Algoritam brzog čišćenja za Eikonalovu jednačinu

Fast Sweeping Algorithm for Eikonal Equation Solution
Herling Gonzalez Alvarez
Instituto Colombiano del Petroleo
Ecopetrol S.A.
9 de mayo de 2021
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The eikonal equation is a non-linear partial differential equation for wave
propagation problems. The eikonal equation will be derived as a
high-frequency approximation of the acoustic wave equation.
∇2 p(~x, t) −
1 ∂ 2 p(~x, t)
=0
c2 ∂t2
Where p(~x, t) is a pressure field, c = c(~x) the propagation velocity on
constant density media. A time-harmonic high-frequency approach is
searched for amplitude P (~x), and the propagation time function φ(~x) and
can be assumed
p(~x, t) = P (~x)eiω{t−φ(~x)}
First, one calculates the time derivatives of the wave equation:
∂p(~x, t)
= iωP (~x)eiω{t−φ(~x)}
∂t
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∂ 2 p(~x, t)
= ω 2 P (~x)eiω{t−φ(~x)}
∂t2
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Now the spatial derivatives follow:
∇p(~x, t) = ∇P (~x)eiω{t−φ(~x)} − iωP (~x)eiω{t−φ(~x)} ∇φ(~x)
= [∇P (~x) − iωP (~x)∇φ(~x)]eiω{t−φ(~x)}
We can take a second derivation ∇ · ∇p = ∇2 p then:
∇2 p
= ∇ · (∇P − iωP ∇φ)eiω{t−φ(~x)} + (∇P − iωP ∇φ) · ∇eiω{t−φ(~x)}
= (∇2 P − iω∇P · ∇φ − iωP ∇2 φ − iω(∇P − iωP ∇φ) · ∇φ)eiω{t−φ(~x)}
= (∇2 P − 2iω∇P · ∇φ − iωP ∇2 φ − ω 2 P (∇φ)2 )eiω{t−φ(~x)}
On account of the vectorial identity
∇ · {a(~x)~b(~x)} = ∇a(~x) · ~b(~x) + a(~x)∇ · ~b(~x)
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The two derivatives inserted into the wave equation result after division by
eiω{t−φ(~x)}
1
2
2
−ω P (∇φ) − 2 − iω(2∇P · ∇φ + P ∇2 φ) + ∇2 P = 0.
c
A division by −ω 2 P then results in
i
1
1
2
(2∇P · ∇φ + P ∇2 φ) − 2 (∇2 P ) = 0
(∇φ) − 2 +
c
ωP
ω P
Since the real and imaginary parts of the equation must be zero
independently, it results:
1
1
(∇φ)2 − 2 − 2 (∇2 P ) = 0
c
ω P
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In the approximation, one assumes that the amplitude P is only
position-dependent, i.e. ∇2 P is limited. Since at the same time neither the
transit time φ nor the amplitude P are frequency dependent, the second
term for very high frequencies is small compared to the first term and the
equation simplifies to:
[∇φ]2 =
1
c2
The solution φ(~x) of the eikonal equation assigns the travel time of the
wave to each point in the space. Lines of equal travel time can be
interpreted accordingly as wave fronts.
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Numerical approximation
Sweeping algorithms such as the fast sweeping method (Zhao, 2004) are
highly efficient for solving Eikonal equations. For numerical approximation,
assume that Ω is discretized into a uniform grid with spacing h on a
cartesian grid xi,j has a travel time value φi,j = φ(xi,j ). A first-order
scheme to approximate the partial derivatives is
h
i2
h
i2
1
−y
+y
−x
+x
max Di,j
φ, −Di,j
φ, 0 + max Di,j
φ, −Di,j
φ, 0 = 2
fi,j
where
±x
φx (xi,j ) ≈ Di,j
φ=
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φi±1,j − φi,j
±h
±y
and φy (xi,j ) ≈ Di,j
φ=
Fast Sweeping Algorithm
φi,j±1 − φi,j
±h
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Numerical approximation
Due to the consistent, monotone, and causal properties of this
discretization (Chacon and Vladimirsky, 2012) it is “easy” to show that if
φxi,j ≡ min [φi−1,j , φi+1,j ], and φyi,j ≡ min [φi,j−1 , φi,j+1 ] as too
|φxi,j − φxi,j | 6 h/fi,j we have
φi,j − φxi,j
h
2
+
φi,j − φyi,j
!2
=
h
1
2
fi,j
Or equivalent
φi,j −
2
φxi,j
+ φi,j −
φyi,j
2
=
h
fi,j
2
The quadratic function (τ − a)2 + (τ − b)2 − c2 = 0 have a solution
p
√
a + b + 2c2 − (a − b)2
a + b + 2c2 − a2 + 2ab − b2
τ=
=
2
2
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Numerical approximation
where
a = φxi,j ,
b = φyi,j ,
c=
h
,
fi,j
and τ = φi,j
√
This solution will always exist as long as |φxi,j − φyi,j | 6 2 h/fi,j is
satisfied and is larger than both, φxi,j and φxi,j , as long as
|φxi,j − φyi,j | 6 h/fi,j .
