Multiscale Full-Waveform Inversion with Land Seismic Field Data: A Case Study from the Jizhong Depression, Middle Eastern China

Abstract

The Jizhong depression contains several geothermal reservoirs that are characterized by localized low-velocity anomalies. In this article, full-waveform inversion (FWI) is used to characterize these anomalies and determine their extent. This is a challenging problem because the reservoirs are quite small and the available data have usable frequencies only down to 5 Hz. An accurate-enough starting model is carefully built by using an iterative travel time tomography method combined with a cycle-skipping assessment method to begin the inversion at 5 Hz. A multiscale Laplace–Fourier-domain FWI with a layer-stripping approach is implemented on the starting model by gradually increasing the maximum offset. The result of overlapping the recovered velocity model on the migrated seismic profile shows a good correlation between the two results. The recovered model is assessed by ray tracing, synthetic seismogram modeling, checkerboard testing and comparisons with nearby borehole data. These tests indicate that low-velocity anomalies down to a size of 0.3 km × 0.3 km at a maximum depth of 2 km can be recovered. Combined with the well log data, the resulting velocity model allows us to delineate two potential geothermal resources, one of which was previously unknown.

1. Introduction

The quantitative imaging of subsurface elastic properties at depth is essential for engineering, exploration and reservoir characterization [1]. The theory of full-waveform inversion has been extensively developed since the pioneering work of [2], and there is a strong interest in applying this high-resolution model-building method to new locations and problems. Here, the application of 2D visco-acoustic full-waveform inversion to delineate geothermal reservoirs in the Jizhong depression is demonstrated.

Geothermal energy is a clean source of energy that is gaining importance as an alternative to hydrocarbons. Seismic imaging and FWI reveal structural and velocity information that is important for the understanding of geothermal systems. Supercritical water reservoirs are considered to be future geothermal energy sources. Both active and passive seismic methods have proven their effectiveness for geothermal exploration [3,4]. Two simulation cases using the full-waveform inversion method with both active and passive seismic data were carried out [5]. Synthetic results show that the supercritical water zone can be well-imaged by combining the FWI method, active seismic sources and appropriate natural earthquakes, as well as a distributed acoustic sensor (DAS) seismic array in the borehole and surface seismic array [5]. Tognarelli et al. (2020) [6] presented a case study of velocity model estimation in geothermal areas with full-waveform inversion of transmitted waves (direct waves, refracted waves and diving waves). The combination of global and local optimization methods allowed for the estimation of velocity to be independent of the a priori interpretation of the subsurface. The input data do not contain reflected waves, so the inverted velocity model does not contain too many high wavenumber portions and it can be used as complementary information to the standard seismic reflection image [6]. For full-waveform inversion in geothermal reservoir exploration, the input data, including reflections that contain high frequencies, can lead to a high-resolution model and even directly delineate the geothermal reservoir on the recovered model. There are several potential geothermal reservoirs in the Jizhong depression, which is located in middle eastern China. There are two horsts in the study area: the Rongcheng High and the Niutuozhen High; the potential geothermal reservoirs are located below these structures [7]. Seismic reflection data were collected to image the geothermal reservoir with the horst in the central part of the survey line. The reservoirs of interest are fractured geothermal reservoirs located under the regional bedrock, which extends to a depth of approximately 1 km. Because of the challenge of using seismic reflection imaging in such heavily fractured areas, it is difficult to delineate the structural scope of fractured geothermal reservoirs directly on the seismic profile. Based on the difference between the velocity of the geothermal reservoir and its surrounding region [8], a high-resolution velocity model was built with full-waveform inversion to help us to understand the scope of fractured geothermal reservoirs.

For FWI in the frequency domain, large offset seismic surveys can be inverted effectively using only a limited number of frequencies, thus reducing the number of forward solvers in a manner not possible with time-domain methods. As a result, the inversion can easily be implemented beginning from the lowest data frequency, thus mitigating the nonlinearities in the seismic inverse problem [9]. Hence, a multiscale approach based on frequency continuation is a common strategy designed to mitigate the nonlinearity of FWI. This strategy is naturally implemented by combining successive inversions of monochromatic datasets of increasing frequency [10]. Although the multiscale frequency strategy is useful to mitigate the nonlinearity of FWI, it may not prevent cycle skipping for waves that propagate over many wavelengths. To decrease the risk of cycle skipping, an accurate starting model should be carefully generated.

Although there are some successful studies of FWI with marine datasets [11,12,13], it is challenging to apply FWI to real land data due to the lack of low-frequency signals and limited offsets. Despite these difficulties, there are some successful applications of FWI to real land data [1,14,15,16,17,18,19]. Plessix et al. [15] used a single vibrator/single receiver dataset to create a distance-separated simultaneous sweeping dataset to study the robustness of full-waveform inversion in this acquisition context. That dataset contained frequencies down to 1.5 Hz and offsets up to 25 km and was thus well-suited for FWI. Although the result was somewhat contaminated by cross-talk between simultaneous sweeps, it was still quite satisfactory. Because a low-frequency vibrator is too expensive for most seismic fieldwork, many surveys instead use an explosive source. The lowest frequency of our field gathers is not lower than 5 Hz, constrained by the 10 Hz geophone. This is too high a starting frequency for existing FWI strategies. Górszczyk et al. (2017) proposed an offset continuation strategy based on a layer-stripping approach [12].

In this paper, aiming at a geothermal reservoir, a 2D case study from the Jizhong depression, which is located in middle eastern China, is presented. The FWI approach increases the resolution, thus reducing the need for more expensive data acquisition, which is important for lower-budget geothermal resources. The input data to full-waveform inversion contain both transmitted waves and reflected waves. First, the geologic background of our research area is introduced. Second, the data and preprocessing are discussed. Third, the methodology adopted for this study is introduced. Fourth, we show the results of synthetic testing with a model designed for the field area (which we call the XA model). Finally, the multiscale strategy is applied to the field data and a detailed interpretation and discussion are given. Our results indicate that FWI can be successfully applied to real land data with a starting frequency of up to 5 Hz through this multiscale strategy. The recovered high-resolution velocity model directly leads to the delineation of geothermal reservoirs.

