Adaptive multi-step Full Waveform Inversion based on Waveform Mode Decomposition

    Adaptive Multi-step Full Waveform Inversion based on Waveform Mode Decomposition Yong Hu, Liguo Han, Zhuo Xu, Fengjiao Zhang, Jingwen Zeng PII: DOI: Reference:

S0926-9851(17)30159-3 doi:10.1016/j.jappgeo.2017.02.017 APPGEO 3221

To appear in:

Journal of Applied Geophysics

Received date: Revised date: Accepted date:

16 July 2016 16 January 2017 14 February 2017

Please cite this article as: Hu, Yong, Han, Liguo, Xu, Zhuo, Zhang, Fengjiao, Zeng, Jingwen, Adaptive Multi-step Full Waveform Inversion based on Waveform Mode Decomposition, Journal of Applied Geophysics (2017), doi:10.1016/j.jappgeo.2017.02.017

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Adaptive Multi-step Full Waveform Inversion based on Waveform Mode

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Decomposition

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Yong Hu, Liguo Han, Zhuo Xu, Fengjiao Zhang, Jingwen Zeng

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College of Geo-exploration Science and Technology, Jilin University

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Abstract

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Full Waveform Inversion (FWI) can be used to build high resolution velocity models, but there

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are still many challenges in seismic field data processing. The most difficult problem is about

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how to recover long-wavelength components of subsurface velocity models when seismic data is lacking of low frequency information and without long-offsets. To solve this problem, we

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propose to use Waveform Mode Decomposition (WMD) method to reconstruct low frequency

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information for FWI to obtain a smooth model, so that the initial model dependence of FWI can be reduced. In this paper, we use adjoint-state method to calculate the gradient for Waveform

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Mode Decomposition Full Waveform Inversion (WMDFWI). Through the illustrative numerical

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examples, we proved that the low frequency which is reconstructed by WMD method is very reliable. WMDFWI in combination with the adaptive multi-step inversion strategy can obtain

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more faithful and accurate final inversion results. Numerical examples show that even if the initial velocity model is far from the true model and lacking of low frequency information, we still can obtain good inversion results with WMD method. From numerical examples of anti-noise test, we see that the adaptive multi-step inversion strategy for WMDFWI has strong ability to resist Gaussian noise. WMD method is promising to be able to implement for the land seismic FWI, because it can reconstruct the low frequency information, lower the dominant frequency in the adjoint source, and has a strong ability to resist noise.

Keywords: Full Waveform Inversion; Waveform Mode Decomposition; Adaptive Multi-step; Low Frequency Reconstruction; Adjoint-state method.

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1. Introduction

With the development of petroleum industry, the stage of exploration from subsurface structures

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gradually changes into lithological characteristics, and the difficulty of exploration has increased

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dramatically. In order to meet the requirements of new era seismic exploration, FWI developed

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rapidly, and it has become the hot topic in the current geophysics exploration. In the 1980s, FWI was first introduced by Lailly (1983) and Tarantola (1984), they proposed to use adjoint-state

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method to calculate the gradient for the FWI objective function, instead of using explicit Jacobian matrix, it saved a large amount of computing time. However, FWI is a strongly

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nonlinear problem, it is sensitive to noise, initial model and low frequency component (Virieux

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and Operto, 2009). These problems have brought huge challenges for FWI in the application of

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field data. When field data are missing low frequency information and the initial velocity model is not good enough, conventional FWI may occur cycle skipping problem. It is known that low-frequency data are important for recovering the long-wavelength background structure and therefore it can reduce the initial model dependence for FWI (Wu, 2013).

In order to obtain the long-wavelength components of subsurface velocity models and overcome the cycle skipping problem, a series of inversion strategies have been proposed. Bunks (1995) developed multi-scale full waveform inversion which is based on low-pass filtering, and it can implement FWI from low frequency to high frequency. However, this multi-scale strategy cannot work very well, when there is no low frequency components in seismic data. Similarly, frequency domain multi-scale FWI (Pratt, 1999) may occur cycle skipping problem when

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seismic data lacks of low frequency components. The acquisition of seismic data generally does not contain the information below 5Hz, but if we want to obtain the low frequency components

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from field acquisition, it is very expensive (Baeten, 2013; Berkhout 2011). Therefore, effort has

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been focused on reconstructing low frequency components from current available data. Envelope

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Inversion (Chi, 2014) can reconstruct ultra low frequency information for seismic waveform inversion by Hilbert transformation, but the waveform of FWI adjoint source will be

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dramatically distorted by mathematical transformation. Bump function (Bharadwaj, 2016) can

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get a Gaussian smooth seismic data, it does not rely on the initial velocity models, and far less sensitive to the cycle skipping problem. Xue (2016) proposed to use smoothing kernels for FWI,

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it can get a good inversion model, even without low frequency components in the modeling

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seismic data, but the layer position is not matching with the real velocity model. Zhang and Han

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(2015) proposed to use seismic data from passive source which contains large mounts of low frequency information to compensate for active source seismic data, however, noise dramatically limits its performance and it also needs a precise initial model to implement passive source full waveform inversion. While there are many other methods can obtain a good initial model rather than reconstruct the low frequency information, such as travel time tomography (Luo and Schuter, 1991; Dines and Lytle, 1978;), migration velocity analysis (Xie and Yang, 2008; Liu and Bleistein, 1995), and Laplace-Fourier domain FWI (Shin and Cha, 2009). There are also many other kinds of methods, which combines FWI with some other types of inversion methods. For example, Zhou et al.(1995) combined travel time with waveform inversion. Biondi and Almomin (2012) combined FWI with wave equation migration velocity analysis.

