBNS8NAFNzUr1q/qh69LBbBU0mkoBeh6MVjBdMW2lA229e6dHcTdl+EEvobvOrJm3j153jwv5jEHLQ6p2HmPd68CWMpLLruh1NZWV1b36hu1ra2d3b36vsHXRslhoPPIxmZfsgsSKHBR4ES+rEBpkIJvXB2nfu9BzBWRPoO5zEEik21mAjOMJN8RS9pa1RvuE23AP1LvJI0SInOqP45HEc8UaCRS2btwHNjDFJmUHAJi9owsRAzPmNTGGRUMwU2SIuwC3qSWIYRjcFQIWkhws+NlClr5yrMJhXDe7vs5eJ/3iDByUWQCh0nCJrnh1BIKA5ZbkTWAtCxMIDI8uRAhaacGYYIRlDGeSYmWS21rA9v+fu/pHvW9Nymd9tqtK/KZqrkiByTU+KRc9ImN6RDfMKJII/kiTw7c+fFeXXevkcrTrlzSH7Bef8CH9eSTw==</latexit><latexit sha1_base64="82gm1PoIiHh8Y0IRmoT67kovAmg=">AAAB93icbVBNS8NAFNzUr1q/qh69LBbBU0mkoBeh6MVjBdMW2lA229e6dHcTdl+EEvobvOrJm3j153jwv5jEHLQ6p2HmPd68CWMpLLruh1NZWV1b36hu1ra2d3b36vsHXRslhoPPIxmZfsgsSKHBR4ES+rEBpkIJvXB2nfu9BzBWRPoO5zEEik21mAjOMJN8RS9pa1RvuE23AP1LvJI0SInOqP45HEc8UaCRS2btwHNjDFJmUHAJi9owsRAzPmNTGGRUMwU2SIuwC3qSWIYRjcFQIWkhws+NlClr5yrMJhXDe7vs5eJ/3iDByUWQCh0nCJrnh1BIKA5ZbkTWAtCxMIDI8uRAhaacGYYIRlDGeSYmWS21rA9v+fu/pHvW9Nymd9tqtK/KZqrkiByTU+KRc9ImN6RDfMKJII/kiTw7c+fFeXXevkcrTrlzSH7Bef8CH9eSTw==</latexit><latexit sha1_base64="82gm1PoIiHh8Y0IRmoT67kovAmg=">AAAB93icbVBNS8NAFNzUr1q/qh69LBbBU0mkoBeh6MVjBdMW2lA229e6dHcTdl+EEvobvOrJm3j153jwv5jEHLQ6p2HmPd68CWMpLLruh1NZWV1b36hu1ra2d3b36vsHXRslhoPPIxmZfsgsSKHBR4ES+rEBpkIJvXB2nfu9BzBWRPoO5zEEik21mAjOMJN8RS9pa1RvuE23AP1LvJI0SInOqP45HEc8UaCRS2btwHNjDFJmUHAJi9owsRAzPmNTGGRUMwU2SIuwC3qSWIYRjcFQIWkhws+NlClr5yrMJhXDe7vs5eJ/3iDByUWQCh0nCJrnh1BIKA5ZbkTWAtCxMIDI8uRAhaacGYYIRlDGeSYmWS21rA9v+fu/pHvW9Nymd9tqtK/KZqrkiByTU+KRc9ImN6RDfMKJII/kiTw7c+fFeXXevkcrTrlzSH7Bef8CH9eSTw==</latexit>