2. Field Dataset

2.1. Geological Setting

The field dataset used in this study was acquired by the Institute of Geophysical and Geochemical Exploration from 2018 to 2020. The research area is called the Xiong’An New Area. It is located in the middle of the Jizhong depression, which is a part of the Bohai Bay basin, located in the middle eastern region of China (Figure 1a). The area has almost no topography, with a difference in elevation of only tens of meters except for the mountainous area in the northwest (Figure 1a). The study area is composed of the Rongcheng High, the southern Niutuozhen High, the southern Niubei Slope and the Baiyangdian Subsag (Figure 1a). The study area is located at the transfer zone between the central and northern Jizhong Depression [20,21]; it is bordered to the southwest by the Baoding Sag, to the northwest by the Xushui Sag, to the southeast by the Baxian Sag and to the south by the Raoyang Sag. At a larger scale, the study area is located on the eastern block of the North China Craton [22]. The NNE-trending thrust structure in this area, which can be seen clearly from the gravity anomalies in Figure 1b, were formed by a series of compression events during the early Yanshanian (a tectonic event primarily during the Jurassic period [23]). Figure 1b shows a regional gravity dataset in which the red (cyan) regions are positive (negative) anomalies corresponding to sags (highs) in the subsurface. We do not annotate the values of the gravity anomalies in Figure 1b for confidentiality reasons. In the late Mesozoic, the tectonic regime of the North China Craton changed from near N-S compression to WNW-ESE extension due to the closure of the Mongolian-Okhotsk Ocean and the subduction of the Paleo-Pacific Ocean [24]. The Bohai Bay Basin began to develop concurrently. Controlled by the Taihang Mountain piedmont fault and several secondary faults, intense extension developed in the study area, and this stretching lasted to the end of the Paleogene [24]. This extension resulted in the formation of extensional sags or subsags in the study area.

The sedimentary stratigraphy in this area contains Paleogene, Neogene and Quaternary strata. The bottom of the Paleogene strata is considered the basement in this area. The shallowest depth to this basement is approximately 800 m, located in the western Rongcheng high. From top to bottom, the stratigraphic system includes the Quaternary Pingyuan Formation consisting of silt and medium sand in the lower part and sandy clay in the upper part, with a thickness of 348–437 m [24]. The Neogene strata are composed primarily of fluvial sandstone and mudstone, of which the Guantao Formation has a thickness of 0–424.5 m and the Minghuazhen Formation has a thickness of 686–947 m. The Paleogene, distributed predominantly in the subdepression, experienced intense erosion on the Rongcheng High and the Niutuozhen High. The Paleogene is composed of, from top to bottom, the Dongying Formation, the Sha-1∼Sha-4 Formation and the Kongdian Formation. The Mesoproterozoic is composed of the Changcheng System and the Jixian System. The Archean strata are dominated by metamorphic rocks, which are distributed extensively in the study area with significant variation in burial depth [24].

The Jizhong depression is enriched in horst geothermal reservoirs [7]. The heat flow varies between 48.7 and 79.7 mW·m⁻² with an average of 59.2 mW·m⁻². These values in subuplift areas of the Jizhong depression are usually higher than those of the subdepression regions [7]. In the study area, there are two geothermal reservoirs, the Nuituozhen geothermal reservoir and the Rongcheng geothermal reservoir, located in the Niutuozhen High and Rongcheng High, respectively. The main thermal strata in the Rongcheng geothermal reservoir include a porosity-type geothermal reservoir of the Neogene Minghuazhen Formation and a bedrock thermal reservoir of the Jixian System and the Changcheng System. Characterizing the extent of the geothermal strata below the top of the Rongcheng High is our goal in this study.

2.2. Data Acquisition

A 2D seismic survey extending from the Taihang Mountain uplift in the northwest to the Dacheng High in the southeast was conducted in this area. The purpose of the seismic survey line is to image the geological structure down to the Moho in this area. Therefore, the geometry has relatively long offsets for a land seismic survey. The sources are large explosive sources because such sources make it relatively easy to generate enough energy to penetrate the entire crust. As mentioned above, the main target of this study is the potential geothermal reservoir located below the top of the Rongcheng High, so we extract a 40 km profile (AA′, see Figure 1a) in the middle part of the line that crosses the Rongcheng High and Niutuozhen High.

The acquisition parameters of our seismic survey are listed in

Table 1. Seismic data acquisition parameters.

Parameters of Seismic Survey
Type of source	Dynamite
Acquisition equipment	Sercel-428XL
Type of geophone	30DX-10 Hz
Geophone combination type	12 geophones (six in series connection and two in parallel)
	linear combination with inner distance 1 m
Data recording type	SEG-D
Recording length	30 s
Sampling rate	2 ms
Geophone spacing	20 m
Source spacing	120 m
Number of receivers per spread	1440
Minimum offset	10 m
Maximum offset	14,390 m
Depth of source activation	approximately 40 m in loose Quaternary areas
	and 15–20 m in rocky areas
Quantity of explosive	16–26 kg (larger quantities in Quaternary areas,
	smaller quantities in rocky areas)

. The length of the spread is 28.8 km, with the source points in the middle, and the maximum offset is 14.4 km. There are 1440 receivers with a spacing of 20 m at each spread. The geometry of the cropped dataset is shown in Figure 2, and the cropped migrated seismic profile is shown in Figure 3. In this area, there are many villages, highways and other human factors affecting the available source locations, resulting in a nonuniform data geometry. These irregularities cause many gaps in the shallow parts of the seismic profile (Figure 3). Although the survey line is not a strict straight line, the data are processed as a 2D line because the deviation is not too large.

In this study, we extracted 93 shots with an associated total receiver length of 40 km distributed across the middle part of the seismic profile, resampled the dataset to a sampling interval of 4 ms and reduced the recording length to 10 s. With the interpretation of horizons from the seismic profile and some geologic information, a model of the research area was built, which we call the XA model. This model was used for synthetic testing before applying our methodology to the field data.

3. Methodology

In this section, a brief overview of our choices for an FWI algorithm to fit our data is given. Of course, the subsurface is complex, and the closer the forward-modeling algorithm is to the physics of the actual wave propagation, the more accurate our inverted model will be. In the sedimentary environment, the applicability of acoustic FWI has been demonstrated [15]. For geothermal reservoirs, which are generally saturated with groundwater, attenuation is expected to be more sensitive to reservoir properties than velocity changes. That said, particularly with land data, recovering sufficiently accurate amplitudes to estimate an attenuation model is challenging [1]. The simultaneous inversion of velocity and attenuation leads to significant cross-talk, which occurs when data residuals caused by an error in the estimate of one parameter are attributed to another [25,26]. For this reason, only the P-wave velocity was inverted. For the application of conventional frequency-domain FWI to field data, one of the key challenges is local minima caused by a lack of low-frequency components. Using a damped wavefield, Laplace–Fourier-domain FWI is considered to be one of the most reliable schemes to alleviate this problem [27]. There is a lack of low frequencies (less than 5 Hz) in our real field data. In order to avoid the local minima, the Laplace–Fourier method was used in this study.