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Considering the lack of low frequency information in the recorded seismic data, and the existing methods have some shortcomings, in this paper, we develop the idea of Wu (2014), and

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propose to use Waveform Mode Decomposition (WMD) method to reconstruct the low

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frequency information for FWI, in time domain. Using the reconstructed low frequency

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information can effectively retrieve the macro structure of velocity models and mitigate the non-linearity problem, especially when low frequency data are missing. To illustrate the

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reliability of the reconstructed WMD waveform, we compared it with original waveform and

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Hilbert envelope waveform. To demonstrated that the WMD method is better than Hilbert transform, we mainly analysis seismic waveform, gradient operator, adjoint source spectrum and

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anti-noise ability. we use adjoint-state theory to conduct the WMDFWI and use the adaptive

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multi-step inversion strategy to obtain a smooth background velocity model, then we use it as a

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good initial model for conventional FWI. Numerical examples show that the WMDFWI can be a good supplement for conventional FWI, even if the initial model is far from the true model and recorded seismic data lacks low frequency information.

2. Waveform Mode Decomposition method 2.1 Definition In seismic data, we can use Cubic Spline Interpolation (CSI) to draw a smooth curve line through all the local maximum points, we call this curve line as upper envelope of the seismic waveform. Similarly, draw a smooth curve line through all the local minimum points, we can get the lower envelope. Finally we summate the upper envelope and lower envelope to obtain the

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demodulation waveform. We call this demodulation signal method as a Waveform Mode Decomposition (WMD) method. The demodulation waveform obtained by WMD method, we

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call it WMD waveform.

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WMD method is similar to Empirical Mode Decomposition (EMD) method. In the digital signal processing, EMD can be used to decompose any complicated data set into a finite and

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often small number of Intrinsic Mode Functions (IMFs). This decomposition method is adaptive.

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Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes (Huang 1998). EMD also can effectively be used for

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signal denoising in a wide range of applications, such as biomedical signals, acoustic signals, and

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ionospheric signals (Tsolis G, Xenos T D,2011) . EMD methods needs to decompose signal into

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many IMFs, until it meets the stopping criteria, then summate all the IMFs to get the denoised signal. So the IMFs which is obtained by EMD method are just intermediate variable values not the final processing results.

WMD method and EMD method are both using cubic spline interpolation approach. But it has many differences in its characteristic and application. EMD method is used to decompose signals into lots of IMFs (IMF1,IMF2,...IMFn). While WMD waveform is the summation of upper and lower envelopes. The relationship of WMD method and EMD method can be expressed by the following equation:

IMF1  x(t ) 

WMD 2

(1)

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Where IMF1 is the first EMD component from the data, x(t ) denotes the original data.

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Although WMD waveform is not the same as the original signal, it reflects the macro

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information of seismic data.

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WMD method has many advantages when it used in the FWI procedure, such as reconstructing low frequency information, lowering the dominant frequency, and strong

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anti-noise ability. In the process of FWI, we only need to match the WMD waveform of synthetic

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data and recorded data, WMD waveform reflects the macro structure of velocity models, so WMD method can be used to mitigate the non-linearity problem and cycle skipping problem for

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2.2 Detailed steps

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FWI by using the reconstructed low frequency information.

In order to reconstruct the low frequency information, we use WMD method to process the seismic data trace by trace. The detailed steps of using WMD method to reconstruct the low frequency information as shown in the following.

Step1: Identify all the local maximum value of seismic data ( D ), and then connect all the local maximum value by a cubic spline line as the upper envelope ( upenv (D) ).

Step2: Repeat the procedure for the local minimum value to obtain the lower envelope ( lowenv (D ) ). The upper and lower envelopes should cover all the data between them.

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D w  upenv( D)  lowenv ( D)

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(2)

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Step3: Summate the upper and lower envelopes, we can obtain the WMD waveform :

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Where D w denotes the WMD waveform, when D w contains abundant low frequency

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information, stop the decomposition. If not, go to step4.

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Step4: When D w still does not have low frequency information, D w is treated as the

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original signal ( D ) to get the further decompose, repeat Step1,2 and 3 .

(3)

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D 2 w  upenv( D w )  lowenv ( D w )

In this paper, we use CSI method to obtain cubic spline line, it ignored most of the details, and reflects the macro information of seismic signals.

2.3 Low frequency

WMD waveform can be calculated by the summation of upper envelope and lower envelopes. But the upper envelope which is obtained by cubic spline interpolation method, so we cannot use specific mathematical formula to express the upper envelope. But from the Fig.1, we can see that the upper envelope and Hilbert Envelope of the harmonic signal are almost the same. So we only need to prove that Hilbert Envelope can reconstruct ultra low frequency information.

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Fig1 Harmonic signal with its Hilbert Envelope and Upper Envelope.

