Full waveform inversion for high resolution seismic imaging: HPC issues on recent applications and ongoing research.

Transcription

1 Full waveform inversion for high resolution seismic imaging: HPC issues on recent applications and ongoing research. L. Métivier 1,2, R. Brossier 2,Q.Mérigot 3, A. Minuissi 4, S. Operto 4,E.Oudet 1,J.Virieux 2 1 LJK, CNRS & Université Grenoble Alpes 2 ISTerre, Université Grenoble Alpes 3 CEREMADE, CNRS & Université Paris-Dauphine 4 Géoazur, CNRS & Université Nice Sophia Antipolis ludovic.metivier@ujf-grenoble.fr 4ièmes Journées Scientifiques Equip@Meso : Sciences de l Univers, Toulouse, CALMIP, 26 et 27 Novembre 2015 SEISCOPE L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

2 Outline 1 Introduction: principle of full waveform inversion 2 Example of application: 3D acoustic frequency-domain FWI 3 Reducing the sensitivity to the initial model accuracy using an optimal transport distance L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

3 Seismic imaging of the subsurface We aim at reconstructing the Earth subsurface mechanical properties from surface measurements of seismic waves Example of a seismic survey in a marine environment. The vessel tows cables containing series of equidistant hydrophones. Air guns are shot at regular interval. The hydrophones record the energy coming back from the subsurface. This signal is interpreted to reconstruction the subsurface mechanical properties. L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

4 Seismic imaging of the Earth Illustration: numerical modeling of seismic wave propagation L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

5 Seismic imaging of the Earth Illustration: numerical modeling of seismic wave propagation L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

6 Seismic imaging of the Earth Illustration: numerical modeling of seismic wave propagation L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

7 Seismic imaging of the Earth Illustration: numerical modeling of seismic wave propagation L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

8 Seismic imaging of the Earth Illustration: numerical modeling of seismic wave propagation L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

9 Seismic imaging of the Earth Illustration: numerical modeling of seismic wave propagation L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

10 Seismic imaging of the Earth Illustration: numerical modeling of seismic wave propagation L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

11 Full waveform inversion : principle Depth (km) Distance (km) Real'Earth' Vp (m/s)?( Measurement' 0 Trace Number Time (s) Observed'data' 2.5 L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

12 Full waveform inversion : principle 0.1 Distance (km) Distance (km) Depth (km) Real'Earth' Depth (km) IniMal'model' Vp (m/s) Vp (m/s)?( Measurement' 0 Trace Number Time (s) Observed'data' 2.5 L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

13 Full waveform inversion : principle 0.1 Distance (km) Distance (km) Depth (km) Real'Earth' Depth (km) IniMal'model' Vp (m/s) Vp (m/s)?( Measurement' Forward' Problem' 0 Trace Number Trace Number Time (s) Observed'data' Time (s) Calculated'data' L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

14 Full waveform inversion : principle 0.1 Distance (km) Distance (km) Depth (km) Real'Earth' Depth (km) IniMal'model' Vp (m/s) Vp (m/s)?( Measurement' Forward' Problem' 0 Trace Number Trace Number Time (s) Observed'data' Time (s) 1.0 Misfit'funcMon' ' 1.5 Δd = d obs d2.0 cal Calculated'data' L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

15 Full waveform inversion : principle 0.1 Distance (km) Distance (km) Depth (km) Real'Earth' Depth (km) Updated'model' Vp (m/s) Vp (m/s)?( Measurement' Inverse' Forward' Problem' Problem' 0 Trace Number Trace Number Time (s) Observed'data' Time (s) 1.0 Misfit'funcMon' ' 1.5 Δd = d obs d2.0 cal Calculated'data' L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

16 Full waveform inversion : principle 0.1 Distance (km) Distance (km) Depth (km) Real'Earth' Depth (km) Final'model' Vp (m/s) Vp (m/s)?( Measurement' Inverse' Forward' Problem' Problem' 0 Trace Number Trace Number Time (s) Observed'data' Time (s) 1.0 Misfit'funcMon' ' 1.5 Δd = d obs d2.0 cal Calculated'data' L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

