Low frequency is a key issue to reduce the nonlinearity of elastic full waveform inversion. Hence, the lack of low frequency in recorded seismic data is one of the most challenging problems in elastic full waveform inversion. Theoretical derivations and numerical analysis are presented in this paper to show that envelope operator can retrieve strong low frequency modulation signal demodulated in multicomponent data, no matter what the frequency bands of the data is. With the benefit of such low frequency information, we use elastic envelope of multicomponent data to construct the objective function and present an elastic envelope inversion method to recover the long-wavelength components of the subsurface model, especially for the S-wave velocity model. Numerical tests using synthetic data for the Marmousi-II model prove the effectiveness of the proposed elastic envelope inversion method, especially when low frequency is missing in multicomponent data and when initial model is far from the true model. The elastic envelope can reduce the nonlinearity of inversion and can provide an excellent starting model.

Over the past several decades, the oil and gas exploration industry has been making great efforts to use full waveform inversion (FWI) to achieve high-resolution quantitative subsurface models by exploiting full information contained in prestack seismic data. However, most applications of FWI to real data have been performed under acoustic approximation. In practice, there is a simultaneous recording of the complicated elastic effects during seismic wave propagation in subsurface media, such as shear-wave reflections, converted waves and amplitude offset variations. Therefore, acoustic FWI suffers from limitations due to the acoustic approximation and may lead to erroneous models if applied to elastic data without specific data preprocessing and inversion preconditions (Brenders and Pratt, 2007; Brossier et al., 2009). Meanwhile, quantitative imaging of the elastic properties of the subsurface is essential for oil and gas-reservoir characterization because elastic parameter quantification can be used to deduce the lithology, fluid content and pore pressure of rocks (Tatham and Stoffa, 1976). Hence, elastic full waveform inversion (EFWI) is urgently required.

Up to now, only a few applications of EFWI have been presented. Aside from the computational challenges, EFWI is a highly ill-posed and nonlinear problem. It is sensitive to the quality of data, the accuracy of starting model and the frequency band of seismic data (Virieux and Operto, 2009). To make things worse, multiparameter EFWI is more ill-conditioned than acoustic FWI because different parameters are more or less coupled and the propagation of the elastic wave is more complicated than acoustic wave.

It is well known that low frequency is essential to reduce the nonlinearity of FWI, and the lack of
ultra-low frequency (frequency below 2

To overcome the lack of low frequency issue in FWI, some other forms of FWI methods are developed by introducing different objective function instead of waveform. For example, Luo and Schuster (1991) used first-arrival traveltime as matching objects and applied cross-correlation to measure the time difference. Zhang et al. (2011) extended this method to update the deep part of subsurface model by using reflection data. Although traveltime cross-correlation misfit function can reduce the nonlinearity of inversion, it requires manual picking and windowing of the arrival time, which make this method time consuming and man-made error dependence. Laplace-domain waveform inversion (Shin and Cha, 2008) can estimate a smooth, large-scale subsurface model by matching weighted waveforms multiplied with a damping function. But it requires long offset seismic data or ultra-low frequency to update deep part of subsurface models and is sensitive to the noise appearing prior to the first arrivals. Bozdağ et al. (2011) discussed instantaneous phase misfit function, envelope misfit function and their applications in the kernel sensitivity analysis of global seismic tomography. Using Hilbert transform, phase and amplitude attribute can be separately used to reduce the nonlinearity of the inverse problem (Bozdağ et al., 2011). However, they just qualitatively compared with different misfit functions and did not show any inversion examples.

Recently, Wu et al. (2014) and Chi et al. (2014) further developed this envelope inversion method
and applied it into acoustic FWI. They showed that by envelope operator, ultra-low frequency
information can be retrieved from acoustic records and hence to reduce the starting-model dependence
of FWI. As Brossier et al. (2009) mentioned in his paper, the starting-model dependence of EFWI is
more severe and the low frequency (frequency below 3.5

In this paper, an elastic envelope inversion method is presented to retrieve low frequency from standard multicomponent data, even the data does not contain any low frequency. To demonstrate the effectiveness of the retrieved low frequency in reducing the nonlinearity of EFWI, some numerical analysis on misfit function are made. Then brief theoretical derivations on how the envelope operator retrieves the low frequency from multicomponent data are proposed. Finally, based on the advantage of elastic envelope inversion, we present two types of numerical elastic inversion tests using a two-step inversion strategy. For the first step, elastic envelope inversion is implemented to recover the long-wavelength components of subsurface models to obtain acceptable background models. Then, these elastic envelope inversion results are used as the starting models for conventional EFWI to recover the short-wavelength components of subsurface models. Final results are presented to show the validity of this inversion strategy.