m̂ = 7<latexit sha1_base64="lduN9/goVdD6ilxNbRcWi4LlKLo=">AAAB/XicbVA9SwNBEN2LXzF+RS1tFoNgFe5EiI0QtLGMYGIgOcLcZpIs2ftgd04IR/BX2GplJ7b+Fgv/i3fnFZr4qsd7M8yb50VKGrLtT6u0srq2vlHerGxt7+zuVfcPOiaMtcC2CFWoux4YVDLANklS2I00gu8pvPem15l//4DayDC4o1mErg/jQI6kAEqlXn8ClPhzfskbg2rNrts5+DJxClJjBVqD6ld/GIrYx4CEAmN6jh2Rm4AmKRTOK/3YYARiCmPspTQAH42b5JHn/CQ2QCGPUHOpeC7i740EfGNmvpdO+kATs+hl4n9eL6bRhZvIIIoJA5EdIqkwP2SElmkXyIdSIxFkyZHLgAvQQIRachAiFeO0nErah7P4/TLpnNUdu+7cnteaV0UzZXbEjtkpc1iDNdkNa7E2EyxkT+yZvViP1qv1Zr3/jJasYueQ/YH18Q0VbZUf</latexit><latexit sha1_base64="lduN9/goVdD6ilxNbRcWi4LlKLo=">AAAB/XicbVA9SwNBEN2LXzF+RS1tFoNgFe5EiI0QtLGMYGIgOcLcZpIs2ftgd04IR/BX2GplJ7b+Fgv/i3fnFZr4qsd7M8yb50VKGrLtT6u0srq2vlHerGxt7+zuVfcPOiaMtcC2CFWoux4YVDLANklS2I00gu8pvPem15l//4DayDC4o1mErg/jQI6kAEqlXn8ClPhzfskbg2rNrts5+DJxClJjBVqD6ld/GIrYx4CEAmN6jh2Rm4AmKRTOK/3YYARiCmPspTQAH42b5JHn/CQ2QCGPUHOpeC7i740EfGNmvpdO+kATs+hl4n9eL6bRhZvIIIoJA5EdIqkwP2SElmkXyIdSIxFkyZHLgAvQQIRachAiFeO0nErah7P4/TLpnNUdu+7cnteaV0UzZXbEjtkpc1iDNdkNa7E2EyxkT+yZvViP1qv1Zr3/jJasYueQ/YH18Q0VbZUf</latexit><latexit sha1_base64="lduN9/goVdD6ilxNbRcWi4LlKLo=">AAAB/XicbVA9SwNBEN2LXzF+RS1tFoNgFe5EiI0QtLGMYGIgOcLcZpIs2ftgd04IR/BX2GplJ7b+Fgv/i3fnFZr4qsd7M8yb50VKGrLtT6u0srq2vlHerGxt7+zuVfcPOiaMtcC2CFWoux4YVDLANklS2I00gu8pvPem15l//4DayDC4o1mErg/jQI6kAEqlXn8ClPhzfskbg2rNrts5+DJxClJjBVqD6ld/GIrYx4CEAmN6jh2Rm4AmKRTOK/3YYARiCmPspTQAH42b5JHn/CQ2QCGPUHOpeC7i740EfGNmvpdO+kATs+hl4n9eL6bRhZvIIIoJA5EdIqkwP2SElmkXyIdSIxFkyZHLgAvQQIRachAiFeO0nErah7P4/TLpnNUdu+7cnteaV0UzZXbEjtkpc1iDNdkNa7E2EyxkT+yZvViP1qv1Zr3/jJasYueQ/YH18Q0VbZUf</latexit><latexit sha1_base64="lduN9/goVdD6ilxNbRcWi4LlKLo=">AAAB/XicbVA9SwNBEN2LXzF+RS1tFoNgFe5EiI0QtLGMYGIgOcLcZpIs2ftgd04IR/BX2GplJ7b+Fgv/i3fnFZr4qsd7M8yb50VKGrLtT6u0srq2vlHerGxt7+zuVfcPOiaMtcC2CFWoux4YVDLANklS2I00gu8pvPem15l//4DayDC4o1mErg/jQI6kAEqlXn8ClPhzfskbg2rNrts5+DJxClJjBVqD6ld/GIrYx4CEAmN6jh2Rm4AmKRTOK/3YYARiCmPspTQAH42b5JHn/CQ2QCGPUHOpeC7i740EfGNmvpdO+kATs+hl4n9eL6bRhZvIIIoJA5EdIqkwP2SElmkXyIdSIxFkyZHLgAvQQIRachAiFeO0nErah7P4/TLpnNUdu+7cnteaV0UzZXbEjtkpc1iDNdkNa7E2EyxkT+yZvViP1qv1Zr3/jJasYueQ/YH18Q0VbZUf</latexit>