17 Time-domain full waveform inversion : mathematical settings Nonlinear optimization problem Minimization of min c f (c) = 1 2 XN s s=1 Z T 0 d cal [s, c](t) d s obs(t) 2 dt (1) where N s is the number of seismic sources and d cal [s, c] iscomputedas d cal [s, c] =Rp[s, c] (2) where R is a restriction operator at the receivers location and p[s, c] isthe time-domain acoustic wavefield satisfying 1 c ttp p = s (3) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

18 Time-domain full waveform inversion : mathematical settings Quasi-Newton technique The minimization is performed following quasi-newton methods: from c 0 build c k such that c k+1 = c k k Q(c k )rf (c k ), (4) Gradient: adjoint state method The gradient is rf (c k )= Ns 2 X ck 3(x) s=1 Z T tt p[s, c k ](x, t) p [s, c k ](x, t)dt (5) where p[s, c](x, t) is the adjoint wavefield solution of 1 c tt p p = R (d cal [s, c] d obs ) (6) Hessian: l-bfgs formula Q(c k )isanapproximationoftheinverse Hessian operator, for instance using the l-bfgs formula (Nocedal, 1980) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

19 Frequency-domain full waveform inversion : mathematical settings Exploiting time redudancy We can work with few discrete temporal frequencies! i of the signal Now d s,! i cal min c f (c) = 1 2 (c) iscomputedas XN s s=1 N freq X! i =1 d cal [s, c](! i ) d s obs(! i ) 2 (7) d cal [s, c](! i )=Rp[s, c](x,! i ) (8) where p[s, c](x,! i )isthefrequency-domain acoustic wavefield satisfying the Helmholtz equation! 2 i c 2 (x) p p = s (9) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

20 Frequency-domain full waveform inversion : mathematical settings Frequency-domain gradient The gradient of the misfit function rf (c k )becomes rf (c k )= 2!2 i c 3 (x) N freq X! i =1 p[s, c](x,! i ) p [s, c](x,! i ) (10) where p[s, c](x,! i )istheadjoint wavefield solution of! 2 i c 2 (x) p p = R (d cal [s, c k ] d s obs) (11) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

21 Outline 1 Introduction: principle of full waveform inversion 2 Example of application: 3D acoustic frequency-domain FWI 3 Reducing the sensitivity to the initial model accuracy using an optimal transport distance L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

22 Case study description: The Valhall oil field (Operto et al., 2015) 4 Y(km) Y (km) X(km) X(km) Depth(km) Shallow-water environment (70m of water). Gas in the overburden, forming locally a gas cloud between 1km and 1.5km depth. Over-pressured, under-saturated Upper Cretaceous chalk reservoir at 2.5km depth (Barkved et al., 2010). Anisotropy as high as 16% hydrophones, 49,954 shots (50m spacing) processed in a reciprocal way during seismic modeling. Covered area: 16km 9km =145km 2 ;Maximumdepthofinvestigation4.5km. L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

23 FWI experimental setup and starting model Ten successive mono-frequency inversions with grid refinement. h(m) f (Hz) N itmax Table: h(m): grid interval. f (Hz): inverted frequencies. N itmax :maximumnumberofiterations. Seismic modeling with a visco-acoustic VTI FDFD method (Operto et al., 2014) Preconditioned steepest-descent with SEISCOPE optimization toolbox (Métivier and Brossier, 2015). Free-surface boundary condition on top of grid. Multiples & ghosts generated by modeling. Mono-parameter inversion for V 0.,, Q P =200, (Gardner law) kept fixed. Source signature estimation alternated with velocity-model update at each FWI iteration. FWI performed with MUMPS full rank and block low rank (" =10 5, 10 4, 10 3 ). Initial model built by VTI reflection traveltime tomography (courtesy of BP). L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

24 FWI results (Operto et al., 2015) X (km) Y (km) z=175m Depth(km) Y(km) x=5575m m/s m/s X (km) z=500m Depth(km) x=6250m m/s m/s X (km) z=1000m Depth(km) x=6475m m/s m/s L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

25 FWI results (Operto et al., 2015) X (km) Y (km) z=175m Depth(km) Y(km) x=5575m m/s m/s X (km) z=500m Depth(km) x=6250m m/s m/s X (km) z=1000m Depth(km) x=6475m m/s m/s L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

