Effects of Conjugate Gradient Methods and Step-Length Formulas on the Multiscale Full Waveform Inversion in Time Domain: Numerical Experiments

Abstract

We carry out full waveform inversion (FWI) in time domain based on an alternative frequency-band selection strategy that allows us to implement the method with success. This strategy aims at decomposing the seismic data within partially overlapped frequency intervals by carrying out a concatenated treatment of the wavelet to largely avoid redundant frequency information to adapt to wavelength or wavenumber coverage. A pertinent numerical test proves the effectiveness of this strategy. Based on this strategy, we comparatively analyze the effects of update parameters for the nonlinear conjugate gradient (CG) method and step-length formulas on the multiscale FWI through several numerical tests. The investigations of up to eight versions of the nonlinear CG method with and without Gaussian white noise make clear that the HS (Hestenes and Stiefel in J Res Natl Bur Stand Sect 5:409–436, 1952), CD (Fletcher in Practical methods of optimization vol. 1: unconstrained optimization, Wiley, New York, 1987), and PRP (Polak and Ribière in Revue Francaise Informat Recherche Opertionelle, 3e Année 16:35–43, 1969; Polyak in USSR Comput Math Math Phys 9:94–112, 1969) versions are more efficient among the eight versions, while the DY (Dai and Yuan in SIAM J Optim 10:177–182, 1999) version always yields inaccurate result, because it overestimates the deeper parts of the model. The application of FWI algorithms using distinct step-length formulas, such as the direct method (Direct), the parabolic search method (Search), and the two-point quadratic interpolation method (Interp), proves that the Interp is more efficient for noise-free data, while the Direct is more efficient for Gaussian white noise data. In contrast, the Search is less efficient because of its slow convergence. In general, the three step-length formulas are robust or partly insensitive to Gaussian white noise and the complexity of the model. When the initial velocity model deviates far from the real model or the data are contaminated by noise, the objective function values of the Direct and Interp are oscillating at the beginning of the inversion, whereas that of the Search decreases consistently.

Appendices

Appendix 1

1.1 Frequency Width of the Ricker wavelet

The expression of the Ricker wavelet in time domain is as follows:

$$ R\left( t \right) = \left( {1 - 2\pi^{2} f_{0}^{2} \left( {t - t_{0} } \right)^{2} } \right)\exp \left( { - \pi^{2} f_{0}^{2} \left( {t - t_{0} } \right)^{2} } \right), $$

(28)

where t is time; $t_{0}$ is the delayed time; $f_{0}$ is the dominant frequency. The Fourier transform of the Ricker wavelet is as follows:

$$ F\left( \omega \right) = \frac{{\sqrt \pi \omega^{2} }}{{\omega_{c}^{3} }}\exp \left( { - \frac{{\omega^{2} }}{{\omega_{c}^{2} }} + i\omega t_{0} } \right), $$

(29)

where $\omega$ is the angular frequency; $\omega_{c} = 2\pi f_{0}$ is the angular frequency corresponding to the maximum amplitude. The amplitude spectrum is

$$ A\left( \omega \right) = \frac{{\sqrt \pi \omega^{2} }}{{\omega_{c}^{3} }}\exp \left( { - \frac{{\omega^{2} }}{{\omega_{c}^{2} }}} \right). $$

(30)

To determine the frequency width of the Ricker wavelet, we take the first derivative of (29) with respect to $\omega$, and after denoting the maximum amplitude of the Ricker wavelet by $A\left( {\omega_{c} } \right)$, we obtain

$$ A\left( \omega \right) = \frac{1}{2}A\left( {\omega_{c} } \right). $$

(31)

Substituting (31) into (30), we have

$$ \frac{{\omega^{2} }}{{\omega_{c}^{2} }}\exp \left( { - \frac{{\omega^{2} }}{{\omega_{c}^{2} }}} \right) = \frac{1}{2}e^{ - 1} , $$

(32)