The EFWI is based on the following second-order linear
elastic wave equation system

The objective function for conventional EFWI is generally defined as

In the time domain, higher power

From the defined elastic envelope objective function (Eq. 4), we can get the corresponding gradient
operator and the adjoint source by the adjoint-state approach, a way of computing a gradient of an
objective functional depending on recursive simulations (Plessix, 2006; Brossier, 2009; Chavent,
2009; Pesssix and Cao, 2011; Gholami et al., 2013). Considering compressional-wave speed

Chi et al. (2014) and Wu et al. (2014) both mentioned the demodulation effect of the envelope operator in acoustic FWI. And elastic envelope operator also works efficiently in retrieving ultra-low frequency from standard multicomponent seismic records even when the data is filtered out by a high-pass filter. In the next section, we shall focus on the role of such low frequency data playing in reducing the nonlinearity of elastic envelope inversion.

Elastic full waveform inversion is an ill-posed, nonlinear problem even for apparently simple models involving few parameters. Low frequency is a key issue to reduce the nonlinearity of EFWI, but mostly it is missing in real seismic data. In Eq. (4), we introduce the elastic envelope objective function. The envelope is extracted from the waveform data by a square-root operator. The square-root operator is a nonlinear operator and can be treated as a nonlinear filter. Hence, the envelope operator is equivalent to a demodulation operator: if we treat the seismogram as a modulated signal, the low-frequency envelope is the modulation signal and the high-frequency reflection waveforms are the carrier signal. And that is the superiority of envelope misfit function to conventional misfit function, which gives us a way to overcome the low frequency issue in EFWI.

To test the ability of the elastic envelope operator in retrieving low frequency from seismic
signal, a rick wavelet signal and its envelope are plotted in Fig. 1. By comparing their spectra,
strong ultra-low frequency can be found in the envelope of wavelet but not shown in the wavelet
itself. More complicate signals are presented in Fig. 2 to further test the envelope operator. An
elastic shot gather is shown in Fig. 2a and b for the horizontal and vertical components of partial
displacement from the synthetic data set of 2-D Marmousi-2 model (see the numerical tests section
for more details). Their envelope data are plotted in Fig. 3 and the corresponding spectra are
presented in Fig. 4. It is easy to notice that the envelope data have stronger ultra-low frequency
and weaker high frequency comparing with the original waveform. Even when we filter the data below
5

Gradient in EFWI determines the direction of the model updating. Next, a simple test is made to
illustrate the superiority of the elastic envelope gradient to conventional gradient. The reference
model is homogeneous with the P-velocity 1.5

In this section, we concentrate on analyzing the nonlinearity of elastic envelope inversion and
conventional EFWI to further reveal the advantage of elastic envelope misfit function. A simple
numerical test is made to show the advantage of elastic envelope misfit function in reducing the
nonlinearity of inversion. Here, we use a 2-D canonical elastic model to investigate the performance
of both envelope and waveform misfit functions with respect to P-wave and S-wave perturbations
around the true model. The reference model is inhomogeneous whose background velocity is
constant-gradient. The P-velocity is 1.5

The results shown in Fig. 11 clearly demonstrate the strong nonlinearity of the waveform functions
on either P-wave or S-wave velocity variations (black line). The waveform functions with respect to

In order to test the sensitivity of the objective functions to the low-frequency components in the
data, we redo the 2-D canonical test by filtering the synthetic data below 6

Due to the robust performance on retrieving low frequency from multicomponent data, the elastic
envelope objective function is superior to waveform objective function to recover the background of
subsurface model. But if the collected multicomponent data contains no low frequency information,
where the low frequency information comes from in the corresponding envelope? In acoustic envelope
FWI, Wu et al. (2014) presented an approximate formula to show how the low frequency comes from, and
in elastic envelope inversion, we can also derive an approximate formula below for multicomponent
data (see Appendix A for more details).