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We choose cosine function x(t ) as an original signal, it can be expressed by equation(4):

(4)

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x(t )  cos(2f mt )  cos(2f z t )

Where f m , f z are the modulation frequency, it must subject to: f m  f z  f m . The Hilbert transform of x(t ) is:

    ~ x (t )  cos 2f mt    cos 2f z t    sin(2f mt )  sin(2f z t ) 2 2  

(5)

So the Hilbert Envelope of x(t ) can be expressed by equation(6):

E[ x(t )]  x(t ) 2  ~ x (t ) 2  2 cost ( f m  f z )  2 cos(ft )

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Where f m  f z  f , E[ x(t )] can be expanded by Fourier series, which contains all harmonic

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components, so equation(6) can be expressed as: (Chi 2014)

E[ x(t )]  A0  A1 cos(ft )  A2 cos( 2 ft )   

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(7)

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From equation(7), we can see that the Hilbert Envelope contains zero and low frequency spectrum. Similarly, the upper envelope also contains much low frequency information. WMD

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(8)

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Ewmd [ x(t )]  Eup[ x(t )]  Elow[ x(t )]

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waveform can be calculated by the summation of upper and lower envelopes:

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Here we can get the Fourier series of upper and lower envelopes, it can be expressed by following equations:

Eup[ x(t )]  Aup0  Aup1 c o s(ft )  Aup2 c o s(2 ft )   

Elow[ x(t )]  Alow0  Alow1 cos(ft )  Alow2 cos( 2 ft )   

(9)

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WMD waveform can be expressed by equation(11):

Ewmd [ x(t )]  Eup[ x(t )]  Elow[ x(t )]  Awmd 0  Awmd1 cos(ft )  Awmd 2 cos( 2 ft )   

(11)

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From equation(11), we can see that the WMD waveform contains zero and low frequency information, which reflects the macro information of seismic data. So we can use the

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reconstructed low frequency information to obtain a good initial model for conventional FWI.

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Seismic waveform is not the harmonic signal, so it always has this condition:

Eup[ x(t )]  Elow[ x(t )]  0

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(12)

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According to equation(12), we do not need to worry the WMD waveform is zero. Seismic data is

x(t )   cos(kft ) 

hft hft  Nft )  sin( ) 2 2 hft 2 sin( ) 2

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k h

sin(

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N

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a complex signal, it can be expressed by:

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~ x (t )  H x(t )   sin(kft )  k h

Where

cos(

hft hft )  cos(  Nft ) 2 2 hft 2 sin( ) 2

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hf Nf is the lowest frequency, is the highest frequency, N  f  1 is the number of 2 2

frequencies.

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Nft ) 2  2 cos( Nft ) 2 2 2 ~ E[ x(t )]  x(t )  x (t )   hft hft 2 sin( ) sin( ) 2 2 sin(

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(15)

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So the envelope of complex frequency signal ( E[ x(t )] ) can be expanded by Fourier series, which

1 A0  A1 cos(ft )  A2 cos( 2 ft )    hft sin( ) 2

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E[ x(t )] 

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contains all harmonic components:

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spectrum.

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From equation(16), we can see that the Hilbert Envelope contains zero and low frequency

Equation(11) and Equation(16) gives mathematical proof about the low frequency in the WMD waveform. Next, we will give some numerical examples to illustrate that WMD method is an effective way to reconstruct the low frequency information. In Fig.2b, it proves that the cubic spline line contains abundant low frequency information, and at the some time the high frequency information has been attenuated. In the process of CSI, low frequency information will be introduced into seismic waveform. This low frequency information was reconstructed from original seismic waveform, and it reflects the long-wavelength components of subsurface velocity models. WMD waveform was obtained from the summation of upper and lower envelope, and it has basically the same phase with the original signal.

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Fig.2 Waveform and spectrum;

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(a) Ricker wavelet and its WMD wavelet; (b) Spectrum.

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From Fig.2a and Fig.3a we can see that the WMD waveform become softer than the original one. And the spectrum shows that the WMD waveform contains abundant low frequency information (Fig.2b and 3b). we can use this low frequency components for WMDFWI to invert background velocity model, and then use it as an initial model for conventional FWI.

It is known that seismic data generally missing information below 5Hz. In order to simulate the actual situation, we using a Butterworth high pass filter to filter out information below 9Hz. After using high-pass filtering, the waveform become sharper (Fig.3a blue dash line). However, WMD still can reconstruct low frequency components (Fig.3b,red line). From Fig.2 and Fig.3, it proves that WMD method not only can reconstruct the low frequency information, but also can protect the waveform from distortion problem.

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Fig.3 Waveform and spectrum; The data are generated from the modified Marmousi model; (b) Spectrum.

2.4 Reliability

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(a) One trace of high-pass filtering waveform and its WMD waveform;

In order to prove that the reconstruction of low frequency information is reliable, we need to compare the WMD waveform with the real seismic waveform. We use Butterworth filter to process the seismic waveform, and filter out information below 9Hz. The waveform without low frequency information is shown in Fig.4a (blue line), and its WMD waveform is expressed by the red line, we can see that the waveform of blue line is sharper than the red line, and WMD waveform varies very slowly, because WMD waveform contains abundant low frequency information and has low dominant frequency. Fig4b is the partial enlarged view from Fig.4a, we can see that the WMD waveform and filtered waveform are similar with each other, just like the

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filtered waveform has been widened after using WMD method. The black arrows shown in

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Fig.4b clearly demonstrate the position of waveform variation.

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The Original waveform and WMD waveform are shown in the Fig.5a, where the WMD

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waveform in Fig.5a is the same as the WMD waveform in Fig4a which is generated from the filtered waveform. From Fig.5a, we can see that the two waveform are similar with each other.

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And the black arrows clearly show that the position of waveform variation. Comparing Fig.4b

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with Fig.5b, we can see that the waveform without low frequency components, and even appears fake peak (dash line). While the WMD waveform which was generated from the filtered

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waveform was not been influenced. It proves that WMD waveform is much closer to the original

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waveform, and WMD method is a good way to reconstruct the low frequency information.