The open-source TOY2DAC FWI code from the SEISCOPE Consortium [28] was used to perform FWI. TOY2DAC conducts a frequency-domain viscoacoustic approximation of the wave equation. It also can be used in isotropic [29] or anisotropic media [30], and it can invert for multiple parameters. For the reasons mentioned above, only the P-wave velocity was recovered under the assumption of isotropy. In addition, the Laplace–Fourier-domain FWI was implemented in TOY2DAC. For numerical optimization of the objective function, the SEISCOPE OPTIMIZATION TOOLBOX [31] was used.

The workflow used in this study is shown in Figure 4 and consists of six stages. In the first stage, a relatively accurate starting model is built. Land field data processing is the second stage. A multiscale FWI algorithm is set up in the third stage. Synthetic testing is involved in the fourth stage. Application of multiscale FWI on the land field data is in the fifth stage, and the result of the FWI is assessed in the last stage.

3.1. Starting Model

The misfit between observed and forward-modeled data is decreased iteratively in full-waveform inversion. If more than a 180° phase shift separates the observed and modeled data, FWI will converge to a local minimum [32]. This is the so-called cycle-skipping problem; the verification that this phase shift is less than 180° is called the half-period criterion. To avoid this, a good starting model is required. The initial model for FWI was built by first-arrival travel time tomography with the open-source code TOMO2D. TOMO2D is an open-source package developed by [33] to apply joint refraction and reflection travel time tomography. In this study, only the refraction travel time was used to invert the starting model. The agreement between the measured and predicted first-arrival travel times was assessed to determine if the model was accurate enough for FWI.

Because the tomography results were used for FWI with TOY2DAC, the node spacing should have been as close to a TOY2DAC model as possible to avoid interpolation errors. As TOY2DAC requires a constant node spacing in the x- and z-dimensions, the node spacing of tomography was set to be the same as that used in TOY2DAC, which was dense for tomography. We compensated for this computational burden by adopting a multiscale strategy by adding more shots progressively to the tomographic inversion. We started with 5 shots evenly spaced over the central region of the model. The travel times of all 93 shots with the recovered velocity model were forward-modeled. Comparing the forward-modeled travel times to the selected times with the half period criterion, the risk of cycle skipping for the tomography result was assessed. For those areas where there was a higher risk of cycle skipping, more shots were added to the inversion and tomography was performed again using the previously recovered velocity model as the new starting velocity model. The workflow for our tomography inversion is shown in Figure 5.

3.2. Land Data Processing

As mentioned above, there is a trade-off between a low starting frequency to avoid cycle skipping and a sufficiently high starting frequency to create usable signals. Having already discussed how the best possible starting model is obtained, we now discuss how to obtain the best signal-to-noise ratio at as low a frequency as possible. The ground roll was muted firstly. To compensate for the amplitude, the data were processed with spherical divergence and a surface-consistent correction. These steps were implemented in the commercial seismic processing software FOCUS 5.4.

In addition, the amplitudes of the real data and the data modeled by TOY2DAC were quite different. To correct this, the shot gathers were forward-modeled with the starting model firstly. Then, a least-squares fitting method was used to scale the amplitude of the real data to match the modeled data. For each shot, the RMS amplitude of each trace was calculated to obtain the matrix $\mathbf{x}$ for real data and $\mathbf{y}$ modeled data. We then found a scaling matrix $\mathbf{C}$ by least-squares fitting $\mathbf{x}$ to $\mathbf{y}$ via $\mathbf{y}=\mathbf{Cx}$. Then, $\mathbf{Cx}$ was used as our scaled field dataset. Next, the dataset was converted to the frequency domain with a range from 0 to 125 Hz. We designed a bandpass filter from 0.005 Hz to 16 Hz with a frequency spacing of 0.005 Hz. Each side of the taper (0.005 to 6 Hz and 10 to 16 Hz) was a Hamming taper. Then, this taper was applied to the frequency domain dataset. After all of the data were converted to the frequency domain, a multiscale FWI dataset for TOY2DAC was selected by frequency and offset.

3.3. Multiscale FWI Algorithm

Full-waveform inversion has been developed for decades by many authors in the time domain [2,34,35] and in the frequency domain [10,14,36,37,38,39,40]. The theory of frequency-domain full-waveform inversion is briefly reviewed here. For more detail, we refer to [32].

In the frequency domain, after discretization, the 2D visco-acoustic wave equation can be written in matrix form as

$$\mathbf{A}\mathbf{U}=\mathbf{S}$$

(1)

where $\mathbf{U}$ is the wavefield, $\mathbf{S}$ is the source and $\mathbf{A}$ is a complex-valued impedance matrix [36],

$$\mathbf{A}=\mathit{K}-{\omega }^2\mathit{M}+i\omega \mathit{Q}.$$

(2)

where $\mathit{K}$ is the stiffness matrix, $\mathit{M}$ is the mass matrix and $\mathit{Q}$ is the damping matrix.

Equation (1) is a sparse linear system. Lower-upper (LU) factorization of the complex impedance matrix allows us to recover its inverse and solve for the forward-modeled seismic wavefield [41]. The inverted impedance matrix obtained through LU factorization additionally solves the adjoint problem and recovers the gradient, which can be used to update the model [42]. For each source, the use of a single LU factorization to solve the forward and adjoint problems further reduces the computational cost in the frequency domain compared to the time domain [43]. TOY2DAC utilizes MUMPS [44] as an LU factorization algorithm.