n = 30, 80, 107<latexit sha1_base64="JvqvgbfEd3rT8bNoqpMyTBQxl30=">AAACBXicbVA9SwNBEN2LXzF+JVraLAbBIoQ7FZJGCNpYRjAfkBxhbzOJS/b2jt05JYTU/gpbrezE1t9h4X/x7rxCE18xPN6bYWaeF0ph0LY/rdzK6tr6Rn6zsLW9s7tXLO23TRBpDi0eyEB3PWZACgUtFCihG2pgvieh402uEr9zD9qIQN3iNATXZ2MlRoIzjKVBsaToBT2zK/1KPSmOXRsUy3bVTkGXiZORMsnQHBS/+sOARz4o5JIZ03PsEN0Z0yi4hHmhHxkIGZ+wMfRiqpgPxp2lp8/pcWQYBjQETYWkqQi/J2bMN2bqe3Gnz/DOLHqJ+J/Xi3BUd2dChRGC4skiFBLSRYZrEWcCdCg0ILLkcqBCUc40QwQtKOM8FqM4pEKch7P4/TJpn1Ydu+rcnJcbl1kyeXJIjsgJcUiNNMg1aZIW4eSBPJFn8mI9Wq/Wm/X+05qzspkD8gfWxzd8U5WQ</latexit><latexit sha1_base64="JvqvgbfEd3rT8bNoqpMyTBQxl30=">AAACBXicbVA9SwNBEN2LXzF+JVraLAbBIoQ7FZJGCNpYRjAfkBxhbzOJS/b2jt05JYTU/gpbrezE1t9h4X/x7rxCE18xPN6bYWaeF0ph0LY/rdzK6tr6Rn6zsLW9s7tXLO23TRBpDi0eyEB3PWZACgUtFCihG2pgvieh402uEr9zD9qIQN3iNATXZ2MlRoIzjKVBsaToBT2zK/1KPSmOXRsUy3bVTkGXiZORMsnQHBS/+sOARz4o5JIZ03PsEN0Z0yi4hHmhHxkIGZ+wMfRiqpgPxp2lp8/pcWQYBjQETYWkqQi/J2bMN2bqe3Gnz/DOLHqJ+J/Xi3BUd2dChRGC4skiFBLSRYZrEWcCdCg0ILLkcqBCUc40QwQtKOM8FqM4pEKch7P4/TJpn1Ydu+rcnJcbl1kyeXJIjsgJcUiNNMg1aZIW4eSBPJFn8mI9Wq/Wm/X+05qzspkD8gfWxzd8U5WQ</latexit><latexit sha1_base64="JvqvgbfEd3rT8bNoqpMyTBQxl30=">AAACBXicbVA9SwNBEN2LXzF+JVraLAbBIoQ7FZJGCNpYRjAfkBxhbzOJS/b2jt05JYTU/gpbrezE1t9h4X/x7rxCE18xPN6bYWaeF0ph0LY/rdzK6tr6Rn6zsLW9s7tXLO23TRBpDi0eyEB3PWZACgUtFCihG2pgvieh402uEr9zD9qIQN3iNATXZ2MlRoIzjKVBsaToBT2zK/1KPSmOXRsUy3bVTkGXiZORMsnQHBS/+sOARz4o5JIZ03PsEN0Z0yi4hHmhHxkIGZ+wMfRiqpgPxp2lp8/pcWQYBjQETYWkqQi/J2bMN2bqe3Gnz/DOLHqJ+J/Xi3BUd2dChRGC4skiFBLSRYZrEWcCdCg0ILLkcqBCUc40QwQtKOM8FqM4pEKch7P4/TJpn1Ydu+rcnJcbl1kyeXJIjsgJcUiNNMg1aZIW4eSBPJFn8mI9Wq/Wm/X+05qzspkD8gfWxzd8U5WQ</latexit><latexit sha1_base64="JvqvgbfEd3rT8bNoqpMyTBQxl30=">AAACBXicbVA9SwNBEN2LXzF+JVraLAbBIoQ7FZJGCNpYRjAfkBxhbzOJS/b2jt05JYTU/gpbrezE1t9h4X/x7rxCE18xPN6bYWaeF0ph0LY/rdzK6tr6Rn6zsLW9s7tXLO23TRBpDi0eyEB3PWZACgUtFCihG2pgvieh402uEr9zD9qIQN3iNATXZ2MlRoIzjKVBsaToBT2zK/1KPSmOXRsUy3bVTkGXiZORMsnQHBS/+sOARz4o5JIZ03PsEN0Z0yi4hHmhHxkIGZ+wMfRiqpgPxp2lp8/pcWQYBjQETYWkqQi/J2bMN2bqe3Gnz/DOLHqJ+J/Xi3BUd2dChRGC4skiFBLSRYZrEWcCdCg0ILLkcqBCUc40QwQtKOM8FqM4pEKch7P4/TJpn1Ydu+rcnJcbl1kyeXJIjsgJcUiNNMg1aZIW4eSBPJFn8mI9Wq/Wm/X+05qzspkD8gfWxzd8U5WQ</latexit>