From Eq. (10), we see clearly where the low frequency in envelope comes from. For the first term in
the right side of Eq. (10), the envelope of high-frequency original wavelet is low-resolution (shown
in Fig. 1); for the second term, because

In the process of EFWI, the inversion shall firstly recover the structure corresponding to the frequencies whose energy are the largest in the data. Therefore, due to the strong low frequency contained in envelope, the elastic envelope inversion shall firstly recover the long-wavelength of the subsurface model, which behaves as the first step of multiscale strategy. That is why we can use results produced by envelope elastic inversion as the starting model for conventional EFWI. According to the advantage of elastic envelope inversion, we present a two-step elastic inversion strategy. In the first step, elastic envelope inversion is used to recover the low-wavenumber components of the subsurface model. Then, the obtained results in first step are used as the starting models for conventional EFWI to recover high-wavenumber components. And some numerical examples are presented in the next section to prove the effectiveness of the two-step elastic inversion strategy.

We test our two-step elastic envelope inversion strategy on a modified 2-D elastic Marmousi-2 model
(Martin et al., 2002). In order to reduce the computation burden, we remove the upper water layer
and keep only the central portion of the P-wave velocity model from the original Marmousi-2
model. Then the S-wave velocity model is regenerated from the original P-wave velocity model using
formula (11). In this formula, we use two

To reduce the nonlinearity of the inversion caused by free-surface effects (Brossier et al., 2009),
an absorbing boundary is placed on the top of the models. To simulate onshore synthetic data on the
Marmousi-2 model, 80 explosive sources along the surface are inspired with the interval of
125

In the first type of test, we use the synthetic 2C data as input to do both conventional EFWI and elastic envelope inversion. Figure 15 shows the revealed results of conventional EFWI after 100 iterations. Because the gradient starting models deviate substantially from the true models and strong ultra-low frequency is unavailable in the data (as shown in Fig. 4), conventional EFWI suffers from cycle-skipping problem and fails to converge to acceptable results. During this inversion, only short wavelength components are updated while at incorrect positions. The inversion process itself is unstable.

Figure 16 shows the results of the elastic envelope inversion. From the results, we can see that the
long-wavelength components of the subsurface models are well recovered. Then, we put the results as
the starting models for conventional EFWI in the second step and the final results are shown in
Fig. 17. We can obviously see that, after the second step inversion, the short wavelength components
of the subsurface models are updated at the right positions and high resolution results are finally
achieved. Comparing results in Fig. 15 with Fig. 17, final results of two-step inversion are much
better than conventional EFWI. The improvements are more obvious on the vertical profiles
(

In the second test, inversions are implemented with the filtered synthetic data whose frequencies
below 5

Envelope operator can demodulate the multicomponent data and extract strong low frequency modulation signal contained in recorded data. After envelope transform, most energy of the corresponding envelope are concentrated on ultra-low frequencies zone. With the benefit of these strong ultra-low frequency information, the long-wavelength components of the subsurface model can be well recovered by elastic envelope inversion. It behaves as a natural multi-scale strategy which starts from the ultra-low frequency. Hence, the nonlinearity and the starting-model dependence of elastic envelope inversion are greatly reduced. The elastic envelope inversion is an effective and robust method to achieve accurate background model using the collected multicomponent data, even when the low frequency information is missing in the data.

Elastic multicomponent displacement for a point source can be represented in frequency domain as

Equation (A5) can be rewritten in time domain as

We can treat Eq. (A10) as a product of two functions: a carrier signal and a modulation signal. For
the first term on the right side, the carrier signal is the source wavelet and the modulator is the
direct propagator; for the second term, the carrier is

The authors gratefully acknowledge the financial supports by the National Science and Technology Major Project of China (Grant No. 2011ZX05005-005-007HZ) and the National Natural Science Foundation of China (Grant No. 41274116, 41474034).