Here, we pose FWI as a least-squares optimization problem that aims to minimize the difference between the observed data and modeled data under an $\mathbf{L}\mathbf{2}$ norm criterion. The misfit vector $\Delta \mathbf{d}\left(\mathbf{m}\right)={\mathbf{d}}_{obs}-{\mathbf{d}}_{cal}\left(\mathbf{m}\right)$ is the difference at the receiver positions between the recorded seismic data ${\mathbf{d}}_{obs}$ and the modeled seismic data ${\mathbf{d}}_{cal}\left(\mathbf{m}\right)$ summed over each source–receiver pair in the seismic survey. Here, ${\mathbf{d}}_{cal}$ can be related to the modeled seismic wavefield $\mathbf{u}$ by a detection operator $\mathcal{R}$, which extracts the values of the wavefields, computed in the full computational domain, at the receiver positions for each source: ${\mathbf{d}}_{cal}$=$\mathcal{R}\mathbf{u}$ [32]. $\mathcal{R}$ is related to the observation geometry.

The weighted least-squares norm of the data misfit in matrix form is given by [12]

$$\mathcal{C}\left(\mathbf{m}\right)=\Delta {\mathbf{d}}^{†}{\mathbf{W}}_d\Delta \mathbf{d}$$

(3)

where † denotes the conjugate transpose and ${\mathbf{W}}_d$ is a data-preconditioning operator that weights each component of $\Delta d$ in the misfit function $\mathcal{C}$ [12]. $\mathcal{C}$ is minimized with an iterative nonlinear method by applying perturbations to a initial model ${\mathbf{m}}_k$:

$${\mathbf{m}}_{k+1}={\mathbf{m}}_k+{\gamma }_k\Delta m_k$$

(4)

where k is the iteration number and ${\gamma }_k$ is a scalar parameter computed through a linear search or a trust-region procedure [45]. The perturbation $\Delta m_k$ is given by

$$\Delta m_k=-{\mathbf{H}}_k^{-1}{\nabla }_m{\mathcal{C}}_k$$

(5)

where $-{\nabla }_m{\mathcal{C}}_k$ is the steepest-descent direction of the kth iteration and ${\mathbf{H}}_k^{-1}$ is the inverse of the Hessian matrix of ${\mathcal{C}}_k$. The gradient ${\nabla }_m$ is computed using an adjoint-state method [42]. Computing the explicit Hessian matrix and its inverse is computationally expensive. An approximation of the inverse Hessian operator is used in many large-scale optimization schemes. The Hessian operator estimated through the l-BFGS method, $H_a\left({\mathbf{m}}_k\right)$, is written as [45]

$$H_a\left({\mathbf{m}}_{\mathbf{k}+\mathbf{1}}\right)=V{\left({\mathbf{m}}_k\right)}^T{H}_a\left({\mathbf{m}}_{\mathbf{k}}\right)V\left({\mathbf{m}}_k\right)+\gamma \left({\mathbf{m}}_k\right)\beta \left({\mathbf{m}}_k\right)\beta {\left({\mathbf{m}}_k\right)}^T$$

(6)

The terms $\gamma \left({\mathbf{m}}_k\right)$ and $V\left({\mathbf{m}}_k\right)$ are defined as [41]

$$\begin{array}{c}\hfill \gamma \left({\mathbf{m}}_k\right)=\frac{1}{y_k^T{\beta }_k},\\ \hfill V\left({\mathbf{m}}_k\right)=I-\gamma \left({\mathbf{m}}_k\right)y\left({\mathbf{m}}_k\right)\beta {\left({\mathbf{m}}_k\right)}^T\end{array}$$

(7)

The terms $\beta \left({\mathbf{m}}_k\right)$ and $\mathbf{y}\left({\mathbf{m}}_k\right)$ are

$$\begin{array}{c}\hfill \beta \left({\mathbf{m}}_k\right)={\mathbf{m}}_{k+1}-{\mathbf{m}}_k,\\ \hfill y\left({\mathbf{m}}_k\right)=\nabla f\left({\mathbf{m}}_{k+1}\right)-\nabla f\left({\mathbf{m}}_k\right)\end{array}$$

(8)

The Hessian matrix, $H_a\left({\mathbf{m}}_k\right)$, is estimated with the l-BFGS method [45]. For the first l-BFGS estimate of the Hessian matrix, an approximate Hessian matrix $H_a^0\left({\mathbf{m}}_k\right)=P_{ϵ}\left({\mathbf{m}}_k\right)$ where $P_{ϵ}\left({\mathbf{m}}_k\right)$ is the preconditioner is used [43]:

$$\begin{array}{c}\hfill P_{ϵ}\left({\mathbf{m}}_k\right)=diag\left(B_k\left(\mathbf{m}\right)\right)+ϵD,\\ \hfill B_k\left(\mathbf{m}\right)=J{\left({\mathbf{m}}_k\right)}^{T}J\left({\mathbf{m}}_k\right),\\ \hfill D=max\left(diag\left(B_k\left(\mathbf{m}\right)\right)\right).\end{array}$$

(9)

where $B_k\left(\mathbf{m}\right)$ is the Gauss–Newton approximation of the Hessian matrix, $J\left({\mathbf{m}}_k\right)$ is the Jacobian matrix and $ϵ$ is a damping term. Smaller values of $ϵ$ weight the gradient toward deeper velocity perturbations, and larger values favor shallower velocity updates [12,14].

Multiscale frequency continuation from low to high is the most common strategy to mitigate the nonlinearity of FWI. This is implemented naturally in the frequency domain through successive inversions of monochromatic datasets of increasing frequency [9,10]. When a range of offsets is available, a computationally efficient frequency selection strategy can be used to reduce the redundancy of the wavenumber coverage [37]. To further mitigate the nonlinearity, the Laplace–Fourier-domain FWI is used [27]. The Fourier transform of a damped signal is given by

$$TF\left(p\left(t\right)e^{-(t-t_0)/\tau }\right)={\int }_{-\infty }^{\infty }p\left(t\right)e^{-i(\omega +i/\tau )t}{e}^{t_0/\tau }dt$$

(10)

where $TF$ means the Fourier transform operation, $t_0$ the first-arrival travel time and $\tau$ the time-damping factor. We apply exponential time damping to control the amount of information that is preserved after the first arrival [27,39]. Increasing the damping factor progressively decreases the degree to which the waveforms are damped. This approach allows us to invert for simpler waveforms first while later attempting to invert for more complex waveforms. Combining the time-damping relaxation and the offset continuation in the Laplace–Fourier domain effectively implements a layer-stripping approach that is implemented by adjusting the time window in the time domain [46,47]. Such an approach implies that data recorded at short offsets and long recording times are not inverted before longer-offset earlier arrivals [12].

