Efficient gradient computation of a misfit function for FWI using the adjoint method

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1 Efficient gradient computation of a misfit function for FWI using the adjoint method B. Galuzzi, department of Earth Sciences A. Desio, University of Milan E. Stucchi, department of Earth Sciences A. Desio, University of Milan 4 Novembre 6 GNGTS 6, Lecce Sessione 3.3: Modelling, aspetti teorici e tecnologie innovative

2 Agenda. Seismic inversion using FWI. Seismic modelling 3. The importance of misfit gradient for FWI 4. Computation of misfit gradient using the adjoint method 5. An example of application

3 Full Waveform Inversion (Tarantola, 986) Estimation of a geological macro-model of subsurface from active seismic data, by means of: ) Seismic modelling algorithm Numerical solution of wave equation (predicted data) ) Misfit function Difference between predicted and observed data 3) Seismic inversion algorithm Optimization of the misfit function estimation of the predicted model Starting model Predicted model modelling algorithm Predicted data Misfit function Observed data vs Predicted data inversion algorithm Stopping criteria: maximum number of iterations, minimum threshold,...

4 The D acoustic wave equation q x, t = v x ) Δp x, t + δ x x / s t p x, t = q x, t Initial conditions: p x, = q(x, ) =, x D ) BCD/) BFD/) = q@,a + dx ) Δ B The seismic modelling and its approximation by an explicit finite difference method p BCD = BCD/) + dt: time sampling dx: space sampling ΔH : approximation of Laplacian nd order in time and 4 nd in space + dtδ x / s B t, T : recording time x D(x, z) R ) space domain x / the location of the source s t the seismic wavelet v x the acoustic wave velocity

5 . Numerical stability: dt < dx v \]^ λ, Numerical considerations. λε.5, : Courant number. Numerical dispersion dx < v \@` f,. v \@` : minimum velocity of the model \]^n. f \]^ : maximum frequency of s(t) 3. n 5: points per wavelength 3. Absorbing boundary conditions (Cerjan, 985): q = p = B q + dtv dx ) Δ B BCD/) + + dtδ x / s B The Gaussian taper factor = V.9, inside the computational domain Inside the absorbing boundary layers

6 Misfit function for seismic inversion The aim of FWI is to find an optimal Earth model m, that minimizes a misfit function: m = min g h Χ(m ) = min \ h p m p / p / : observed data p m : predicted data (using seismic modelling) M : the set of geological models A classical functional for FWI is the L distance between the observed and the synthetic seismograms: Χ m = `t `s v nr : number of receivers p p q[p m, t, x s, x t, m p / (t, x s, x t )u ) ns : number of the sources dt, T : registration time twd swd / x s : position of r th receiver x t : position of the s th source Due to space discretization, the geological models mεm are discretized with components = [v z v t ), ), ] i :,, nx j :,, nz k:,, ny ) Number of unknowns = (number of grid nodes) * (number of geological parameters) ) Χ(m) is a generally a complicated non-linear functional of m

7 Optimization algorithms for seismic inversion Local iterative minimization algorithms Starting model m / m ycd = m y + γ y h y h y is the descend direction γ y > is the step length No Computation of h y = \ Χ m y Computation of γ y Χ m ycd < Χ m y Predicted model m ycd A descend direction is given by: Χ m ycd <tol D or \ Χ m ycd <tol )? h y = \ Χ m y the gradient at the k-th iteration The Misfit gradient is useful to guide the minimization algorithm to a local minimum of Χ m How compute the gradient \ Χ m y?

8 The adjoint method Powerful tool to shorten the time required for computing the gradient (Fichner, ) Acoustic equation+l misfit p is the regular solution of the wave equation p t vž Δp t = δ x x / s t with: p = p ) =, x D. Χ( Š = m q p t) p / v p v is the solution of the adjoint equation p t vž Δp t = g t with: p T = p T) =, x D. g v t = p t p / t δ x s To compute the time integral à simultaneously p and p for each time step, in all the domain D Regular solution p )Wave equation scheme, forward in time (from to T), to compute the adjoint source (g ) to store p only inside the absorbing boundaries )Wave equation scheme, reverse in time (from T to ) to re-compute p inside the domain using the storing p inside the absorbing boundaries Adjoint solution p Wave equation scheme, backward in time (from T to ) to satisfy the final conditions (t=t) after the adjoint source computation N.B The storing of p inside the absorbing boundaries to prevent numerical instability FD > ) during backward propagation