d = 1, 3<latexit sha1_base64="e7O8EIRbqVmEVhR6ToC3DZi5MlI=">AAAB+3icbVC7SgNBFJ2NrxhfUUubwSBYhLCrgjZC0MYygnlIEsLs5CYOmZldZu4KYclX2GplJ7Z+jIX/4u66hSae6nDOvdxzjx9KYdF1P53C0vLK6lpxvbSxubW9U97da9kgMhyaPJCB6fjMghQamihQQic0wJQvoe1PrlO//QjGikDf4TSEvmJjLUaCM0yk+yG9pF61Vz0dlCtuzc1AF4mXkwrJ0RiUv3rDgEcKNHLJrO16boj9mBkUXMKs1IsshIxP2Bi6CdVMge3HWeAZPYosw4CGYKiQNBPh90bMlLVT5SeTiuGDnfdS8T+vG+Hooh8LHUYImqeHUEjIDlluRNIE0KEwgMjS5ECFppwZhghGUMZ5IkZJNaWkD2/++0XSOql5bs27PavUr/JmiuSAHJJj4pFzUic3pEGahBNFnsgzeXFmzqvz5rz/jBacfGef/IHz8Q0OvpNS</latexit><latexit sha1_base64="e7O8EIRbqVmEVhR6ToC3DZi5MlI=">AAAB+3icbVC7SgNBFJ2NrxhfUUubwSBYhLCrgjZC0MYygnlIEsLs5CYOmZldZu4KYclX2GplJ7Z+jIX/4u66hSae6nDOvdxzjx9KYdF1P53C0vLK6lpxvbSxubW9U97da9kgMhyaPJCB6fjMghQamihQQic0wJQvoe1PrlO//QjGikDf4TSEvmJjLUaCM0yk+yG9pF61Vz0dlCtuzc1AF4mXkwrJ0RiUv3rDgEcKNHLJrO16boj9mBkUXMKs1IsshIxP2Bi6CdVMge3HWeAZPYosw4CGYKiQNBPh90bMlLVT5SeTiuGDnfdS8T+vG+Hooh8LHUYImqeHUEjIDlluRNIE0KEwgMjS5ECFppwZhghGUMZ5IkZJNaWkD2/++0XSOql5bs27PavUr/JmiuSAHJJj4pFzUic3pEGahBNFnsgzeXFmzqvz5rz/jBacfGef/IHz8Q0OvpNS</latexit><latexit sha1_base64="e7O8EIRbqVmEVhR6ToC3DZi5MlI=">AAAB+3icbVC7SgNBFJ2NrxhfUUubwSBYhLCrgjZC0MYygnlIEsLs5CYOmZldZu4KYclX2GplJ7Z+jIX/4u66hSae6nDOvdxzjx9KYdF1P53C0vLK6lpxvbSxubW9U97da9kgMhyaPJCB6fjMghQamihQQic0wJQvoe1PrlO//QjGikDf4TSEvmJjLUaCM0yk+yG9pF61Vz0dlCtuzc1AF4mXkwrJ0RiUv3rDgEcKNHLJrO16boj9mBkUXMKs1IsshIxP2Bi6CdVMge3HWeAZPYosw4CGYKiQNBPh90bMlLVT5SeTiuGDnfdS8T+vG+Hooh8LHUYImqeHUEjIDlluRNIE0KEwgMjS5ECFppwZhghGUMZ5IkZJNaWkD2/++0XSOql5bs27PavUr/JmiuSAHJJj4pFzUic3pEGahBNFnsgzeXFmzqvz5rz/jBacfGef/IHz8Q0OvpNS</latexit><latexit sha1_base64="e7O8EIRbqVmEVhR6ToC3DZi5MlI=">AAAB+3icbVC7SgNBFJ2NrxhfUUubwSBYhLCrgjZC0MYygnlIEsLs5CYOmZldZu4KYclX2GplJ7Z+jIX/4u66hSae6nDOvdxzjx9KYdF1P53C0vLK6lpxvbSxubW9U97da9kgMhyaPJCB6fjMghQamihQQic0wJQvoe1PrlO//QjGikDf4TSEvmJjLUaCM0yk+yG9pF61Vz0dlCtuzc1AF4mXkwrJ0RiUv3rDgEcKNHLJrO16boj9mBkUXMKs1IsshIxP2Bi6CdVMge3HWeAZPYosw4CGYKiQNBPh90bMlLVT5SeTiuGDnfdS8T+vG+Hooh8LHUYImqeHUEjIDlluRNIE0KEwgMjS5ECFppwZhghGUMZ5IkZJNaWkD2/++0XSOql5bs27PavUr/JmiuSAHJJj4pFzUic3pEGahBNFnsgzeXFmzqvz5rz/jBacfGef/IHz8Q0OvpNS</latexit>