26 HPC strategy Each inversion iteration requires to solve more than wave equation problems with the same operator and di erent right hand sides (sources) The total number of inversion iterations reaches 200 which makes wave propagation problems solution for one inversion We would like to rely on e problems cient strategies for multiple right hand sides We should use a direct solver L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

27 HPC strategy Each inversion iteration requires to solve more than wave equation problems with the same operator and di erent right hand sides (sources) The total number of inversion iterations reaches 200 which makes wave propagation problems solution for one inversion We would like to rely on e problems cient strategies for multiple right hand sides We should use a direct solver However, we need to control the memory requirement of such a strategy. This is performed through an appropriate choice of 1 the PDE to be solved 2 the discretization scheme which is used 3 the type of direct solver which is used L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

28 VTI acoustic wave equation for FDFD modeling (Operto et al., 2014) Step 1: eliminateparticlevelocitiesfromthe1st-order system (e.g. Duveneck and Bakker, 2011) 8 <! 2 p apple 0 h +(1+2 )(X + Y) p h + p 1+2 Z p v =!2 s h s, apple 0 :! 2 p apple v + p 1+2 (X + Y) p 0 h + Z p v =!2 s v s, apple 0 with X x b@ x, Y b@ỹ and Z z b@ z. Step 2: Explicitexpressionofp v as a function of p h (first equation) p v = 1 p 1+2 p h + 2( )apple 0! 2p 1+2 (X + Y) p h + s v 1 p 1+2 s h «s.! 2 Step 3: Eliminatep v from the system for a 4 th -order equation for p h 2 3 Nearly elliptic z } {! 2 6 +(1+2 )(X + Y)+ p Z p 7 4 apple p h + Anelliptic z } { 2 p 1+2 Z apple 0( ) p 1+2 (X + Y) p h = b with b =!4 s h s! apple 2 p Z s v p s h s. Step 4: p =1/3(2p h + p v ). L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

29 A compact mixed-grid stencil (Operto et al., 2007; Brossier et al., 2010; Operto et al., 2014) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

30 MUMPS direct solver with block low rank compression (Weisbecker, 2013; Amestoy et al., 2015b,a) Computer nodes: two 2.5GHz Intel Xeon IvyBridge E5-2670v2 CPUs. 10 cores/cpu. 64GB memory/node. Frequency band (Hz) Grid dimensions n pml #u #n #MPI #th #rhs x 130 x x 181 x x 258 x Table: #u(10 6 ): number of unknowns. #n: numberofnodes.#mpi :numberofmpiprocess.#th: numberofthreadspermpiprocess. F(Hz)/h(m) " M LU (Gb) T LU (s) T s (s) T ms (s) T g (mn) #g FR 61(1.0) 96.2(1.0) Hz/70m (1.7) 51.2(1.9) (2.0) 47.6(2.0) (2.5) 42.8(2.2) FR 206(1.0) 381(1.0) Hz/50m (1.9) 168(2.26) (2.3) 145(2.6) (2.9) 129(3.0) FR 712(1.0) 1298(1.0) Hz/35m (2.2) 541(2.4) (2.7) 488(2.7) (3.4) 420(3.1) Table: M LU (Gb): Memory for LU factors (Gbytes). T LU (s): Elapsed time for LU. Ts (s): Elapsed time for 1 solve. Tms (s): Elapsed time for 4604 solves. Tg (mn): time for 1 gradient. #g: Number of gradients. L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

31 Outline 1 Introduction: principle of full waveform inversion 2 Example of application: 3D acoustic frequency-domain FWI 3 Reducing the sensitivity to the initial model accuracy using an optimal transport distance L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

32 Sensitivity to the initial model Numerical optimization point of view Local optimization techniques for nonlinear minimization problem only converge to the nearest local minimum (Nocedal and Wright, 2006) Geophysics point of view Sensitivity to the large scale variation of the velocity only in the Fresnel zone (Woodward, 1992; Operto et al., 2006) Phase matching with a L 2 norm only if the initial model predicts the data within half a phase: this is a cycle skipping issue (Jannane et al., 1989) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