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According to the waveform analysis and the compared results, it proves that the WMD waveform is very reliable, and we can use this reliable low frequency information which is reconstructed by WMD method to invert a background velocity models for FWI.

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Fig.4 One trace of seismic waveform; (a) Filtered waveform without low frequency information and its WMD waveform;

(b) Partial

enlarged view from (a).

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Fig.5 One trace of seismic waveform; (a) Original waveform and WMD waveform;

(b) partial enlarged view from (a).

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In order to demonstrate WMD method is an effective way to reconstruct low frequency information simulate the actual situation, we use high pass filtering to filter out information

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below 9Hz from the recorded seismic data. From Fig.6, which gives an example of recorded

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seismic data (Fig.6a) and its WMD waveform (Fig.6c), the spectrum are shown in Fig.6b and 6d.

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From Fig.6b, we can see that the effective frequency band of recorded seismic data is ranging from 9Hz to 40Hz, while the effective frequency band of the WMD waveform is ranging from

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0-25Hz. From the numerical examples (Fig.6), we can see that WMD method not only can lower

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the dominant frequency of recorded seismic data, but also effectively reconstruct low frequency

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information.

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Fig6. One shot waveform and its spectrum;

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(a) A shot profile of high-pass filtering waveform; (b) Filtered waveform spectrum;

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(c) A shot profile of WMD waveform; (d) WMD waveform spectrum.

Wu and Chi et al.(2014) proposed Hilbert Envelope FWI which is based on Hilbert transform. But seismic waveform is not harmonic signals, so Hilbert Envelope (HE) is not exactly wrapping the seismic waveform (Fig.7a green line). For this reason, Wu et. al use mathematical transformation with respect to Hilbert Envelope to obtain the adjoint source. While the WMD waveform (Fig.7a red line) directly reflects the macro information of seismic data, and has the similar phase information with original waveform. We use WMD method to reconstruct the low frequency information and then use it for WMDFWI, so WMD method not only can dramatically simplify mathematics of the HEFWI, but also obtain more accurate macro information from seismic data for FWI.

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Fake peak

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Fig.7 Waveform and its spectrum; The data are generate from two-layer model. (a) Original waveform and its WMD waveform and Hilbert Envelope waveform; (c) Real seismic data;

(b) Spectrum.

(d) Partial enlarged view from (c)..

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From Fig.7, we can see that the HE waveform has many fake peak (dash circle), and the WMD waveform (red line) reflects the macro information of seismic data better (Fig.7a, and 7c).

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The WMD waveform are softer than HE waveform, and the spectrum (Fig.7b) shows that WMD

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method also can reconstruct low frequency information for seismic inversion. WMD waveform

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and its spectrum indicate that WMD is a good way to reconstruct low frequency information

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3. The gradient and adjoint source

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from recorded seismic data, and it can keep waveform unchanged.

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3.1 Conventional FWI

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FWI finds a subsurface parameter ( m ) by minimizing the data residual, between the synthetic data ( u ) and the recorded data ( d ). The 2-D constant density acoustic wave equation:

2 p 2 p 1 2 p    f (t ) ( x  xs ) x 2 z 2 v 2 t 2

(17)

where p denotes acoustic wave field, f (t ) denotes source function, v denotes velocity model. For conventional FWI objective function can be defined by:

E c (v ) 

T 1 u  d 2 dt   0 2 s r

(18)

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And the objective function( E c (v) ) derivative with respect to synthetic data ( u ) can be expressed





 E c (v )  (u  d ) u

(19)

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fs 

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(Bharadwaj, 2016)

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by:

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It is known that the FWI gradient can be calculated by zero-lag correlation of the forward wave-fields and the backward propagated residual wave-fields (Lailly, 1983; Tarantola, 1984;

 2 Pf

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t 2

  s

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 Pb dt

(20)

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E c (v) 2  3 v v

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Bunks et al., 1995). The gradient of multi-source can be expressed by following:

where Pf denotes incident wave-field, Pb denotes back-propagation adjoint source wave-field. s denotes the number of source,

r denotes the number of receiver,

T denotes the

total recording time.

3.2 Waveform mode decomposition FWI

In this paper, we use u w to denote the WMD waveform of synthetic data, and use d w to denote the WMD waveform of recorded data. It can be expressed by:

u w  Wu

(21)

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d w  Wd

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(22)

In the above equation, W is a WMD operator. After using WMD method, we can obtain

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uw , d w which demonstrate the macro information of seismic data. In order to find a macro

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structure of velocity models by minimizing the difference between u w and d w . We can define an

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objective function for WMDFWI, which only need to match u w with d w , so the objective

1 2

  u T

s

r

0

 d w  dt 2

w

(23)

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E w (v ) 

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function can be expressed by the following equation:

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In this paper, we use adjoint-state method to calculate the gradient for WMDFWI. The objective function of WMDFWI aims to minimize the difference between u w and d w . So WMD method is a kind of pre-processing techniques. After using the pre-processing techniques of WMD method, we treat d w as the original recorded data, and u w as the original synthetic data. the the WMDFWI adjoint source can be calculated by the following equation:

f sw 





 E w (v )  (uw  d w ) uw

(24)

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 2 Pf

0

t 2

  s

r

 Pbw dt

(25)

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E w (v) 2  3 v v

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So the gradient of WMDFWI can be expressed by following:

where Pf denotes incident wave-field, Pbw denotes back-propagation WMDFWI adjoint source

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wave-field.