log10 i ⇠ N (�5, 1)<latexit sha1_base64="L9R3ZhwEZG4ovi9M6VUfYABr6YQ=">AAACHnicbVC7TsNAEDzzJrwClDQnIiRAENkIBCWChgoFiRCkOLLWxxJOOdunuzVSZPkf+AS+ghYqOkQLBf+CHVLwmmo0s6udnVAracl1352R0bHxicmp6crM7Nz8QnVx6cImqRHYFIlKzGUIFpWMsUmSFF5qgxCFClth77j0W7dorEzic+pr7ETQjeW1FECFFFQ3fZV0g8xz88zvgdYQyJz7Vkbcj4BuBKjsNF/f3tvyNoJqza27A/C/xBuSGhuiEVQ//KtEpBHGJBRY2/ZcTZ0MDEmhMK/4qUUNogddbBc0hghtJxv8lPO11AIlXKPhUvGBiN83Mois7UdhMVkGtb+9UvzPa6d0fdDJZKxTwliUh0gqHByywsiiLORX0iARlMmRy5gLMECERnIQohDTor1K0Yf3+/u/5GKn7rl172y3dng0bGaKrbBVts48ts8O2QlrsCYT7I49sEf25Nw7z86L8/o1OuIMd5bZDzhvn8kbocQ=</latexit><latexit sha1_base64="L9R3ZhwEZG4ovi9M6VUfYABr6YQ=">AAACHnicbVC7TsNAEDzzJrwClDQnIiRAENkIBCWChgoFiRCkOLLWxxJOOdunuzVSZPkf+AS+ghYqOkQLBf+CHVLwmmo0s6udnVAracl1352R0bHxicmp6crM7Nz8QnVx6cImqRHYFIlKzGUIFpWMsUmSFF5qgxCFClth77j0W7dorEzic+pr7ETQjeW1FECFFFQ3fZV0g8xz88zvgdYQyJz7Vkbcj4BuBKjsNF/f3tvyNoJqza27A/C/xBuSGhuiEVQ//KtEpBHGJBRY2/ZcTZ0MDEmhMK/4qUUNogddbBc0hghtJxv8lPO11AIlXKPhUvGBiN83Mois7UdhMVkGtb+9UvzPa6d0fdDJZKxTwliUh0gqHByywsiiLORX0iARlMmRy5gLMECERnIQohDTor1K0Yf3+/u/5GKn7rl172y3dng0bGaKrbBVts48ts8O2QlrsCYT7I49sEf25Nw7z86L8/o1OuIMd5bZDzhvn8kbocQ=</latexit><latexit sha1_base64="L9R3ZhwEZG4ovi9M6VUfYABr6YQ=">AAACHnicbVC7TsNAEDzzJrwClDQnIiRAENkIBCWChgoFiRCkOLLWxxJOOdunuzVSZPkf+AS+ghYqOkQLBf+CHVLwmmo0s6udnVAracl1352R0bHxicmp6crM7Nz8QnVx6cImqRHYFIlKzGUIFpWMsUmSFF5qgxCFClth77j0W7dorEzic+pr7ETQjeW1FECFFFQ3fZV0g8xz88zvgdYQyJz7Vkbcj4BuBKjsNF/f3tvyNoJqza27A/C/xBuSGhuiEVQ//KtEpBHGJBRY2/ZcTZ0MDEmhMK/4qUUNogddbBc0hghtJxv8lPO11AIlXKPhUvGBiN83Mois7UdhMVkGtb+9UvzPa6d0fdDJZKxTwliUh0gqHByywsiiLORX0iARlMmRy5gLMECERnIQohDTor1K0Yf3+/u/5GKn7rl172y3dng0bGaKrbBVts48ts8O2QlrsCYT7I49sEf25Nw7z86L8/o1OuIMd5bZDzhvn8kbocQ=</latexit><latexit sha1_base64="L9R3ZhwEZG4ovi9M6VUfYABr6YQ=">AAACHnicbVC7TsNAEDzzJrwClDQnIiRAENkIBCWChgoFiRCkOLLWxxJOOdunuzVSZPkf+AS+ghYqOkQLBf+CHVLwmmo0s6udnVAracl1352R0bHxicmp6crM7Nz8QnVx6cImqRHYFIlKzGUIFpWMsUmSFF5qgxCFClth77j0W7dorEzic+pr7ETQjeW1FECFFFQ3fZV0g8xz88zvgdYQyJz7Vkbcj4BuBKjsNF/f3tvyNoJqza27A/C/xBuSGhuiEVQ//KtEpBHGJBRY2/ZcTZ0MDEmhMK/4qUUNogddbBc0hghtJxv8lPO11AIlXKPhUvGBiN83Mois7UdhMVkGtb+9UvzPa6d0fdDJZKxTwliUh0gqHByywsiiLORX0iARlMmRy5gLMECERnIQohDTor1K0Yf3+/u/5GKn7rl172y3dng0bGaKrbBVts48ts8O2QlrsCYT7I49sEf25Nw7z86L8/o1OuIMd5bZDzhvn8kbocQ=</latexit>