Now, if |φxi,j − φyi,j | > h/fi,j lower-dimensional update must be performed
by assuming one of the partial derivatives is 0:
i
h
h
φi,j = min φxi,j , φyi,j +
fi,j
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Fast Sweeping 2D Algorithm
We execute the following instructions four times (sweeps) for each cell φi,j .
Tx ← min [φi−1,j , φi+1,j ], Ty ← min [φi,j−1 , φi,j+1 ]
Tmin ← min [Tx , Ty ]
h
c←
fi,j
d ← |Tx − Ty |
if d ≥ c then
U ← Tmin + c
else
√
Tx + Ty + 2c2 − d2
U←
2
end if
φnew
i,j ← min [φi,j , U ]
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Fast Sweeping 2D Algorithm
The Fast Sweeping Method (FSM) is an iterative algorithm which
computes the travel-time map by successively sweeping the whole grid
following a specific order. FSM performs Gauss-Seidel iterations in
alternating directions. We sweep the whole domain with four alternating
orderings repeatedly
1)
2)
3)
4)
i=0:Nx-1,
i=Nx-1:0,
i=Nx-1:0,
i=0:Nx-1,
for i ← 0 to Nx − 1 do
l ← (N x − 1) − i
for j ← 0 to Ny − 1 do
φnew
l,j ← φl,j
end for
end for
j=0:Ny-1
j=0:Ny-1
j=Ny-1:0
j=Ny-1:0
For example for sweep 2:
Initial and boundary conditions must be taken into account.
φi,j = ∞ for all
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xi,j ∈ Ω,
Fast Sweeping Algorithm
φ(~x0 ) = 0
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Fast Sweeping 2D Algorithm
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Fast Sweeping Algorithm
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Fast Sweeping 2D Algorithm
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Fast Sweeping Algorithm
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Fast Sweeping 2D Algorithm
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Fast Sweeping Algorithm
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Fast Sweeping 3D Algorithm
Now we apply the fast sweeping algorithm to solve the Eikonal equation
with a point source in a 3-D medium, using the first-order Godunov upwind
finite difference scheme (Rouy and Tourin, 1992), we discretize as follow:
φi,j,k − φxi,j,j
h
2
+
φi,j,k − φyi,j,k
h
!2
+
φi,j,k − φzi,j,k
h
2
=
1
2
fi,j
φxi,j,k = min [φi−1,j,k , φi+1,j,k ]
φyi,j,k = min [φi,j−1,k , φi,j+1,k ]
φzi,j,k = min [φi,j,k−1 , φi,j,k+1 ]
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Fast Sweeping 3D Algorithm
According to Huang and Bellefleur (2012) for each cell φi,j,k we have:
Tx ← φxi,j,k , Ty ← φyi,j,k , Tz ← φzi,j,k
F ← h/fi,j
Require: Tx < Ty < Tz
U ← Tx + F
if U > Ty then q
Ty + Tx +
2F 2 − (Ty2 + Tx2 ) + 2Ty Tx
U←
2
if U > Tz then
U ← (Tx +qTy + Tz )/3
4(Tx + Ty + Tz )2 − 12(Tx2 + Ty2 + Tz2 − F 2 )
U ←U+
6
end if
end if
Ensure: φnew
i,j,k ← min [φi,j,k , U ]
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Fast Sweeping 3D Algorithm
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3D Algorithm (Sweeping)
We sweep the whole domain with eight alternating orderings repeatedly
The TRUE value reverses the
direction of the sweep.
1)
2)
3)
4)
5)
6)
7)
8)
i
0
0
0
0
1
1
1
1
j
0
0
1
1
0
0
1
1
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k
0
1
0
1
0
1
0
1
For example sweeping 4
for i ← 0 to Nx − 1 do
for j ← 0 to Ny − 1 do
m ← (ny − 1) − j
for k ← 0 to Nz − 1 do
l ← (nz − 1) − k
φnew
i,m,l ← φi,m,l
end for
end for
end for
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3D Traveltime: Proposed algorithm
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3D Traveltime: From Pascal Podvin (fdtimes.h)
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References
Chacon, A. and Vladimirsky, A. (2012). Fast two-scale methods for eikonal
equations. SIAM Journal on Scientific Computing, 34:A547–A578.
Huang, J.-W. and Bellefleur, G. (2012). Joint transmission and reflection
traveltime tomography using the fast sweeping method and the
adjoint-state techniqu. Geophysical Journal International, 188:570–582.
Rouy, E. and Tourin, A. (1992). A viscosity solutions approach to
shape-from-shading. SIAM Journal on Numerical Analysis, 29:867–884.
Zhao, H. (2004). A fast sweeping method for eikonal equations.
Mathematics of Computation, 74:603–627.
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