Our multiscale FWI strategy was performed after Górszczyk et al. (2017) [12], who successfully applied multiscale FWI to OBS data. This strategy contains three aspects: offset continuation from short to long, damping factors from small to large and frequency progressive continuation from low to high. This can be implemented as a three-loop strategy. The external loop is a frequency progression loop from low to high. The middle loop is implemented over Laplace constant, and the inner loop is implemented over maximum offset. The algorithm is shown in Algorithm 1.

Algorithm 1: Multiscale full-waveform inversion algorithm.

Require: ${\mathit{v}}_{\mathit{p}},{\mathit{Q}}_{\mathit{p}},\rho$ Require: input files Require: n_iter    $\mathbf{For}$ f $\mathbf{in}$ freq_vector 2:      update the frequencies for this offset range          $\mathbf{For}$ $\tau$ $\mathbf{in}$ tau_vector 4:            update the Laplace constant                $\mathbf{For}$ o $\mathbf{in}$ offset_vector 6:                   set the offset range                       set a corresponding preconditioner 8:                   set a gradient preconditioner                       run n_iter iterations 10:                 update velocity model        $output$←$param\_vp\_final$

▹ initial model ▹ set up all the input files that are needed by TOY2DAC ▹ set up the number of iterations for inversion ▹ copy the result to the initial velocity model ▹ the final result of multiscale FWI

4. Synthetic Testing

Based on the cropped migrated seismic profile in the depth domain shown in Figure 3, we created horizons for a synthetic model we called the XA model. To build the P-wave velocity model, we use the geological information and petrophysical properties from this area [48] were used. The resulting model is shown in Figure 6a. The relatively flat horizontal sedimentary layers from the surface to a depth of 1 km consist of three strata: the Quaternary, Paleogene Minghuazhen and Paleogene Guantao Formations. Below the unconformity at a depth of approximately 1 km, there are five structural units. From left to right, these units are the eastern Xushui Sag, the Rongcheng High, the Baiyang Lake-Dahezhen Subsag, the Niutuozhen High and the western Baxian Sag. The model has strong lateral velocity changes, dominated by two high-velocity horsts. The synthetic density model was built from the P-velocity (Figure 6a) with Gardner’s relation [49] and remained unchanged during the inversion.

Our model includes 201 nodes in the z-dimension and 2001 nodes in the x-dimension with an equal spacing of 20 m in both dimensions for a total length of 40 km in x and 4 km in z. The same synthetic acquisition geometry was used as that of our extracted real land data, which contained 93 shots with 1440 receivers per shot evenly distributed on both sides of the shot point (see Figure 2). The shot points were distributed nonuniformly (as mentioned above, there were many human factors dictating source locations) from 12 km to 25 km in the central part of the model.

The initial model for the multiscale FWI, which was the same as that used for the real land data FWI, was recovered by first-arrival travel time tomography as discussed above. The inversion parameters for the multiscale strategy are shown in

Table 2. The multiscale FWI strategy used for the XA model.

Loop One—Progressive Frequency Groups in Multiscale FWI
Groups	Frequency/Hz	Groups	Frequency/Hz
1	[2.0, 2.5, 3.0]	2	[2.5, 3.0, 3.5]
3	[3.0, 3.5, 4.0]	4	[3.5, 4.0, 4.5]
5	[4.0, 4.5, 5.0]	6	[5.0, 5.5, 6.0]
7	[6.0, 6.5, 7.0]	8	[7.0, 7.5, 8.0]
9	[8.0, 8.5, 9.0]	10	[9.0, 9.5, 10.0]
11	[10.0, 10.5, 11.0]	12	[11.0, 11.5, 12.0]
13	[12.0, 12.5, 13.0]	14	[13.0, 13.5, 14.0]
15	[14.0, 14.5, 15.0]	16	[15.0, 16.0, 17.0, 18.0, 19.0, 20.0]
Loop Two	Loop Three
Laplace Constant	Max Offset (km)	Gradient Preconditioner	Model Smoothing
0.5	5	1 × 10−2	(1.2, 0.6)
1.0	14.4	1 × 10−6	(0.6, 0.3)

.

Figure 6b shows the result of the multiscale FWI with the XA model. Figure 6c shows the difference between the result and the true model. Above the basement (the surface at a depth of approximately 1 km) in this area, the flat horizons are recovered accurately. For those horizons that are below the basement (horizons in the subsag and below the top of the horst), the result has lower resolution. This is because there is a strong contrast between the layer above the basement and the layer below the basement where the velocity jumps from approximately 3000 m/s to approximately 5000 m/s. In Figure 6b, the maximum depth at which the model is recovered is approximately 3.6 km. It is accurate down to approximately 2 km depth at x = 4 km and 2.5 km at x = 24 km (Figure 7). It is not accurate below 2.5 km because of the intense variation in horizontal velocity. However, the boundary of the two high structures can be delineated. In general, we expect to be able to recover a model to a depth of approximately 7.2 km because our maximum offset is 14.4 km, and in general, FWI can recover a good model down to a depth of approximately half the maximum offset [37]. We compare vertical profiles at different distances in Figure 7. In the sedimentary strata (Figure 7a,c), the FWI results match the exact model very well above 2 km and, in the horst area (Figure 7b), the models match well above 1 km. The results degenerate for deeper depths because of the strong velocity contrast mentioned above.

5. Field Data Results

For this study, we extracted 93 shots with source points distributed from 12 km to 25 km of the model (see Figure 2). The receivers were distributed from $\mathbf{A}$ to ${\mathbf{A}}^{\prime }$ in Figure 1, with 20-m spacing. In time, the data were resampled to 4 ms, and the total recording length was cut to 10 s.

5.1. Starting Model

The initial model (see Figure 8a) for tomography was built by smoothing our XA model (see Figure 6a). As mentioned in the previous section, the x- and z-node spacing of tomography were set to 20 m, which was dense for tomography but appropriate for FWI. We started from five shots evenly distributed in the middle of the model. After the first travel time tomography inversion, the travel times of all 93 shots were forward-modeled with the recovered velocity model. Comparing the forward-modeled travel times to the selected times with the half-period criteria, the risk of cycle skipping of the tomography result was assessed (see Figure 8c,d). For those areas where there was a higher risk of cycle skipping, more shots were added to the tomographic inversion and the tomography was repeated, with the initial model being the result of the previous inversion.

