GEOP510 – Seismic Data Analysis

Part 2

Finite-difference (FD) solutions of the wave equation

Abdullatif A. Al-Shuhail

Associate Professor of Geophysics

Earth Sciences Department

College of Sciences

ashuhail@kfupm.edu.sa

mailto:wailmousa@kfupm.edu.sa

2

Introduction

• When a source is fired at the surface in the field, waves will reflect, refract, diffract,

disperse, and attenuate according to very complicated wave equations (WE) (e.g., poro-

viscoelastic WE) and we record the superposition of all these effects at the surface.

• Apparently, it is a challenging task to image the subsurface owing to the complexity of the

WE involved and the need to know many subsurface parameters.

• Therefore, we simplify the problem by assuming that only certain waves are important (e.g.,

P-waves), establish the corresponding WE (e.g., acoustic), and solve it numerically using

one of many methods (e.g., finite-difference).

• Solving the resulting WE means estimating the wavefield as it propagated from the source

into the earth and back to the receivers (modeling).

3

Finite Difference Method (FDM)

• The finite difference method (FDM) provides a numerical way of estimating the derivatives

of a function.

• The starting point in FDM is the Taylor series approximation of a function f(u) of one

variable u about the point u0:

𝑓 𝑢 ≅ 𝑓 u0 +𝑑𝑓

𝑑𝑢. 𝑢 − u0 +

1

2.𝑑2𝑓

𝑑𝑢2. 𝑢 − u0

2 +1

6.𝑑3𝑓

𝑑𝑢3. 𝑢 − u0

3 +1

24.𝑑4𝑓

𝑑𝑢4. 𝑢 − u0

4 + ... (0)

• Next, we put u=𝑥 + ∆𝑥 and u0=x in equation (0) for a small constant value Dx:

𝑓 𝑥 + ∆𝑥 ≅ 𝑓 𝑥 +𝑑𝑓

𝑑𝑥. ∆𝑥 +

1

2.𝑑2𝑓

𝑑𝑥2. ∆𝑥 2 +

1

6.𝑑3𝑓

𝑑𝑥3. ∆𝑥 3 +

1

24.𝑑4𝑓

𝑑𝑥4. ∆𝑥 4 + ... (1)

• Next, we put u=𝑥 − ∆𝑥 and u0=x in equation (0) for a constant Dx:

𝑓 𝑥 − ∆𝑥 ≅ 𝑓 𝑥 −𝑑𝑓

𝑑𝑥. ∆𝑥 +

1

2.𝑑2𝑓

𝑑𝑥2. ∆𝑥 2 −

1

6.𝑑3𝑓

𝑑𝑥3. ∆𝑥 3 +

1

24.𝑑4𝑓

𝑑𝑥4. ∆𝑥 4 − ... (2)

• See this site for more on Taylor series: https://en.wikipedia.org/wiki/Taylor_series.

https://en.wikipedia.org/wiki/Taylor_series

4

FDM

• Subtracting equation (2) from (1), solving for 𝑑𝑓

𝑑𝑥, and discarding any terms with powers of

Dx higher than 1 yields the central approximation of the first derivative at x:𝑑𝑓

𝑑𝑥≅

𝑓 𝑥+∆𝑥 −𝑓 𝑥−∆𝑥

2.∆𝑥. (3)

• Adding equations (1) from (2), solving for 𝑑2𝑓

𝑑𝑥2, and discarding any terms with powers of Dx

higher than 2 yields the central approximation of the second derivative at x:𝑑2𝑓

𝑑𝑥2≅

𝑓 𝑥−∆𝑥 −2.𝑓 𝑥 +𝑓 𝑥+∆𝑥

(∆𝑥)2. (4)

• Equation (4) is the second-order approximation of the second derivative.

• Higher-order approximations of 𝑑𝑓

𝑑𝑥and

𝑑2𝑓

𝑑𝑥2are also possible by including more terms of

equations (1) and (2).

5

FDM

• The main use of these formulas is to substitute them in differential equations and use them

to solve these equations numerically. (See example)

• Equation (4) can be generalized to functions of several variables by using partial instead of

total derivatives in equations (1) and (2).