33 Di erent approaches to reduce the sensitivity to the initial model accuracy Hierarchical (or multi-scale) approaches Frequency and illumination through time windowing and o set selection (Pratt, 1999; Bunks et al., 1995; Sirgue and Pratt, 2004; Shipp and Singh, 2002; Wang and Rao, 2009; Brossier et al., 2009) Hybridization with tomography methods Increase the sensitivity to first-arrival time detrimental to the resolution (Luo and Schuster, 1991; Dahlen et al., 2000; Montelli et al., 2004; Tromp et al., 2005; Nolet, 2008) Extended domain strategies Kinematic control through the maximization of the energy refocusing in a fictitious dimension introduced in the imaging condition (o set, angle, time-lag) (Symes and Kern, 1994; Sava and Biondi, 2004a,b; Sava and Fomel, 2006; Symes, 2008; Biondi and Almomin, 2013) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

34 Di erent approaches to reduce the sensitivity to the initial model accuracy Hierarchical (or multi-scale) approaches Frequency and illumination through time windowing and o set selection (Pratt, 1999; Bunks et al., 1995; Sirgue and Pratt, 2004; Shipp and Singh, 2002; Wang and Rao, 2009; Brossier et al., 2009) Hybridization with tomography methods Increase the sensitivity to first-arrival time detrimental to the resolution (Luo and Schuster, 1991; Dahlen et al., 2000; Montelli et al., 2004; Tromp et al., 2005; Nolet, 2008) Extended domain strategies Kinematic control through the maximization of the energy refocusing in a fictitious dimension introduced in the imaging condition (o set, angle, time-lag) (Symes and Kern, 1994; Sava and Biondi, 2004a,b; Sava and Fomel, 2006; Symes, 2008; Biondi and Almomin, 2013) What we propose: using an optimal transport measure of the distance (aka Earth s mover distance or Wasserstein distance) between observed and predicted data L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

35 Wasserstein 1 distance between f and g Primal formulation 8 Z >< W 1 (d cal, d obs )=min M Z >: where 8A X, x2x x2a kx M(x)kd cal (x)dx, Z d obs (x)dx = d cal (x)dx. M(x)2A (12a) (12b) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

36 Wasserstein 1 distance between f and g Primal formulation 8 Z >< W 1 (d cal, d obs )=min M Z >: where 8A X, x2x x2a kx M(x)kd cal (x)dx, Z d obs (x)dx = d cal (x)dx. M(x)2A (12a) (12b) Dual equivalent formulation W 1 (d cal, d obs )= max ' 2Lip 1 Zx2X where Lip 1 is the space of 1-Lipschitz functions, such that '(x)(d cal (x) d obs (x)) dx, (13) 8(x, y) 2 X, '(x) '(y) applekx yk 1. (14) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

37 Wasserstein 1 distance between f and g Dual equivalent formulation W 1 (d cal, d obs )= max ' 2Lip 1 Zx2X where Lip 1 is the space of 1-Lipschitz functions, such that '(x)(d cal (x) d obs (x)) dx, (12) 8(x, y) 2 X, '(x) '(y) applekx yk 1. (13) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

38 Wasserstein 1 distance between f and g Dual equivalent formulation W 1 (d cal, d obs )= max ' 2Lip 1 Zx2X where Lip 1 is the space of 1-Lipschitz functions, such that '(x)(d cal (x) d obs (x)) dx, (12) 8(x, y) 2 X, '(x) '(y) applekx yk 1. (13) Extension to non conservation of the mass and non positivity of the signals fw 1 (d cal, d obs )= max ' 2BLip 1 Zx2X '(x)(d cal (x) d obs (x)) dx. (14) where BLip 1 is the space of bounded 1-Lipschitz functions, such that 8(x, y) 2 X, '(x) '(y) applekx yk 1, 8x 2 X, '(x) < c (15) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

39 Numerical computation of f W 1 Naive discretization: linear programming problem with O(N 2 ) global constraints 8 >< fw 1 (d cal [m], d obs )=max ' XN r 8(i, j), ' ij < c, >: 8(i, j), (k, l) ' ij ' kl < (x r ) i (x r ) k + t j t l. i=1 XN t j=1 ' ij ((d cal [m]) ij (d obs ) ij ) t x r (16a) (16b) (16c) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

