Keywords: seismic tomography; seismic waves; seismic attenuation; three-dimensional approaches.

Introduction

Since 1980s, the development of the quantity and quality of seismic data and the computer powers has motivated scientists to generate and improve the field of seismic tomography. Similar to the medical Computer-Aided Tomography, known as CAT-scanning, tomography in seismology is usually conducted by assessed travel times of seismic waves to conclude the seismic wave velocity structure of the Earth. On the other hand, the amplitudes of seismic waves at specific frequencies have been used to speculate the Earth’s attenuation structure, that is, to point out regions where seismic waves lose energy more or less than normal.

Seismic tomography used seismic records to retrieve 2D and 3D images of subsurface anomalies by solving large inverse problems by matching the create models and the observed data. Different approaches are used to resolve anomalies in the crust and lithosphere, shallow mantle, whole mantle, and core, dependent on the available seismic data (passive or active data) and the types of seismic waves (body or surface waves) that penetrate the region at a suitable wavelength for feature resolution. The accuracy of the created model is restricted to the quantity and quality of the seismic data, wave type used, and assumptions made in the model. Figure (1) shows an example of the necessity of the quantity seismic data observations in imaging borders of the heterogeneity structures of the earth.

P-wave data are commonly utilized in the local and global models in regions of dense seismic data observations. However S-wave and surface wave data are utilized in global models in case of insufficient seismic observations. First-arrival times are the most commonly utilized, however, models using reflected and refracted phases are used in more complicated models, for example; those imaging the core. In addition, differential travel times between wave phases or types are also utilized. Figure (2) shows an actual seismic tomography of the crust and mantle underneath North America.

Depending on the type of seismic waves, the body wave tomography and surface wave tomography were known, while local, regional and global tomography were classified depending on the scales of the study regions (Zhao, 2001). Aki and Lee (1976) and Aki et al., (1977) were the leader scientists in the body wave tomography for the local and regional studies, while Dziewonski (1984) for the global tomography. On the other hand, Nakanishi and Anderson (1982), Woodhouse and Dziewonski (1984) and Tanimoto and Anderson (1984) started the surface wave tomography.

Imaging seismic attenuation of the crust and upper mantle is necessary for many purposes. First, seismic attenuation can be of utmost importance for differentiation between temperature and volatile content variations within the Earth in comparison to seismic velocity (Karato, 1993; Karato, 2003; Priestley and McKenzie, 2006). Information of both velocity and attenuation structure are also critical for precisely anticipating ground motion from forthcoming huge earthquakes (Komatitsch et al., 2004; Power et al., 2008). Second, the seismic velocities present in sedimentary basins exaggerate seismic waves, however, attenuation decreases these amplitudes as well, and declining to consider attenuation can result in larger ground motion amplitude predictions (e.g., Olsen et al., 2006). Third, by stripping the crustal and upper mantle attenuation from body wave records, it can be attainable in further researches to examine the attenuation structure deeper within the Earth in a better way.

Attenuation has been evaluated tomographically with different techniques and assumptions related to the frequency dependence of attenuation (Kjartansson, 1979; Stainsby and Worthington, 1985; Brzostowski and McMechan, 1992; Quan and Harris, 1997; Liu et al., 1998).

Seismic attenuation can be described and quantified using the quality factor Q. A straightforward postulation is that the seismic attenuation is frequency-dependent however the quality factor Q is frequency-independent. This hypothesis is accurate in the frequency range of the applications of the exploration geophysics.

In this article, the 3-D attenuation tomography is classified based on the tomographic technique and the estimation method of the Q-model. This classification includes the seismic data type/phase (QP, QS, QC, QLg, or Q-surface waves) and frequency dependence (seismic amplitude decay or seismic frequency shift). Valuable development in the seismic observations technology, the computed power and storage disc capacity have driven to good deployment of seismic data and high accuracy models of 3-D seismic attenuation tomography. Through this article, I shall provide an outline of the seismic tomography field, seismic attenuation tomography, common and new approaches in the 3-D seismic attenuation tomography.

SEISMIC TOMOGRAPHY

Seismic tomography is one of the central technologies to check the 3-D dissemination of the physical characteristics, which influence seismic-wave propagation: elastic, inelastic, and anisotropic specifications, and density. Tomographic models usually have an important part in investigating the subsurface – lithology, temperature, fracturing, fluid content, etc. Since its start in 1970s, seismic tomography has developed to be one of the essential techniques of the current seismology. It is most frequently linked to the roots of seismic tomography of Keiiti Aki, who published a crucial research in 1976 on 3D velocity precision beneath California from local earthquakes (Aki and Lee, 1976). In 1977, an equally important research was published, which applied tele-seismic tomography to image the 3D velocity structure beneath the Norwegian Seismic Array (Norsar) in southeast Norway (Aki et al., 1977). In the same year, Adam Dziewonski conducted a research applying global tomography, in which they utilized P wave travel time residuals to image the velocity structure of the Earth’s mantle, delineated by a spherical harmonic characterization (Dziewonski et al., 1977).