• For example, to estimate the second partial derivatives of f(x,z,t), equation (4) becomes:𝜕2𝑓

𝜕𝑥2≅

𝑓 𝑥−∆𝑥,𝑧,𝑡 −2.𝑓 𝑥,𝑧,𝑡 +𝑓 𝑥+∆𝑥,𝑧,𝑡

(∆𝑥)2(5a)

𝜕2𝑓

𝜕𝑧2≅

𝑓 𝑥,𝑧−∆𝑧,𝑡 −2.𝑓 𝑥,𝑧,𝑡 +𝑓 𝑥,𝑧+∆𝑧,𝑡

(∆𝑧)2(5b)

𝜕2𝑓

𝜕𝑡2≅

𝑓 𝑥,𝑧,𝑡−∆𝑡 −2.𝑓 𝑥,𝑧,𝑡 +𝑓 𝑥,𝑧,𝑡+∆𝑡

(∆𝑡)2(5c)

• Next, we apply this method to numerically solve the 2-D acoustic wave equation.

6

FD Solution of Acoustic Wave Equation

• The 2-D acoustic wave equation in a medium with constant density is:

𝑐 𝑥, 𝑧 2.𝜕2𝑝(𝑥,𝑧,𝑡)

𝜕𝑥2+

𝜕2𝑝(𝑥,𝑧,𝑡)

𝜕𝑧2−

𝜕2𝑝(𝑥,𝑧,𝑡)

𝜕𝑡2= 𝑠(𝑥, 𝑧, 𝑡) (6)

where:

p(x,z,t): pressure field

c(x,z): velocity field

s(x,z,t): source wavelet.

• Our goal is to use the FDM to estimate p(x,z,t) in the region:

0 ≤ x ≤ xmax

0 ≤ z ≤ zmax

0 ≤ t ≤ tmax.

• Inserting equations (5) into (6) and solving for p(x,z,t+Dt) yields the following numerical

solution used to estimate future values of p (i.e., p(x,z,t+Dt)) given its past (i.e., p(x,z,t-Dt))

and present (i.e., p(x,z,t)) values:

𝑝 𝑥, 𝑧, 𝑡 + ∆𝑡 = 𝑟2 𝑝 𝑥 − ∆𝑥, 𝑧, 𝑡 − 2. 𝑝 𝑥, 𝑧, 𝑡 + 𝑝 𝑥 + ∆𝑥, 𝑧, 𝑡

+𝑟2 𝑝 𝑥, 𝑧 − ∆𝑧, 𝑡 − 2. 𝑝 𝑥, 𝑧, 𝑡 + 𝑝 𝑥, 𝑧 + ∆𝑧, 𝑡

−𝑝 𝑥, 𝑧, 𝑡 − ∆𝑡 + 2. 𝑝 𝑥, 𝑧, 𝑡 − ∆𝑥 2. 𝑟2. 𝑠 𝑥, 𝑧, 𝑡 . (7)

where r = c.Dt/Dx = c.Dt/Dz and Dx=Dz and Dt are user-selected space and time intervals (Figure 2.1).

• Initial and boundary conditions

• The following initial and boundary conditions are usually provided:

• 𝑝 𝑥, 𝑧, 0 = f(x,z) (initial condition) (8a)

•𝜕𝑝(𝑥,𝑧,0)

𝜕𝑡= g(x,z) (initial condition) (8b)

• 𝑝 0, 𝑧, 𝑡 = 𝑝 𝑥𝑚𝑎𝑥, 𝑧, 𝑡 = 𝑝 𝑥, 𝑧𝑚𝑎𝑥, 𝑡 =0. (boundary condition) (8c)

where f(x,z) and g(x,z) are known functions of x and z.

7

FD Solution of Acoustic Wave Equation

8

FD Solution of Acoustic Wave Equation

• Attempting to use equation (7) with the initial and boundary conditions of equation (8) at t=0

to calculate p(x,z,Dt) will require values of p(x,z,-Dt) and p(x,z,0). But p(x,z,0)=f(x,z), which

is known.