The solution of (32) is equivalent to the root of the following equation:

$$ x^{2} \exp \left( { - x^{2} } \right) = \frac{1}{2}e^{ - 1} , $$

(33)

where $x = \omega /\omega_{c}$ and ‘e’ is Euler’s number. The solution of (33) leads to the well-known Lambert W function (Lambert, 1758), and with the help of the graphical method (see Fig. 21), the two roots of (33) are

$$ \left\{ \begin{aligned} \omega_{l} /\omega_{c} = c_{l} \hfill \\ \omega_{h} /\omega_{c} = c_{h} \hfill \\ \end{aligned} \right., $$

(34)

where $c_{l} = 0.482$ and $c_{h} = 1.637$ are two constants. These two limit frequencies define the frequency width of the Ricker wavelet.

Fig. 21

Lambert function (solid line). The two circles indicate the points intersected by the horizontal dashed line that marks the value of $e^{ - 1} /2$, which defines the frequency bandwidth of the Ricker wavelet

Appendix 2

2.1 Frequency Band Selection Strategy

According to multiscale strategy, the seismic data and wavelets are decomposed by scale. To proceed to an adequate frequency-band selection, so that the frequency or wavelength information contained in adjacent frequency bands is less redundant, we set the equality

$$ A_{i - 1} \left( {c_{l} f_{0}^{i} } \right) = A_{i} \left( {c_{l} f_{0}^{i} } \right), $$

(35)

where $c_{l}$ is the constant given in Appendix 1; $f_{0}^{i}$ is the dominant frequency of the target Ricker wavelet within the higher frequency bands; $A_{i - 1}$ and $A_{i}$ are the amplitude spectra of the target Ricker wavelet within the lower and higher frequency bands, respectively. The subscript or superscript i denotes the frequency band index. Substituting (30) into (35), we obtain the following equation:

$$ \left( {\frac{{\omega_{c}^{i} }}{{\omega_{c}^{i - 1} }}} \right)^{3} \exp \left( { - c_{l}^{2} \left( {\frac{{\omega_{c}^{i} }}{{\omega_{c}^{i - 1} }}} \right)^{2} } \right) = \exp \left( { - c_{l}^{2} } \right), $$

(36)

where $\omega_{c}^{i - 1}$ and $\omega_{c}^{i}$ are the most energetic angular frequencies corresponding to the lower and higher frequency bands, respectively. The solution of (36) is equivalent to the root of the following equation:

$$ x^{3} \exp \left( { - c_{l}^{2} x^{2} } \right) = \exp \left( { - c_{l}^{2} } \right), $$

(37)

where $x = \frac{{\omega_{c}^{i} }}{{\omega_{c}^{i - 1} }}$. Also with the help of the graphical method (see Fig. 22), the two roots of (37) and, therefore, of (36) are

$$ \omega_{c}^{i} /\omega_{c}^{i - 1} = c_{1,2} , $$

(38)

where $c_{1} = 1$ and $c_{2} = 4.533$. Obviously, $c_{2}$ is the desired solution, so that

$$ f_{0}^{i - 1} = f_{0}^{i} /4.533. $$

(39)

Fig. 22

$x^{3} \exp \left( { - c_{l}^{2} x^{2} } \right)$ function (solid line). Here, the value $c_{l}$ corresponds to the lower frequency point of the two points that define the frequency bandwidth of the Ricker wavelet (see Fig. 21). The two circles indicate the points intersected by the horizontal dashed line that marks the value of $\exp \left( { - c_{l}^{2} } \right)$, which are the two roots of the equation $x^{3} \exp \left( { - c_{l}^{2} x^{2} } \right) = \exp \left( { - c_{l}^{2} } \right)$

Here, $f_{0}^{i - 1}$ is the dominant frequency within the lower frequency band and $f_{0}^{i}$ is the dominant frequency within the higher frequency band. This relationship expresses the recursive relation between dominant frequencies of the target wavelet within adjacent frequency bands, thus allowing the correct frequency selection to implement the multiscale strategy.