Broadly, seismic tomography depends on solving a large inverse problem to get a heterogeneous seismic model homogenous with the measurements. More precisely, it depends on the establishment of an approximate relationship d = g (m) between seismic data d and seismic structure m. Thus, for a given model m we can speculate d, then the seismic tomography problem amounts to prediction of m such that d explains the data observations dobs. In the majority of conditions, d and m are distinct vectors of high dimension, which implicates that most data reports are utilized to generate a comprehensive model. Essentially, this design has to be applicable for both vertical and lateral structure.

The categorization of seismic tomography lay on the extent of the defined areas and on the seismic data categories as well. There are three main categories of data utilized in the interrelated tomographic methods: body-wave travel time, surface wave dispersion, and free-oscillation (normal-mode) spectral measurements.

There were three tomographic scales classified as local, regional and global seismic tomography. Subsequent to the early research of Aki and Lee (1976), local earthquake tomography has turned to be a common technology for imaging subsurface structure in seismogenic areas. Despite that the theoritical concept of local earthquake tomography has not actually differed since Aki’s pioneer work, many developments have been established. These include full 3D ray tracing and iterative non-linear inversion (Eberhart-Phillips, 1990); direct inversion for VP/VS or QP/QS ratio (e.g. Walck, 1988); and improvement of ways for checking 3D anisotropic velocity variations (Hirahara, 1988; Eberhart-Phillips and Henderson, 2004). Attenuation structure (Sanders, 1993; Tsumura et al., 2000); and double difference tomography (Zhang and Thurber, 2003; Monteiller and Got, 2005), which targets to notably enhance hypocenter relocation, are included as well. In subduction zone settings, modern technologies incorporate tomographic inversion of shear wave splitting assessments for anisotropic fabric (e.g. Abt and Fischer, 2008), and of velocity and attenuation anomalies for water content, temperature and composition (Shito et al., 2006).

Contrary to the local tomography, regional and global tomography researches more popularly use data from established networks, which occupy wide continental areas or most of the globe, added to all accessible information from temporary arrays. Aims include the upper mantle, whole mantle or the entire Earth. Since the original publication of Dziewonski et al. (1977), who utilized the travel times of P-waves, attempts have been directed to developing resolution by applying a constantly increasing volume of reported data. Present global P-wave mantle models that accomplish travel time data usually check structures at a scale length of a few 100km or less using millions of paths (Zhao, 2004; Burdick et al., 2008). Moreover, to direct P-waves, other phases such as PcP and PKP are usually utilized to enhance scope, especially in the core (Vasco and Johnson, 1998; Boschi and Dziewonski, 2000; Karason and van der Hilst, 2001).

Body wave tomography utilizing S-waves is also popular in regional and global records (e.g. Grand et al., 1997; Vasco and Johnson, 1998; Widiyantoro et al., 2002), and can either be accomplished alone or in combination with P-waves to retrieve VP/VS ratio similarly to local tomography.

Surface waves and normal modes are possibly utilized to create tomographic images of the Earth’s interior, as well. In comparison to body waves, surface waves have the privilege that they are capable of illustrating the upper mantle beneath ocean basins at enough density to get well-established models of oceanic lithosphere. However, they are not capable to probe into the deep mantle at high resolution, and they are hardly elucidating crustal structure. Various approaches have been used to retrieve data from the surface Wave train. Some global researches utilized long paths and aimed to assess phase velocity straightly for the crucial mode for each path (e.g. Ekstrom et al., 1997; Laske and Masters, 1996). Group velocities may be obtained by filter analysis, and have been utilized to develop maps at regional scales (e.g. Ritzwoller and Levshin, 1998; Danesi and Morelli, 2000; Pasyanos and Nyblade, 2007), and global scales, as well (Shapiro and Ritzwoller, 2002).

Normal modes or free oscillations of the Earth that could be shown as extremely long duration surface waves, also propose a method to obtain seismic structure. Separate peaks of the distinct range are usually separate due to Earth rotation, ellipticity and lateral heterogeneity. Excluding the latter effect makes both mantle structure (Li et al., 1991; Resovsky and Ritzwoller, 1999) and core structure (Ishii and Tromp, 2004) liable to be imaged.