• To estimate p(x,z,-Dt): we follow these steps:1. Use equation (3) w.r.t. time:

𝜕𝑝(𝑥,𝑧,0)

𝜕𝑡=

𝑝 𝑥,𝑧,∆𝑡 −𝑝 𝑥,𝑧,−∆𝑡

2.∆𝑡(9a)

2. Solve (9a) for p(x,z,-Dt):

𝑝 𝑥, 𝑧, −∆𝑡 = 𝑝 𝑥, 𝑧, ∆𝑡 − 2. ∆𝑡.𝜕𝑝(𝑥,𝑧,0)

𝜕𝑡(9b)

3. Put (9b) into (7) and solve the resulting equation for p(x,z,Dt):

𝑝 𝑥, 𝑧, ∆𝑡 = (1

2)𝑟2 𝑝 𝑥 − ∆𝑥, 𝑧, 0 − 2. 𝑝 𝑥, 𝑧, 0 + 𝑝 𝑥 + ∆𝑥, 𝑧, 0

+(1

2)𝑟2 𝑝 𝑥, 𝑧 − ∆𝑧, 0 − 2. 𝑝 𝑥, 𝑧, 0 + 𝑝 𝑥, 𝑧 + ∆𝑧, 0

+∆𝑡.𝜕𝑝(𝑥,𝑧,0)

𝜕𝑡+ 𝑝 𝑥, 𝑧, 0 − (

1

2) ∆𝑥 2. 𝑟2. 𝑠 𝑥, 𝑧, 0 . (10)

• Therefore, we use equation (10) to calculate the field at the first time step and equation (7)

to calculate the field at all other time steps.

9

FD Solution of Acoustic Wave Equation

• Stability and dispersion conditions• The dispersion condition requires that:

∆𝑥 = ∆𝑦 = ∆𝑧 ≤𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

20.𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦(11)

in inhomogeneous media in order for the FD solution to be accurate (non-dispersive).

• The Courant-Friedrichs-Lewy (CFL) stability condition requires that:

𝑟 =𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦.∆𝑡

∆𝑥≤ 1 (for a wave propagating in 1-D) (12a)

𝑟 =𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦.∆𝑡

∆𝑥=

𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦.∆𝑡

∆𝑧≤

1

2(in 2-D) (12b)

𝑟 =𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦.∆𝑡

∆𝑥=

𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦.∆𝑡

∆𝑦=

𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦.∆𝑡

∆𝑧≤

1

3(in 3-D) (12c)

in order for the FD solution to be stable (we will always assume that ∆𝑥 = ∆𝑦 = ∆𝑧).

• We usually start by calculating Dx according to equation (11) and plug the answer in equation

(12) to calculate the corresponding Dt.

• The above Dx and Dt are used only for FD modeling. They do not affect acquisition geometry.

10

FD Solution of Acoustic Wave Equation

• Suppressing boundary effects• Application of boundary conditions at the model edges will lead to spurious reflections.

• It is important to attenuate these reflections to prevent them from interfering with the actual

waves propagating in the medium.

• There are many methods to do this including the following two common ones:

1. Sponge boundary condition (SBC)

• A 50-point zone is added beyond the model edges.

• Within this zone, the wave amplitude is multiplied by an exponential factor with an

amplitude of one at the first and zero at the last point of this zone.

2. Annihilating boundary condition (ABC)

• The wavefields obtained by equation (7) at the model edges are re-calculated using

these formulas:

𝑝 1, 𝑧, 𝑡 + ∆𝑡 = 𝑟 𝑝 2, 𝑧, 𝑡 − 𝑝 1, 𝑧, 𝑡 + 𝑝 1, 𝑧, 𝑡 at left edge,

𝑝 𝑥𝑚𝑎𝑥, 𝑧, 𝑡 + ∆𝑡 = −𝑟 𝑝 𝑥𝑚𝑎𝑥, 𝑧, 𝑡 − 𝑝 𝑥𝑚𝑎𝑥 − ∆𝑥, 𝑧, 𝑡 + 𝑝 𝑥𝑚𝑎𝑥, 𝑧, 𝑡 at right edge,𝑝 𝑥, 𝑧𝑚𝑎𝑥, 𝑡 + ∆𝑡 = 𝑟 𝑝 𝑥, 𝑧𝑚𝑎𝑥, 𝑡 − 𝑝 𝑥, 𝑧𝑚𝑎𝑥 − ∆𝑧, 𝑡 + 𝑝 𝑥, 𝑧𝑚𝑎𝑥, 𝑡 at bottom edge,and no suppression is used for the top edge (i.e., free surface).

11

FD Solution of Acoustic Wave Equation

• Code• See code for FD wave modeling using the method described here.