The tomography result with five shots was shown in Figure 8b. Based on the difference between the forward-modeled and selected first breaks, four levels were set up to assess the likelihood of cycle skipping at the specified frequency. There are differences less than $T/4$, greater than $T/4$ and less than $T/2$, greater than $T/2$ and less than $3T/4$ and greater than $3T/4$, where T is the period of the wave. For 1.5 Hz (see Figure 8c), almost all of the receivers have a low risk of cycle skipping except for the source at approximately 20 km and receiver locations from 20 km to 22 km. For 2.5 Hz (see Figure 8d), the receivers with midrange offsets have a high risk of cycle skipping. Although the result of tomography with five shots is geologically meaningful (Figure 8b), it is not acceptable for our multiscale FWI even for a starting frequency as low as 2.5 Hz.

We then increased the number of shots to ten and then twenty, performed tomography inversion and reassessed the results. The final result is shown in Figure 9a. At both 1.5 and 2.5 Hz, the risk of cycle skipping at all of the receivers is minimal. For our standard land data with an explosive source, the lowest frequency that contains a high-quality signal is 5–6 Hz. Therefore, we focus on the assessment for 5 Hz. As shown in Figure 9d, the result is acceptable for midrange offsets. There is a higher risk of cycle skipping for sources between 18 km and 24 km. The two high-risk zones are on the top of the boundary between the horst and the subsag. This is because of the intense velocity variations in the horizontal direction at these locations. There are only minor improvements when comparing the result inverted with 20 shots to the result inverted with 10 shots. The rays are overlaid on the final model shown in Figure 10. The ray coverage extends as deep as 3.5 km. We thus adopt Figure 9a as our starting model for multiscale FWI.

5.2. Real Data Preprocessing

In this section, the processing steps applied to our data are described. Shot 1 is illustrated as an example. Figure 11a shows the raw data. Figure 11b shows the modeled data in TOY2DAC with the starting model (Figure 10a). The method we used to process the amplitude of real land data to match the modeled data is explained above, and here, we show the result. As shown in Figure 11c, the RMS amplitudes of the real and modeled data match each other well except in the vicinity of the source. For each trace in each shot, we subtracted the RMS amplitude of the field and modeled data to obtain the RMS amplitude difference. Then, the standard deviation of this RMS amplitude difference was calculated and normalized by dividing by the maximum standard deviation over all of the traces in a single shot. The normalized standard deviation of the amplitude difference is relatively low except around the source. The amplitude-corrected data are shown in Figure 11d. Then, the dataset was converted to the frequency domain and tapered with a frequency range from 0.005 Hz to 16 Hz with a frequency spacing of 0.005 Hz; this frequency filter removes ground roll and improves the signal-to-noise ratio of the data. The result is shown in Figure 12. Comparing the field data to the modeled dataset in the frequency domain, the field data are lacking at frequencies lower than 5 Hz.

5.3. Multiscale FWI

The lower the starting frequency is, the lower the risk of cycle skipping, but there is little signal at frequencies lower than 5 Hz in our field data. Thus, there is a trade-off between adequate signal quality and a low starting frequency. Referring to the assessment of cycle skipping (Figure 11) and the available frequencies in the data (Figure 12), we adopted 5 Hz as our starting frequency. Twelve progressive frequency groups from 5 Hz to 12 Hz and two offset groups were set up in our strategy. The parameters of our multiscale FWI strategy are listed in

Table 3. The multiscale FWI strategy used for the real land data.

Loop One—Progressive Frequency Groups in Multiscale FWI
Frequency Group 1	5.0, 5.25, 5.5
Frequency Group 2	5.0, 5.25, 5.5, 5.75, 6.0
Frequency Group 3	5.0, 5.5, 5.75, 6.0, 6.25, 6.5
Frequency Group 4	5.0, 5.5, 6.0, 6.25, 6.5, 6.75, 7.0
Frequency Group 5	5.0, 5.5, 6.0, 6.5, 6.75, 7.0, 7.25, 7.5
Frequency Group 6	5.0, 5.5, 6.0, 6.5, 6.75, 7.0, 7.25, 7.5, 7.75, 8.0
Frequency Group 7	5.0, 6.0, 7.0, 7.25, 7.5, 7.75, 8.0, 8.25, 8.5
Frequency Group 8	5.0, 6.0, 7.0, 7.25, 7.5, 7.75, 8.0, 8.25, 8.5, 8.75, 9.0
Frequency Group 9	5.0, 6.0, 7.0, 7.25, 7.5, 7.75, 8.0, 8.25, 8.5, 8.75, 9.0, 9.25, 9.5
Frequency Group 10	5.0, 6.0, 7.0, 7.25, 7.5, 7.75, 8.0, 8.25, 8.5, 8.75, 9.0, 9.25, 9.5, 9.75, 10.0
Frequency Group 11	5.0, 6.0, 7.0, 7.25, 7.5, 7.75, 8.0, 8.25, 8.5, 8.75, 9.0, 9.25, 9.5, 9.75, 10.0,
	10.25,10.5,10.75,11.0
Frequency Group 12	5.0, 6.0, 7.0, 7.25, 7.5, 7.75, 8.0, 8.25, 8.5, 8.75, 9.0, 9.25, 9.5, 9.75, 10.0,
	10.25, 10.5, 10.75, 11.0, 11.25, 11.5, 11.75, 12.0
Loop Two	Loop Three
Laplace Constant	Max Offset (km)	Gradient Preconditioner	Model Smoothing
0.5	5	1 × 10−2	(1.2, 0.6)
1.0	14.4	1 × 10−6	(0.6, 0.3)

.

The starting velocity model is shown in Figure 9a, and the density model was generated from the starting velocity model with Gardner’s formula [49] and was not updated in the FWI. A simplistic attenuation model was built following the technique of Górszczyk et al. [12]. In a marine setting, the attenuation model consists of two values, one for the water column and one for the subsurface. For our dataset, a single value of $Q=200$ was used, which was not updated in the inversion. During the inversion, we estimate the source wavelet for each gather. This can be implemented by setting a source-estimating parameter included in TOY2DAC.