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3.3 Characteristic of WMDFWI and HEFWI

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In order to obtain the adjoint source of HEFWI (Chi 2014), we need to use Hilbert

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transformation and chain rule with respect to seismic data, which could distort the FWI adjoint source waveform. While the WMDFWI adjoint source can be obtained by equation(24), it only

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needs to reconstruct WMD waveform from synthetic data and recorded data. The waveform of HEFWI adjoint source (Fig.8a) has a great difference from the original one, and the waveform of HEFWI adjoint source is sharper than the original one, at the same time, the dominant frequency of HEFWI adjoint source is higher than the desirable value (Fig.8c). From the spectrum of adjoint source (Fig.8c), we can see that HE method can reconstruct low frequency information, but also brings waveform distortion problem (Fig.8a). It is known that the distorted waveform and high dominant frequency may cause cycle skipping problem. From Fig.8b, we can see that the waveform of WMD adjoint source becomes softer than the original one, and the spectrum shows that WMD adjoint source contains abundant low frequency information.The dominant frequency becomes lower than before and also loss lots of high frequency information, so it can effectively mitigate the cycle skipping problem.

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Fig.8 Adjoint source and its spectrum; the data are generated from the modified Marmousi model with blended source; (a) Original and HE adjoint source ; (b) Original and WMD adjoint source ; (c) spectrum.

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Fig.9 One shot waveform of adjoint source and its spectrum;

(c) One shot waveform of HEFWI adjoint source; (d) spectrum;

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(e) One shot waveform of WMD adjoint source; (e) spectrum;

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(a) One shot waveform of high-pass filtering adjoint source; (b) spectrum;

In time domain FWI, gradient can be calculated by zero-lag correlation of the forward

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wave-fields and the backward propagated residual wave-fields. If the adjoint source does not

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contain low frequency component, the zero-lag correlation results also do not have low frequency component, therefore we can not obtain the macro information of velocity models.

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Similarly, if the dominant frequency of adjoint source is very high, the zero-lag correlation

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results are most high frequency information, the high dominant frequency could influence the

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inversion results, and cause severe cycle skipping problem. So the frequency components of adjoint source determine inversion precision. In order to solve the cycle skipping problem, we need to reconstruct the low frequency components and lower the dominant frequency for the adjoint source. We propose to use WMD method to process the recorded data and synthetic data. To demonstrate WMD method is better than HE method, we use high passing filter to filter out components below 9Hz from recorded data, the filtered waveform of adjoint source and its spectrum are shown in Fig.9a and 9b, respectively. We take filtered data to compare with HE data about the adjoint source frequency components. From the spectrum (Fig.9b), we can clearly see that there is no signal below 9Hz in the filtered data,. while from the adjoint source of HEFWI (Fig.9c and 9d), we can see that the adjoint source of HEFWI does contain abundant low frequency information, but the dominant frequency is higher than desirable value (Fig.9d), and the high dominant frequency adjoint source may cause cycle skipping problem. According to

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equation (24) we can obtain adjoint source of WMDFWI, it does not need to use chain rule and mathematics transformation, just like the way we use adjoint source of the conventional FWI.

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From Fig.9e and 9f, it proves that WMD method is an effective way to reconstruct low frequency

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components, and also can lower the dominant frequency.

According to equation (23) and (24), we can use the WMD waveform to implement for

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WMDFWI. The waveform contains abundant low frequency information and with low dominant

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frequency in the adjoint source, so WMDFWI can get a good smooth velocity model and then

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use it as an initial model for conventional FWI to get the final inversion result.

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4. Adaptive multi-step inversion strategy for WMDFWI

Multi-scale strategy in time domain is similar to that in frequency domain. Using frequencies from low to high to invert the velocity models from large scale to small scale, resulting in very good inversion results. Similarly, in this paper, we propose a new adaptive multi-step inversion strategy for WMDFWI which is easier to implement and also has the advantages of multi-scale strategy. The detailed descriptions are shown as following .

Step1: Compute the WMD waveform, and use it for WMDFWI:

uw  upenv(u)  lowenv (u)

(26)

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If noise is very strong, we can use equation (3) for further decomposing.

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Step2: Use inversion result in Step1 as an initial model, and then adaptively decrease the

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u  rupenv(u)  lowenv (u)

(27)

r 1

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uwr 

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proportion of u w in FWI:

where r is a weight coefficient, in the equation(27), we can adjust r to control the proportion

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of WMD waveform in the process of FWI. In this paper, according to slope of the objective function and iteration numbers to adjust the value of r . By adaptively adjusting the value of r ,

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we can obtain an adaptive multi-step inversion strategy.

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Step3: Use inversion result in Step2 as an initial model for conventional FWI.

WMD method uses Cubic Spline Interpolation to obtain the WMD waveform, and it has many different characteristics with EMD method, as we have said. But WMD method and EMD method have the same mathematical theoretical basis. EMD method was effectively used for signal denoising (Tsolis G, Xenos T D,2011) , so WMD method has the same ability to resist noise. According to equation (25), (26), and Fig.10, we can see that when adaptively decrease the proportion of WMD waveform in the process of FWI, the modified waveform is closer to original waveform, and also contains abundant low frequency information. Using the adaptive multi-step inversion strategy, we can get a kind of multi-scale FWI results, and this adaptive multi-step strategy can effectively mitigate the cycle skipping problem, even with strong Gaussian noise.

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Fig.10 Waveform and its spectrum;

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The data are generated from two-layer model.

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(a) Original waveform and its WMD waveform and modified waveform; (b) Spectrum.