i<latexit sha1_base64="wa9Efg+EX7aDylR7wZJWmebZVVk=">AAAB+nicbVC7TsNAEDyHVwivACXNiQiJKrIREpQRNJRBIg8psaL1ZRNOOdunuzVSFPITtFDRIVp+hoJ/wTYuIGGq0cyudnYCraQl1/10Siura+sb5c3K1vbO7l51/6Bt48QIbIlYxaYbgEUlI2yRJIVdbRDCQGEnmFxnfucBjZVxdEdTjX4I40iOpABKpW5/AlrDQA6qNbfu5uDLxCtIjRVoDqpf/WEskhAjEgqs7XmuJn8GhqRQOK/0E4saxATG2EtpBCFaf5bnnfOTxALFXKPhUvFcxN8bMwitnYZBOhkC3dtFLxP/83oJjS79mYx0QhiJ7BBJhfkhK4xMi0A+lAaJIEuOXEZcgAEiNJKDEKmYpM1U0j68xe+XSfus7rl17/a81rgqmimzI3bMTpnHLliD3bAmazHBFHtiz+zFeXRenTfn/We05BQ7h+wPnI9v0vqUgA==</latexit><latexit sha1_base64="wa9Efg+EX7aDylR7wZJWmebZVVk=">AAAB+nicbVC7TsNAEDyHVwivACXNiQiJKrIREpQRNJRBIg8psaL1ZRNOOdunuzVSFPITtFDRIVp+hoJ/wTYuIGGq0cyudnYCraQl1/10Siura+sb5c3K1vbO7l51/6Bt48QIbIlYxaYbgEUlI2yRJIVdbRDCQGEnmFxnfucBjZVxdEdTjX4I40iOpABKpW5/AlrDQA6qNbfu5uDLxCtIjRVoDqpf/WEskhAjEgqs7XmuJn8GhqRQOK/0E4saxATG2EtpBCFaf5bnnfOTxALFXKPhUvFcxN8bMwitnYZBOhkC3dtFLxP/83oJjS79mYx0QhiJ7BBJhfkhK4xMi0A+lAaJIEuOXEZcgAEiNJKDEKmYpM1U0j68xe+XSfus7rl17/a81rgqmimzI3bMTpnHLliD3bAmazHBFHtiz+zFeXRenTfn/We05BQ7h+wPnI9v0vqUgA==</latexit><latexit sha1_base64="wa9Efg+EX7aDylR7wZJWmebZVVk=">AAAB+nicbVC7TsNAEDyHVwivACXNiQiJKrIREpQRNJRBIg8psaL1ZRNOOdunuzVSFPITtFDRIVp+hoJ/wTYuIGGq0cyudnYCraQl1/10Siura+sb5c3K1vbO7l51/6Bt48QIbIlYxaYbgEUlI2yRJIVdbRDCQGEnmFxnfucBjZVxdEdTjX4I40iOpABKpW5/AlrDQA6qNbfu5uDLxCtIjRVoDqpf/WEskhAjEgqs7XmuJn8GhqRQOK/0E4saxATG2EtpBCFaf5bnnfOTxALFXKPhUvFcxN8bMwitnYZBOhkC3dtFLxP/83oJjS79mYx0QhiJ7BBJhfkhK4xMi0A+lAaJIEuOXEZcgAEiNJKDEKmYpM1U0j68xe+XSfus7rl17/a81rgqmimzI3bMTpnHLliD3bAmazHBFHtiz+zFeXRenTfn/We05BQ7h+wPnI9v0vqUgA==</latexit><latexit sha1_base64="wa9Efg+EX7aDylR7wZJWmebZVVk=">AAAB+nicbVC7TsNAEDyHVwivACXNiQiJKrIREpQRNJRBIg8psaL1ZRNOOdunuzVSFPITtFDRIVp+hoJ/wTYuIGGq0cyudnYCraQl1/10Siura+sb5c3K1vbO7l51/6Bt48QIbIlYxaYbgEUlI2yRJIVdbRDCQGEnmFxnfucBjZVxdEdTjX4I40iOpABKpW5/AlrDQA6qNbfu5uDLxCtIjRVoDqpf/WEskhAjEgqs7XmuJn8GhqRQOK/0E4saxATG2EtpBCFaf5bnnfOTxALFXKPhUvFcxN8bMwitnYZBOhkC3dtFLxP/83oJjS79mYx0QhiJ7BBJhfkhK4xMi0A+lAaJIEuOXEZcgAEiNJKDEKmYpM1U0j68xe+XSfus7rl17/a81rgqmimzI3bMTpnHLliD3bAmazHBFHtiz+zFeXRenTfn/We05BQ7h+wPnI9v0vqUgA==</latexit>