The multiscale FWI results are shown in Figure 13. Consistent with the synthetic testing of the XA model, the recovered model was also updated only in the areas where we had source coverage. On both sides of the model, the model updates were small. To focus on our area of interest, the model was cropped from 10 km to 30 km. To ensure that we were neither over- nor undersmoothing the data, the effect of the model smoothing factor was tested. The results of this test are shown in Figure 13. All of the results successfully recover the location of the interface with the basement, but the three models are quite different. The model in Figure 13a has intermediate smoothing parameters and shows reasonable detail without introducing too much clutter. The second model, in Figure 13b, is quite smooth but loses details. The model shown in Figure 13c has less smoothing applied, and the resulting detail at depth seems unlikely to be well-constrained given our synthetic results. We thus used the intermediate smoothing parameters shown in Figure 13a.

5.4. Assessment of Results

To assess our multiscale FWI results, in Figure 14, the forward-modeled shot gathers of the starting model and the recovered model were compared with the field data. We also forward-modeled the first arrival travel times in the starting model and the recovered model with TOMO2D and overlay them on the shot gather.

There are more reflected waves on the predicted gather than on the starting gather, especially at early times (less than 1.2 s). For the shallow subsurface (less than 1.2 s in time), the reflected events match well. Other events also agree reasonably well. Furthermore, frequency domain traces for the modeled and predicted and starting datasets are provided at 5 Hz and 8 Hz (Figure 15). The predicted data match the observed data better than the starting data. Additionally, the predicted data match the observed data well not only for 5 Hz but also for 8 Hz.

To combine the recovered model with the seismic profile, we apply Kirchhoff prestack depth migration to the shot gathers with the FWI velocity model (Figure 16b) and the initial model (Figure 16a). The Pre-SDM result shows significant improvement in the deeper part of the image. The common image gathers comparison of CDP points 9036 and 9044 are shown in Figure 16c–f. The common imaging gathers also show improvement within our area of interest (1.0–2.0 km), particularly at deeper depths, which indicates that the FWI velocity model is more accurate than the initial model.

To assess the resolution of our results, a series of checkerboard tests were performed [50]. The checkerboard sensitivity test was first used to assess the ability of tomographic inversion to resolve structural details on Earth. However, it can be applied to any inversion procedure without knowledge of the internal operation of the inversion. For our multiscale FWI, the idea is to superimpose a small perturbation (typically a smoothed grid pattern) onto the inverted velocity model, compute synthetic seismic data with the geometry used in the original FWI and then invert the synthetic seismic data in the same manner as the actual data. The ability of the multiscale FWI to quantitatively recover the perturbed model is then an estimate of the ability of the original inversion to recover similar details in the real Earth data.

According to the estimates from [12], we used checkerboards with the following sizes: 1.0 km × 0.5 km, 0.5 km × 0.5 km, 0.4 km × 0.4 km and 0.3 km × 0.3 km. To smooth the boundary of the checkerboard patterns, a sinusoidal taper was applied. Setting the value of the perturbation to $\pm 5\%$, we add this perturbation to the the recovered model (Figure 14a,c,e,g) to generate the perturbed model. The perturbed model was used as the exact model and the recovered model was used (Figure 13a) as the initial model. Then, the multiscale FWI with the same strategy as we applied to the real data was performed except that only the last frequency group in Table 3 was used. The number of iterations was increased to 20 per offset portion of the data to compensate for the fact that only the last frequency group was used. The recovered perturbations are shown in Figure 17.

In the horizontal dimension, the results of all four pattern sizes show that the model between 10 km and 30 km can be recovered, where we have good source coverage (see Figure 2). We cropped the resulting images to this region. In the vertical dimension, the deepest depth at which we have a good recovery is approximately 2 km (Figure 17b), but the capacity of inversion can reach 4 km. This is limited by the maximum offset of our acquisition geometry and the strong velocity contrast. The resolution becomes lower as the depth increases. For the 1.0 km × 0.5 km size, the deepest recoverable depth reaches nearly 4 km, while for the 0.5 km × 0.5 km, it is approximately 3 km; for the 0.4 km × 0.4 km, it is approximately 2.5 km and for the 0.3 km × 0.3 km, it is approximately 2 km. In the horizontal dimension, there is lower resolution in areas where there are intense velocity variations. When the checkerboard pattern is decreased to 0.3 km × 0.3 km, the perturbations down to a depth of approximately 2 km are recovered, but the accuracy is reduced, leading us to conclude that we can recover structures down to approximately 0.3 km × 0.3 km at depths shallower than 2 km.

Having verified our resolution and compared the modeled and field data, the inverted result (Figure 13a) was compared to the log data of well D13 (see Figure 1). This comparison is shown in Figure 18, where the red line is the velocity converted from the sonic well data, the green line is the initial velocity used in the multiscale FWI and the blue line is the recovered velocity from the multiscale FWI. The FWI result matches the trend of the borehole data well, especially at depths between 1100 m and 2000 m. We also observe a correspondence of the FWI model with resistivity data (Figure 18c) and stratum temperature data (Figure 19a).

5.5. Interpretation

In this section, a detailed interpretation of the results discussed above is given. The geological structure of the area is interpreted using a combination of the migrated section and the recovered velocity model. These results, along with our geologic interpretation, are shown in Figure 20b. As mentioned in the geologic background section, for the sedimentary sequences above the basement of this area, there are three primary strata marked in Figure 20b: the Quaternary (from the surface to approximately 400 m in depth x = 20 km in the migrated seismic profile), the Minghuazhen Formation of the Neogene (from approximately 400 m to approximately 1 km in depth = 20 km) and the Guantao Formation of Neogene (from approximately 1 km to approximately 1.2 km at x = 20 km). Above the two highs, part of the Guantao Formation is eroded. Shown in the overlain velocity model with the migrated seismic profile (Figure 20a), the velocity variation matches the seismic events well, which also verifies the reliability of the inverted result.

From the temperature logs of eight boreholes and the thermal conductivity measurements of 108 rock samples, ref. [51] studied the characteristics of the geothermal field in this area. From the borehole data, the lithology above the top of the Rongcheng High is sand and mudstone, which has a geothermal gradient of 35.5 °C·km⁻¹ and low thermal conductivity. Below the top of the Rongcheng High, the lithology is dolomite, which has a geothermal gradient of 4.2 °C·km⁻¹ and high thermal conductivity. If it has good permeability and high thermal conductivity, dolomite can be regarded as a good geothermal reservoir. The compressional velocity of dolomite varies from 3500 m/s to 6500 m/s, with the main control being the fluid saturation. The convection of groundwater, the combination of strata and well-developed faults contribute to the high potential of these geothermal reservoirs. In summary, the research area is rich in porous and fault-type geothermal resources below the basement ([51], their Figure 7). These resources are expected to appear as low-velocity anomalies. As shown in Figure 20, there is indeed a low-velocity anomaly in the area between 13 km and 19 km in the horizontal dimension and between 1.5 km and 2.0 km in depth. It is interpreted as being related to the Rongcheng geothermal reservoir and the Rongcheng fault.