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Sometimes, we do not need to reconstruct strong ultra low frequency components, we just need to reconstruct low frequency information, even if the reconstructed low frequency components are very weak. We can use low-passing filter to implement for multi-scale FWI (Bunks,1995) to obtain good inversion results. So the best way to reconstruct low frequency components is to protect the seismic waveform from distortion, at the same time, get the low frequency components. For this purpose, the WMD method meet the requirements of reconstructing low frequency components without dramatically changing the original waveform.

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5. Numerical examples

We apply WMDFWI to the modified Marmousi model as shown in Fig.11b. The grid size of

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this modified Marmousi model is 69 192 ,with a grid interval of 12.5 m, and the velocity value

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ranges from 1.5km/s to 4km/s.. The initial velocity model is built by linear model (Fig.11a).

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Compared with the two models, we can see that linear initial model is far from the true model, even do not satisfy the variation tendency of velocity value with the true model, so it is very

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difficult for conventional FWI. We use dynamic random blended source encoding strategy (Boonyasiriwat, 2010) to accelerate the process of FWI, and the blended source is random

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distributed on the surface of the modified Marrmousi model, so it can dramatically improve the

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water depth is 62.5m.

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computational efficiency for FWI. There are 192 receivers equally spaced on the surface, and the

a

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Fig.11 Velocity models; (a) Linear initial velocity model; (b) True modified Marmousi model .

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5.1 WMDFWI without Low frequency information

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In order to prove that WMD method can reconstruct the low frequency components and lower the dominant frequency, we choose Ricker wavelet as a seismic source function, with

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dominant frequency of 22 Hz (Fig.12). Recording time of forward modeling is 2.5s with a time

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interval of 1ms. To demonstrate our methods can mitigate cycle skipping problem, even if there is no low frequency information in seismic recorded data, we use Butterworth high passing filter

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to filter out the information below 9Hz. The filtered Ricker wavelet waveform and its spectrum are shown in Fig.12, and we can see that the Ricker wavelet waveform without low frequency

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components becomes sharper than the original one (Fig.12a red line).

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Fig.12 Ricker wavelet and its spectrum; (a) Ricker wavelet waveform and its filtered waveform without information below 9Hz; (b) Spectrum.

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It is known that low frequency information is very important for recovering the macro structure of velocity models. In this section, we use high passing filter to test the inversion result

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influenced by the low frequency. From the Fig.13a we can see that multi-scale strategy (Bunks)

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can effectively avoid the cycle skipping problem when the ricker wavelet contains abundant low

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frequency information (Fig.12a blue line). While when we use high passing filter to filter out information below 9Hz (Fig.12a red line), we can see that there is an obvious mistake in the

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multi-scale inversion result (Fig.13b,). Multi-scale strategy can effectively lower the dominant

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frequency of FWI adjoint source, but it can not work when seismic data miss low frequency information, especially when the initial model is not good enough. So we must reconstruct the

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low frequency information to invert the macro structure of velocity models. Next, we will give

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some detailed examples to show that the WMDFWI can get good inversion result, even if there is

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no low frequency information in the seismic data.

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Fig.13 FWI results; (a) multi-scale FWI (0-15Hz); (b) multi-scale FWI (9-15Hz), without low frequency information.

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Conventional FWI result is shown in Fig.14a, which is start from the linear initial model (Fig.11a), the inversion result is seriously deviated from the true model. WMDFWI result is

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shown in Fig.13b, we can see that the macro structure of modified Marmousi model has been

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retrieved. Using WMDFWI result as an initial velocity model, and then use conventional FWI to

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invert the final velocity model, the final inversion result is shown in Fig.14d. Compared Fig.14a with Fig.14d, which makes a clear demonstration that the WMDFWI+conventional FWI result

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has a good convergence with the true model and does not depend on initial model, even if there

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is no low frequency information in recorded seismic data. To demonstrate the inversion accuracy of velocity value, we extract one trace from the velocity model, and the location is marked by red

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line as shown in Fig.14, then plot the inversion velocity value with true model and initial model

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(Fig.15). Compared Fig.15a with Fig.15d, we can see that the velocity value of

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WMDFWI+conventional FWI result is closer to the true velocity model, and it is much more accurate than conventional FWI result.

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a

d

Fig.14 FWI results

(a) The conventional FWI result; (b) The WMDFWI inversion result; (c) the WMDFWI+conventional FWI with 35 iterations result; (d) The WMDFWI+conventional FWI final result .

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a

d

Fig.15 FWI result; The trace is extracted at the 1.25km in the distance of modified Marmousi model as shown in Fig.14. (a) The conventional FWI result;

(b) The WMDFWI inversion result;

(c) the WMDFWI+conventional FWI with 35 iterations result;

(d) The

WMDFWI+conventional FWI final result .

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5.2 Adaptive multi-step WMDFWI and noise test

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FWI is a strongly nonlinear problem and sensitive to noise, when recorded data contain much noise, it is difficult for us to use synthetic data to match the recorded data. A series of

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denoising methods have already been proposed, but if we use denoising methods to dispose the

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recorded data such as high-passing filter, it may cause waveform distortion problem and destruct the low frequency information. In this paper, we use WMD method to alleviate the influence of

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the noise on FWI. Where WMD is a method which can not only denoise but also be able to reconstruct low frequency information. In order to make sure the original waveform is not

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remoulded, in the process of FWI, we do not use any denoising method. But if we use WMDFWI

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result as an initial model, and then directly using it for conventional FWI, it may also occur cycle

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skipping problem, when there is strong noise in the recorded data. To solve this problem, we use a new adaptive multi-step inversion strategy for WMDFWI, which is gradually decreases the proportion of WMD waveform in the seismic data.