Another field of ongoing research in global seismology is attenuation tomography, in which lateral variations in the inelastic parameter Q are recorded. A fundamental interrogation with this technology is to precisely obtain the inelastic signal from the reported waveform, influenced by elastic effects. Until date, researches favour to utilize surface waves and, thus, concentrate on the upper mantle (Romanowicz, 1995; Billien and Leveque, 2000; Selby and Woodhouse, 2002; Gung and Romanowicz, 2004; Dalton and Ekstrom, 2006; Dalton et al., 2008). However, body wave researches have been conducted, as well (Bhattacharyya et al., 1996; Reid et al., 2001; Warren and Shearer, 2002). One of the interests of attenuation tomography is the high susceptibility to temperature changes, and, thus, the possibility to screen hot spots, mantle plumes and subduction areas.

ATTENUATION TOMOGRAPHY

As seismic waves propagate via a medium, a gradual attenuation of the wave amplitude is noticed prior to diminution of the wave. This observation is known as the attenuation of seismic waves. The attenuation of seismic waves happens because of geometrical spreading, scattering of seismic energy and inelasticity of the propagation medium. Geometrical spreading delineates seismic energy attenuation because of the spread of wave energy over the surface of the spherical wave front. However, scattering attenuation happens because of the wave propagation medium heterogeneity. When the seismic wave propagates via a medium, it inter-reacts with the medium heterogeneity and the wave energy is repositioned, leading to diminution because of many reflections inside this medium. Nevertheless, intrinsic attenuation happens because of inelastic properties of the medium, depending on the immeasurable quality factor Q that is inversely correlated to the amount of attenuation (Q-1).

Knopoff13 characterized wave attenuation in terms of energy as:

Q^(-1)=ΔE⁄2πE

where ΔE is the loss of energy per cycle and E is the total wave energy.

This approximation is much more suitable for seismic purposes:

A(x)=A_0 (-wx)/e^2cQ

where x is estimated through the direction of propagation and c is the wave velocity.

This equation obviates that for a constant value of Q, the more the frequency the stronger the attenuation. This is due to that for a known distance; the high frequency wave propagates more oscillations than a low frequency wave. As the wave goes far away from the origin, the pulse gets wider at subsequent distances. As the wave travels, diminution ameliorates the high frequency portion of the pulse.

This equation can be phrased in terms of time, as well. This is much more suitable if we considered seismic applications, as the wave is travelling forward in time:

A(t)=A_0 e^(-wt/2Q)

Seismic waves’ attenuation contributes crucial independent constraints on Earth characteristics due to their senstivity to rock constitution, liquid component, temperature, and other characteristics distinct from those obtained by P-wave and S-wave velocities.

Attenuation has been investigated tomographically with different methods and speculations regard to the frequency dependence of attenuation (Kjartansson, 1979; Stainsby and Worthington, 1985; Brzostowski and McMechan, 1992; Quan and Harris, 1997; Liu et al., 1998). The most of these techniques started by a hypothesis that attenuation coefficient rises proportionately to the frequency. Earlier, the rise time associated with the pulse-broadening effect has been used for estimation of the attenuation (Kjartansson, 1979), where the attenuation of the high-frequency degrees of direct wave amplitude spectra has been utilized to specify the whole path attenuation, quantiﬁed by the frequency-independent attenuation operator t∗ values.

Generally, the variations of amplitude of the recorded seismic data are utilized for tomographic inversions to identify the 3-D attenuation structure similarly to the velocity tomography. Based on the identification of the model parameterization, the attenuation values t∗ from a sequence of earthquakes observed by number of seismic stations are inverted to the inverse of quality factor Q for the 3-D attenuation structure, by tracing the ray paths through initial velocity model, preferred a 3-D model (Eberhart-Phillips and Chadwick, 2002).

Q tomography algorithms have been categorized into two main classes. One is ray-based tomography (Quan and Harris, 1997; Plessix, 2006; Rossi et al, 2007), with two peculiar constituents of a ray-based Q tomography algorithm: 1) a straightforward but precise correlation between seismic data and Q models for generating the mathematical model for Q tomography, 2) a respectable and powerful boosting algorithm for resolving this mathematical dilemma.

The second class of Q tomography algorithms is wave-equation-based tomography (Liao and McMechan, 1996; Hicks and Pratt, 2001, Pratt et al, 2003; Watanabe et al, 2004; Gao et al, 2005). Wave-equation-based tomography is substantially more precise but much more costly and not practical for 3D cases.