9 Scheme for the adjoint gradient computation ) Computation of the p, forward in time (Storing p inside the absorbing layers for each time step) ) Computation of g using p and p 3) Computation of p and p, backward in time (using the stored p, inside the absorbing layers) Computation of the time integral using p and p only 3 computation of wave equation is required: forward modelling+ backward modelling

10 .Observed data Example of application depth (km) Spread layout Portion of Marmousi model 9 receivers 6 sources The time grid dt =.s T=6s The modelling grid nx = 9 nz = 48 dx = dz = 4m Local optimization method m ycd = m y + γ y h y \ Χ m y is the gradient at the k-th iteration β y = \Χ m ycd ) ) \ Χ m y ) ) γ y obtained by a line search procedure on y = Χ(m y + γ y h y ) (linear or quadratic interpolation) h ycd = \ Χ m y + β y h y T \z s *(T \z (h y )+T \z (γ y )) s *(about 4-5 modellings)

11 3.The starting model (smooth version of the true model) depth (km) Starting model The gradient emphasizes the main differences between the starting and the true model Note the high values of the gradient near the sources The gradient of the misfit function can guide the inversion procedure to the true model

12 Results depth (km) depth (km) depth (km).5 True model Starting model Final model L Misfif Misfit evolution Iter Note the misfit reduction from to about.5 The final model isclose the global minimum solution The inversion procedure has highlighted the main geological structures that were not clear in the smooth model The main differences between the final and true model are at the border where the illumination is poor

13 Conclusion. We have implemented the adjoint method to compute the gradient of a misfit function in an efficient way. The low computational time required by the adjoint method and the low storage memory employed make this implementation of particular interest especially in applications such as the Full Waveform Inversion 3. As an example of application, we carried out a local FWI on a portion of the Marmousi model. The finalmodel, after the localinversion fairly match the true model 4. As a future work we plan to apply this method for real data Thank you for your attention!

14 The finite difference approximation How compute the misfit gradient? Computation of the partial derivatives by using a finite difference approximation: Š Χ m D,D + ϵδm D,D, m D,), Χ m D,D ϵδm D,D, m D,), m D,D ϵ with a small ϵ > Problems:. The choice of ϵ is not obvious. To obtain the gradient is necessary to compute two forward modelling for each spatial derivative 3. This method is impracticable when the number of unknowns is large and the forward modelling is computationally expensive

15 .The observed data Example of seismic acquisition Spread layout 9 receivers sources depth (km) Marmousi model km/s

16 .The forward modelling nx = 384 nz = dx = dz = 4m dt =.s 4.The predicted model 3.The inversion procedure Misfit function: L difference Starting model: D True model: Marmousi Optimization algorithm : steepest descend m ycd = m y γ y \ Χ m y Marmousi model D model.5.5 depth (km).5 depth (km) Velocity (km/s) Velocity (km/s) We need to compute the gradient of the misfit function for the predicted model

17 Finite difference vs adjoint method.5 Marmousi model.5 D model The two methods produce very similar solutions. depth (km) Velocity (km/s) depth (km) Velocity (km/s) The computational time of the adjoint method is considerably reduced: Adjoint method: forward modelling + backward modelling Classical gradient method: x384x forward modelling

18 Formula for the adjointstate method Misfit function: Χ v = D ) v [p t, x, v p / (t, x )u ) š / dtdx = Χ v, with Χ v = D ) [ p t, x, v p / (t, x ) ) Wave equation: p x, t v ) Δp x, t = f(x, t) L p v, x, t, v = f(x, t) With L linear operator

19 Formula for the adjointstate method Misfit function gradient: Χ v = z Χ p(v, x, t) p(v, x, t), Wave equation gradient: ) L p v, x, t, v = f(x, t) ) L p v, x, t, v + z L p v, x, t, v p v, x, t = 3) L p v, x, t, v g + z L p v, x, t, v p v, x, t g = 4) L p v, x, t, v g + z L p v, x, t, v p v, x, t g =

20 Formula for the adjointstate method Combining the two relation: \ Χ m = z Χ p m, x, t \ p m, x, t + \ L p m, x, t, m g + z L p m, x, t, m \ p m, x, t g \ Χ m = z Χ p m, x, t + z L p m, x, t, m g m p(m, x, t) + \ L p m, x, t, m g To remove the m p(m, x, t), we use the adjoint term g, imposing: z L p m, x, t, m g = z Χ p m, x, t If L is a linear operator L g m, x, t, m = z Χ p m, x, t, that is is the adjoint equation