To test the influence factor of adaptive multi-step WMDFWI, such as anti-noise ability, independence of initial model and the requirement of low frequency components. We use Butterworth filter to filter out information below 9Hz(Fig.12a, red line), and gradually add Gaussian noise into the blended recorded data as shown in Fig.16. The signal-to-noise ratio (SNR) can be defined by：

2

SNR  10  lg(

d data ) noise 2

(28)

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From Fig.16c, we can see that Gaussian noise almost cover up all valid information, and we can only see a little about direct wave. Where SNR  7.27 denotes that the power spectrum of

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Gaussian noise is much larger than valid signal.Comparing Fig.16d with Fig.17d, it is clear that

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the WMD method has a strong ability of denoising, and the waveform basically remain

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unchanged. Therefore WMD methed has a great advantage of resist noise, maybe it is suitable

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for land seismic FWI.

c

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Fig16 Blended recorded data with Gaussian noise. (a) No noise; (b) SNR=0.01; (c) SNR=-7.27; (d) SNR=-13.88.

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Fig17 Blended WMD data. The data are reconstructed from Fig.16abcd.

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d Fig18 One trace from seismic data.

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(a) No noise waveform; (b) Waveform with SNR=-12.98 ; (c) First step WMD result; (d) Second

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step WMD result.

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To further prove that WMD method has the strong ability of denoising, we extract one trace

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from seismic data with SNR=-12.98. From 18b, we can see that Gaussian noise almost cover up all valid information(Fig.18a), while after using WMD method to decompose the signal which

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contains much Gaussian noise, the valid waveform could be seen in Fig.18c as marked in red line

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circle, but it still contains a lot of noises. After the first disposition, we use equation (3) to conduct further decomposition, the result is shown in Fig.18d, the valid waveform was retrieved,

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and effectively suppress Gaussian noise.

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We use linear initial velocity model as an initial model (Fig.11a), and use the adaptive multi-step WMDFWI to implement for seismic inversion. We gradually lower the SNR, such as SNR=0.01, SNR=-7.27, SNR=-13.88 and SNR=-16.91. The adaptive multi-step WMDFWI with strong Gaussian noise inversion results are shown in Fig.19, and the final inversion results are shown in Fig.20. From the inversion results with Gaussian noise, we can see that the adaptive multi-step WMDFWI has a great advantage of anti-noise.

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Fig.19 WMDFWI results with different signal-noise ratio; (a) SNR=0.01; (b) SNR=-7.27; (c) SNR=-13.88; (d) SNR=-16.91.

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Fig.20 WMDFWI+Multi-step FWI results with different signal-noise ratio;

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5.3 Advantages of WMDFWI

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(a) SNR=0.01; (b) SNR=-7.27; (c) SNR=-13.88; (d) SNR=-16.91.

Both of WMD method and HE method can reconstruct low frequency information for FWI and the waveform both reflect the macro information of seismic data. But when we use HE method to reconstruct the low frequency information, and then use the chain rule to obtain the adjoint source for HEFWI, it may cause very high dominant frequency in the adjoint source, while the high dominant frequency may cause cycle skipping problem.

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In order to compare HEFWI with WMDFWI, we use the same objective function, which is

  e T

s

r

0

0

 e dt 2

(29)

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1 2

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E H (v ) 

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built by Chi et al.(2014):

e denotes the Hilbert Envelope of recorded data. So the adjoint

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Envelope of synthetic data，and

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where s, r denote source number and receiver number, respectively. e0 denotes the Hilbert

H

denotes Hilbert transform , u denotes synthetic data, u~ denotes Hilbert transform of

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where

(30)

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e  e   e0  e ~  fs  u 0 u  H e  0   e0 

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source can be defined by:

u . According to the equation (30), if e0  0 , the HEFWI adjoint source may be not stable. Even if we put a small number  on the denominator, the

 e0  e    still  e0   

may be a sharp pulse.

From Fig.9d, we can see that the dominant frequency of the HEFWI adjoint source is very high, especially near the seismic source. It is known that the seismic data near the seismic source are mainly reflected wave, while the reflected wave are corresponding to the detailed structure of underground velocity models. So the reflected wave of synthetic data and recorded data may e e

have severe waveform mismatch problem (  e0      (t ) ), this severe waveform mismatch 

0



phenomenon leads to the high dominant frequency of HEFWI adjoint source ( f s  u (t )  H  (t )u~).

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Because the dominant frequency of HEFWI adjoint source is very high, we must use the Ricker wavelet with low dominant frequency. We use the same parameter with the previous

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paper (Chi,2014) to test the inversion ability of HEFWI. The source function is Ricker wavelet

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with dominant frequency of 8 Hz. Recording time of forward modeling is 2.5s with the time

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interval of 1ms. And we use Butterworth high pass filter to filter out information below 5Hz. The grid size of the modified Marmousi model is 69 192 ,and the grid interval is 15 m. The initial

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velocity model for this inversion test is built by linear model (Fig.11a).

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Fig.21 FWI results; (a) HEFWI result;

(b) WMDFWI result.

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In this paper, we mainly solve the cycle skipping problem, if we obtain a good initial velocity model, we can avoid conventional FWI trapped in local minima. So we only focus on

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the macro structure of the true velocity model. HEFWI result is shown in Fig.21a. It proves that

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HEFWI can mitigate the cycle skipping problem. WMDFWI result is shown in Fig.22b. It looks

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like the HEFWI result is better than WMDFWI result, but in this test, we use ricker wavelet with very low dominant frequency (10Hz), and the dominant frequency of WMDFWI adjoint source

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is lower than 8Hz, so it only can get the more macro structure of the true velocity model, if we

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use the adaptive multi-step strategy, it also can get better result, like that in Fig14b. Therefore when ricker wavelet has low dominant frequency, HEFWI and WMDFWI both can get good

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inversion result.