! 2 ⌦<latexit sha1_base64="cr8ghsIcj3/XjmzvAewaNSz7vqk=">AAACBXicbVC7TsNAEDzzDOGVQElzIkKiimyEBGUEDR1BIg8ptqL1ZRNOOT90twZFUWq+ghYqOkTLd1DwL9jGBSRMNZrZ1c6OHytpyLY/raXlldW19dJGeXNre2e3Ut1rmyjRAlsiUpHu+mBQyRBbJElhN9YIga+w448vM79zj9rIKLylSYxeAKNQDqUASqV+pepGAY6AuzLk7nVG+5WaXbdz8EXiFKTGCjT7lS93EIkkwJCEAmN6jh2TNwVNUiicld3EYAxiDCPspTSEAI03zaPP+FFigCIeo+ZS8VzE3xtTCIyZBH46GQDdmXkvE//zegkNz72pDOOEMBTZIZIK80NGaJl2gnwgNRJBlhx5WoAADUSoJQchUjFJSyqnfTjz3y+S9kndsevOzWmtcVE0U2IH7JAdM4edsQa7Yk3WYoI9sCf2zF6sR+vVerPef0aXrGJnn/2B9fENTe2X9w==</latexit><latexit sha1_base64="cr8ghsIcj3/XjmzvAewaNSz7vqk=">AAACBXicbVC7TsNAEDzzDOGVQElzIkKiimyEBGUEDR1BIg8ptqL1ZRNOOT90twZFUWq+ghYqOkTLd1DwL9jGBSRMNZrZ1c6OHytpyLY/raXlldW19dJGeXNre2e3Ut1rmyjRAlsiUpHu+mBQyRBbJElhN9YIga+w448vM79zj9rIKLylSYxeAKNQDqUASqV+pepGAY6AuzLk7nVG+5WaXbdz8EXiFKTGCjT7lS93EIkkwJCEAmN6jh2TNwVNUiicld3EYAxiDCPspTSEAI03zaPP+FFigCIeo+ZS8VzE3xtTCIyZBH46GQDdmXkvE//zegkNz72pDOOEMBTZIZIK80NGaJl2gnwgNRJBlhx5WoAADUSoJQchUjFJSyqnfTjz3y+S9kndsevOzWmtcVE0U2IH7JAdM4edsQa7Yk3WYoI9sCf2zF6sR+vVerPef0aXrGJnn/2B9fENTe2X9w==</latexit><latexit sha1_base64="cr8ghsIcj3/XjmzvAewaNSz7vqk=">AAACBXicbVC7TsNAEDzzDOGVQElzIkKiimyEBGUEDR1BIg8ptqL1ZRNOOT90twZFUWq+ghYqOkTLd1DwL9jGBSRMNZrZ1c6OHytpyLY/raXlldW19dJGeXNre2e3Ut1rmyjRAlsiUpHu+mBQyRBbJElhN9YIga+w448vM79zj9rIKLylSYxeAKNQDqUASqV+pepGAY6AuzLk7nVG+5WaXbdz8EXiFKTGCjT7lS93EIkkwJCEAmN6jh2TNwVNUiicld3EYAxiDCPspTSEAI03zaPP+FFigCIeo+ZS8VzE3xtTCIyZBH46GQDdmXkvE//zegkNz72pDOOEMBTZIZIK80NGaJl2gnwgNRJBlhx5WoAADUSoJQchUjFJSyqnfTjz3y+S9kndsevOzWmtcVE0U2IH7JAdM4edsQa7Yk3WYoI9sCf2zF6sR+vVerPef0aXrGJnn/2B9fENTe2X9w==</latexit><latexit sha1_base64="cr8ghsIcj3/XjmzvAewaNSz7vqk=">AAACBXicbVC7TsNAEDzzDOGVQElzIkKiimyEBGUEDR1BIg8ptqL1ZRNOOT90twZFUWq+ghYqOkTLd1DwL9jGBSRMNZrZ1c6OHytpyLY/raXlldW19dJGeXNre2e3Ut1rmyjRAlsiUpHu+mBQyRBbJElhN9YIga+w448vM79zj9rIKLylSYxeAKNQDqUASqV+pepGAY6AuzLk7nVG+5WaXbdz8EXiFKTGCjT7lS93EIkkwJCEAmN6jh2TNwVNUiicld3EYAxiDCPspTSEAI03zaPP+FFigCIeo+ZS8VzE3xtTCIyZBH46GQDdmXkvE//zegkNz72pDOOEMBTZIZIK80NGaJl2gnwgNRJBlhx5WoAADUSoJQchUjFJSyqnfTjz3y+S9kndsevOzWmtcVE0U2IH7JAdM4edsQa7Yk3WYoI9sCf2zF6sR+vVerPef0aXrGJnn/2B9fENTe2X9w==</latexit>