Our focus area is in the eastern part of the Rongcheng High because most of our source points are located in this area, and our assessments indicate that our results are well-constrained in this region. Below the top of the Rongcheng High, at distances between 13 km and 19 km and depths between 1 km and 1.8 km, a potential geothermal reservoir is delineated by combining our results with borehole data. The depth between 1100 m and 1200 m is interpreted from the borehole data as the geothermal reservoir, which has low velocity (Figure 18b), low resistivity (Figure 18c) and a low geothermal gradient (Figure 19a). From the characteristics of the geothermal reservoir in this area, another geothermal reservoir between approximately 1700 m and 1800 m is conjectured, although there are no stratum temperature data at these depths. The size of the anomaly is approximately 6.0 × 0.5 km, which is large enough to be resolved in the recovered model. Although it seems slightly abrupt, it is a reliable anomaly. The two reservoirs are separated by a high-velocity region. The velocity of the two reservoirs varies from approximately 3500 m/s to 4200 m/s (Figure 20b). Referring to our borehole data, the difference in velocity is caused by different groundwater saturations within the dolomite strata and that the two reservoirs correspond to the geothermal strata of the Wumishan Formation of the Jixian System (shallower anomaly) and Gaoyuzhuang Formation of the Changcheng System (deeper anomaly).

6. Discussion

It is possible to use FWI to recover a geologically meaningful velocity model in a complicated region with standard dynamite source land field data. As the field data are very different from synthetic data, quality control should be conducted at every step of multiscale FWI. There are two critical challenges in land field data FWI.

The first challenge is the avoidance of cycle skipping. Cycle skipping is a common issue in any waveform inversion that is based on a local optimization method. To avoid this, low-frequency data are needed, which is not generally available for this type of data. Without these low-frequency data, an accurate starting model is needed. To obtain this starting model, tomography is the most natural choice. A multiscale tomography method is adopted to generate a sufficiently accurate starting model for a starting frequency of 5 Hz. The result shows an acceptable risk of cycle skipping for most offsets, but the areas where the structural unit boundaries are located still exhibit some cycle skipping. One drawback of first-arrival tomography is the incomplete illumination of deep structures by diving waves [12]. This problem is mitigated by the acquisition of long-offset data. Laplace–Fourier-domain full-waveform inversion is another effective method used to reduce cycle skipping. Even if there is no low-frequency signal, the Fourier transform of the damped signal can contain low-frequency information or even zero-frequency information [52]. As not all land field data have as high a signal-to-noise ratio as the dataset used in this study, another direct solution for cycle skipping may be the extrapolation of low frequencies [53,54].

The second challenge is the denoising and long-offset acquisition of real data. A high signal-to-noise ratio is needed for any inversion method. Several denoising methods should be used to attenuate ground roll, random noise and linear noise. Long-offset acquisition is beneficial not only for the low wavenumbers of the model but also for the deep portion of the model. Transmitted-wave full-waveform inversion can be used to estimate the velocity model of geothermal areas [6]. The estimated velocity model has a low resolution because only the transmitted waves are included in the inversion. The multiscale full-waveform inversion with both transmitted waves and reflected waves is utilized. The frequency of the reflected wave is higher than that of the diving wave in the long offset and can be used to invert the high wavenumber of the model. To assess the resolution of the inverted model, the checkerboard test method which is routinely applied in tomography analysis is used. The resolution of our result is approximately 0.3 km. The results show a relatively lower resolution in the areas where the model has large lateral velocity variations.

Prestack depth migration can be used to evaluate the inverted velocity model. A comparison of the migrated profile and gathers shows the improvement of the FWI result not only in the shallow part but also in the deeper part (to 3 km). The continuity and resolution of seismic events in the profile migrated with the FWI velocity model are better than those migrated with the initial velocity model. This also shows that the recovered velocity is correct and updates considerably in the deeper part.

The geothermal reservoir in this area is a combination of porous- and fault-type reservoirs. It is very difficult to obtain high-resolution imaging in this area because of the high wave impedance of the basement and the porous and fractured geological environment. In terms of seismic velocity properties, the geothermal reservoir filled with groundwater is characterized by low velocity. Therefore, in the recovered velocity model, combined with geological information, two banded low-velocity anomalies are delineated as geothermal reservoirs. The average thickness of the first low-velocity anomaly (Figure 20b at depths between 1.1 km and 1.2 km) is approximately 100 m. Compared to the velocity of the basement (approximately 4500 m/s), the velocity of the first banded area is very low (approximately 3500 m/s). The velocity of the second banded area (Figure 20b at depths between 1.7 km and 1.8 km) is approximately 4200 m/s, while the velocity of the surrounding rock is approximately 5500 m/s.

It may be difficult to evaluate the geothermal potential of this area just through the seismic profile because of the poor quality of the seismic events. Generally, the uplift structure under the sedimentary structure should have a high velocity and high resistivity. The low resistivity attribute can be used as a direct indicator of fluid, but it has a relatively low resolution. A high-resolution velocity model can be obtained through multiscale full-waveform inversion with both transmitted and reflected waves. In the uplift structure of this area, combining the resistivity attribute with the high-resolution velocity attribute can help delineate the geothermal reservoir.

7. Conclusions

With the goal of imaging a geothermal reservoir, a multiscale Laplace–Fourier-domain FWI strategy was applied to a standard land field seismic dataset acquired with an dynamite source, which lacks frequencies lower than 5 Hz. Data with this lack of low frequency are typically challenging for FWI, but an excellent starting model for FWI can be recovered by using a careful tomography strategy. Even though we do not have much energy below 5 Hz, a good velocity model with a lateral resolution down to approximately 0.3 km can still be recovered. The variations in the recovered velocity model match the seismic events of the migrated seismic profile and sonic data well. By interpreting with a combination of the migrated seismic profile and the recovered velocity model, two potential geothermal reservoirs with a relatively low-velocity anomaly corresponding to low resistivity below the top of the horst were delineated. The second reservoir was not imaged with conventional techniques but was well-delineated with our FWI method.