40 Numerical computation of f W 1 Naive discretization: linear programming problem with O(N 2 ) global constraints 8 >< fw 1 (d cal [m], d obs )=max ' XN r 8(i, j), ' ij < c, >: 8(i, j), (k, l) ' ij ' kl < (x r ) i (x r ) k + t j t l. i=1 XN t j=1 ' ij ((d cal [m]) ij (d obs ) ij ) t x r (16a) (16b) (16c) Simplification: reduction to only O(N) local constraints 8 >< fw 1 (d cal [m], d obs )=max ' 8(i, j), ' ij < c, XN r i=1 XN t j=1 8(i, j), ' i+1j ' ij < (x r ) i+1 (x r ) i = x r >: 8(i, j), ' ij+1 ' ij < t j+1 t j = t. ' ij ((d cal [m]) ij (d obs ) ij ) t x r (17a) (17b) (17c) (17d) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

41 Numerical computation of f W 1 Nonsmooth concave maximization problem W 1 (d cal [m], d obs )=max ' XN r XN t i=1 j=1 ' ij ((d cal [m]) ij (d obs ) ij ) t x r i K (A'), (16) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

42 Numerical computation of f W 1 Nonsmooth concave maximization problem W 1 (d cal [m], d obs )=max ' XN r XN t i=1 j=1 ' ij ((d cal [m]) ij (d obs ) ij ) t x r i K (A'), (16) where K is the unit hypercube of R 3N K = n o x 2 R 3N, x i apple 1, i =1,...3N. (17) the indicator function of K is denoted by i K, i K (x) = 0 if x 2 K +1 if x /2 K. and the matrix A represents the constraints ' k for k =1,...,N c ' k+1 ' k (A') k = for k = N +1,...,2 N x r ' k+nr ' k for k =2 N +1,...,3 N. x t (18) (19) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

43 Numerical computation of f W 1 Proximal splitting algorithm We use the SDMM method (Simultaneous Descent Method Multipliers, Combettes and Pesquet (2011)) to solve the maximization problem This iterative method requires the solution at each iteration of a linear system involving the matrix which can be proved to be equivalent to Q = I + A T A, (16) Q =2I (17) where is a second-order finite di erence discretized Laplace operator with Neumann boundary conditions L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

44 Numerical computation of f W 1 Proximal splitting algorithm We use the SDMM method (Simultaneous Descent Method Multipliers, Combettes and Pesquet (2011)) to solve the maximization problem This iterative method requires the solution at each iteration of a linear system involving the matrix which can be proved to be equivalent to Q = I + A T A, (16) Q =2I (17) where is a second-order finite di erence discretized Laplace operator with Neumann boundary conditions A Poisson equation must be solved at each iteration of the SDMM algorithm. FFT based solvers (O(N logn), Swarztrauber (1974)) or multigrid based solvers (O(N), Adams (1993)) can be used. L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

45 Example of application: the 2D Marmousi model L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

46 Marmousi 2 iterative reconstruction using a L 2 distance L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

47 Marmousi 2 iterative reconstruction using the optimal transport distance L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

48 Conclusion and perspectives A direct solver strategy amenable for moderate size 3D acoustic problems Frequency-domain strategies with sparse direct solvers are amenable for moderate size 3D acoustic problems using appropriate choice of PDE, discretized scheme, and solver Low rank compression will help us push up the limit of this strategy (Amestoy et al., 2015a), however 3D elastodynamics still seems out of reach: time-domain methods are still preferred for this problem Beyond the L 2 distance: mitigating the sensitivity to the initial model accuracy Standard FWI still relies on the assumption that the initial model is accurate enough: the sensitivity to the accuracy of the initial model could be mitigated by the choice of an optimal transport distance instead of the standard L 2 norm (Métivier et al., 2016) L. Métivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

49 Acknowledgments Thank you for your attention National HPC facilities of GENCI-IDRIS under grant Grant Local HPC facilities of CIMENT-SCCI (Univ. Grenoble) SEICOPE sponsors : L. Me tivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/ / 27

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