Fracture E['] (mm2s�1) E['̂] (mm2s�1)

8 ⇠ 17.445 ⇠ 17.486

33 ⇠ 8.838 ⇠ 8.796

62 ⇠ 9.165 ⇠ 9.055

105 ⇠ 8.006 ⇠ 7.932

Fracture �['] (mm2s�1) �['̂] (mm2s�1)

8 ⇠ 9.843 ⇠ 9.521

33 ⇠ 4.719 ⇠ 4.533

62 ⇠ 5.579 ⇠ 5.399

105 ⇠ 5.213 ⇠ 5.093<latexit sha1_base64="edFvzWT6Nq1a8qvjDeBCvOFVSBc=">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</latexit><latexit sha1_base64="edFvzWT6Nq1a8qvjDeBCvOFVSBc=">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</latexit><latexit sha1_base64="edFvzWT6Nq1a8qvjDeBCvOFVSBc=">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</latexit><latexit sha1_base64="edFvzWT6Nq1a8qvjDeBCvOFVSBc=">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</latexit>

ea (mm2s�1) er ew

Mean Val. ⇠ 0.924 ⇠ 0.110 ⇠ 0.018

97th perc. ⇠ 3.310 ⇠ 0.385 ⇠ 0.069

max ⇠ 20.994 ⇠ 8.980 ⇠ 0.430<latexit sha1_base64="7GVJxA6LdB4xag8y7JOzOO8O2yc=">AAADsXicdVLbbtNAEHViLm24pfDIy4oYVCSw7CTQ5K2CF3hAKpekRXESxptpsop3be2uKZHrP+NHeOBfWAe3SdN0bGnHZ2bOWc9MmERMac/7U6nat27fubuzW7t3/8HDR/W9x30Vp5Jij8ZRLE9CUBgxgT3NdIQniUTgYYTH4fx9ET/+iVKxWHzTiwSHHKaCnTIK2kDjvcrvQKLAMxpzDmKSBSAlLJSWqOksz3y3ndeCEKdMZBrCNAKZZ+f03LzFY2KzQrlGyAviBGEcTdSCmyPDfAzBq33OR001yl77+UtnS4rcBp45QXDBWx6fEATpQ+SSq1YUK8aJ53abbWft0/c9ZyPD8zvOZeFKwekejDI9yx2SoKQ3CbTcVsG4Ymt13ji7a+Rvu846acDhl0NuIGua63bNdQ1QIh2321nSXxK2W96qDWsctQDNlC5GMa43jPbSyHXHL52GVdrRuP43mMQ05Sg0jUCpge8lepiB1IxGaMaZKkyAzmGKA+MK4KiG2XLNcvI8VaDjok+ERWQJ4npFBlwVYzSZHPRMbcYKcFtskOrTzjBjIkk1CloIaRbhUkhRycz+Ipkws5IaipsjYYJQkKA1SkaAUgOmZqFrph/+5t9fd/pN1zcr8rndOHxXdmbHemo9s/Yt3zqwDq0P1pHVs2i1Uf1Y/VL9arfs7/YPO/yfWq2UNU+sK2bP/wGyBwx0</latexit><latexit sha1_base64="7GVJxA6LdB4xag8y7JOzOO8O2yc=">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</latexit><latexit sha1_base64="7GVJxA6LdB4xag8y7JOzOO8O2yc=">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</latexit><latexit sha1_base64="7GVJxA6LdB4xag8y7JOzOO8O2yc=">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</latexit>

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