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When we use higher dominant frequency (20Hz) for ricker wavelet, the inversion results are shown in the Fig22. We can see that, the HEFWI result (Fig.22a) has severe cycle skipping problem, while the WMDFWI result is better. From the comparison between Fig22a and Fig22b, it proves that WMDFWI can obtain better inversion result, even if the dominant frequency of seismic data is very high. The dominant frequency of field seismic data may reach 40Hz, so that the WMDFWI is more applicable to the field data processing.

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Fig.22 FWI results;

(b) WMDFWI result.

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(a) HEFWI result;

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HEFWI has the strong ability of anti-noise (Luo 2015). According to the comparison results (Fig.21, Fig.22), if we want to compare the two methods about the anti-noise ability, we have to avoid HEFWI appear cycle skipping problem, so we use the Ricker wavelet with low dominant frequency(8 Hz). The Gaussian noise is added into the recorded data, with SNR=-2.53(very strong noise, may be stronger than Luo 2015). From Fig23a, we can see that HEFWI affected by noise seriously, while WMDFWI is not sensitive to the Gaussian noise. The WMDFWI result is not good, because the dominant frequency is too low,and it only can retrieve the macro structure of the velocity models. When we use higher dominant frequency, the inversion results is shown in Fig20b, it proves that WMDFWI has the strong anti-noise capability.

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(b) WMDFWI result.

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(a) HEFWI result;

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Fig.23 FWI results;

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Comparison of Fig.22 and Fig.23 shows that WMDFWI is superior to HEFWI in some ways.

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In all, WMD is a good method to reconstruct the low frequency information which is very reliable, it can lower the dominant frequency of the seismic data, and it also has the strong ability of resisting noise.

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6. Conclusion

The Waveform Mode Decomposition method is used to reconstruct the low frequency

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information for Full Waveform Inversion to mitigate the severe cycle skipping problem. The

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WMDFWI is a fully data driven approach which requires less prior information. We mainly

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focus on the dominant frequency of the adjoint source and reliability of the WMD waveform with the purpose of demonstrating why the WMDFWI performs better than conventional FWI

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and HEFWI when the initial velocity model is far from the true model and low frequency data are missing. The WMD waveform reflects the macro information of the seismic data, therefore

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using reconstructed recorded data to implement for FWI, it can get a good initial velocity model

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and it is not likely to be influenced by the cycle skipping problem. Gaussian noise test shows that

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the adaptive multi-step WMDFWI has a strong ability to resist noise, can effectively mitigate influence of noise, maybe WMDFWI is more suitable for land seismic data. The numerical experiments demonstrate the applicability of this approach, especially when the initial model is far from the true model and low frequency data are missing, even if it contains much noise. In all, the adaptive multi-step WMDFWI can be a good supplement for conventional FWI.

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Acknowledgments

We want to thank the National Natural Science Foundation of China (grant number: 41674124

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and number: 41304086) and the national ‘863’ Project (number: 2014AA06A605) for their

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support of this work. We would also like to thank the support from the Key Laboratory of

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Applied Geophysics, the Ministry of Land and Resources of China. We also acknowledge the contributions of five anonymous reviewers. Their comments were extremely insightful, greatly

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improving the clarity and overall quality of the manuscript.

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Ru-Shan Wu, Jingrui Luo, and Bangyu Wu (2013) Ultra-low-frequency information in seismic data and envelope inversion. SEG Technical Program Expanded Abstracts 2013: pp. 3078-3082. http://library.seg.org/doi/abs/10.1190/segam2013-0825.1

Zhiguang Xue, Nick Alger, and Sergey Fomel (2016) Full-waveform inversion using smoothing kernels. SEG Technical Program Expanded Abstracts 2016: pp. 1358-1363.doi: 10.1190/segam2016-13948739.1 http://library.seg.org/doi/abs/10.1190/segam2016-13948739.1

Yong Hu, Liguo Han, Pan Zhang, Zhongzheng Cai, and Qixin Ge (2016) Multistep full-waveform inversion based on waveform-mode decomposition. SEG Technical Program Expanded Abstracts 2016: pp. 1501-1505. doi: 10.1190/segam2016-13712251.1 http://library.seg.org/doi/abs/10.1190/segam2016-13712251.1

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Zhang P, Han L G, Zhou Y, et al. Passive-source multitaper-spectral method based low-frequency data reconstruction for active seismic sources[J]. Applied Geophysics, 2015, 12(4): 585-597.

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http://manu09.magtech.com.cn/jweb_yydqwl/EN/abstract/abstract355.shtml

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Zhou C, Cai W, Luo Y, et al. Acoustic wave-equation traveltime and waveform inversion of crosshole seismic data[J]. Geophysics, 1995, 60(3): 765-773.

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http://geophysics.geoscienceworld.org/content/60/3/765.abstract

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Research Highlights

1. Ultra low frequency components of seismic data can be reconstructed by waveform mode

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decomposition method.

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2. Waveform mode decomposition method is a good way to mitigate the cycle skipping for full

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waveform inversion.

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ability to resist Gaussian noise.

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3. Adaptive multi-step strategy combined with waveform mode decomposition FWI has a strong

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