A major challenge with Q tomography is how to establish a link of Q models and seismic data with the least approximations and with the most flexibility. An extensively utilized technique is dependent on the correlation between Q and seismic amplitude decay. Another approach utilizes the seismic frequency downshift in estimation of the quality factor Q. This is thought to be more powerful as it does not depend on the geometrical spreading influence and reflection/transmission scattering effect.

The most clear and smooth technique in estimating Q model is the spectral ratio methodology (Spencer et al., 1982; Tonn, 1991). In this method, the logarithm of the spectra ratio between two seismic waveforms is estimated as a function of frequency, which is approximated by a linear function of frequency, whose slope is treated as the accumulated seismic attenuation and is eventually related to the Q values along the wave propagation path. This method is ideal in removing the influence of geometrical radial spread and reflection/transmission scatter effect with the hypothesis that these influences are not dependent on the frequency. Rickett (2006) suggested a tomographic expansion of the spectral ratio technique assisted by time-frequency analysis methodology. It was concluded that the seismic wavelet amplitude spectrum shape is nearly entirely influenced by the quality factor Q, and a peak frequency variation technique was generated for Q assessment (Zhang, 2008).

Quan and Harris (1997) invented a unique technique where the data over the whole frequency band of seismic waveforms is utilized to estimate the centroid frequency downshift and to subsequently correlate with the centroid frequency shift to the Q profile along the ray path by a straightforward competent design formulation. This methodology is essentially resistant to geometrical spread and reflection/transmission scattering.

3-D Q-tomography Approaches

Over the past three decades, the development of three-dimensional models of the Earth’s seismic velocity structure has been accelerated by several factors, among them an expanding inventory of earthquake data, rapid growth in computational power, and improvements in wave propagation theory and model parameterization schemes.

In this article, the 3-D attenuation tomography approaches are classified based on the tomographic technique and the estimation method of the Q-model. This classification includes the seismic data type/phase (QP, QS, QC, QLg, or Q-surface waves) under the frequency dependence of seismic amplitude decay or seismic frequency shift methods.

A) Seismic amplitude decay

Seismic amplitude decay is a common technique to estimate Q-tomography. It has been done using different types of seismic observations to image 3-D seismic attenuation structure since 1980s around Japan Arc. Umino and Hasegawa (1984); Sekiguchi (1991) and Tsumura et al.(1996) used data of micro-earthquake network and interpreted that the high-Qs zones correspond to an oceanic plate. Seismic intensity data was used by Hashida and Shimazaki (1985) and concluded that a high-Qs zone is related to the oceanic plate. Peak ground acceleration calculated from a large quantity of seismic intensity data was used by Hashida and Shimazaki (1987) to show that the low-Qs zone is due to the distribution of volcanoes. Nakamura and Uetake (2004) used strong motion observations to illustrate that the low-Qs zones agree with the distribution of volcanoes.

Fergany et al. (2012) used the coda wave attenuation (Q_C^(-1)) to reveal 3-D attenuation tomography for Egypt using 397 local and regional earthquakes recorded by Egyptian National Seismic Network (1997-2008) to study the origin of earthquake activity and tectonic structure of Egypt. Based on the single-scattering theory, the decay rate of the coda amplitudes (Q_C^(-1)) was estimated and the 3-D was calculated for each central frequency ranged from 1.0 to 24 Hz avoiding the unreliable results of the marginal area.

Lin (2014) retrieved the frequency-independent three-dimensional P-wave attenuation model Qp for the crust and uppermost mantle of northern and central California. Figures (3 and 4) shows examples of using t∗ values for retrive Q cross-section models. The results of Qp model provides a significant complement to the existing regional-scale velocity models for interpreting the structural heterogeneity and fluid saturation of rocks in the study region. Komatsu et al. (2017) used the amplitude decay rate of the P- and S-wave spectra to retrive 3-D P- and S-wave attenuation (Q_P^(-1) and Q_S^(-1)) structures of the crust around the 2016 Kumamoto earthquakes source zone.

Surface wave records from regional/teleseismic earthquakes have been used traditionally at both scales of global and regional to investigate the seismic velocity and attenuation structure of the Earth (e.g., Mitchell, 1995 ; Romanowicz, 2002; Yang et al. 2007; Yang and Forsyth, 2008). Amplitudes of long-period (50–300 s) Rayleigh waves were considered primarily for global studies of upper mantle attenuation structure. The first 3-D global model of shear attenuation in the mantle using the surface wave Q maps at 80–300 s was formed by Romanowicz, (1995). Selby and Woodhouse (2000) noticed that Rayleigh wave amplitudes contain a considerable amount of signal from elastic focusing in the period range 73–171 s, and they concluded that the attenuation maps obtained by inverting their amplitude data set were contaminated by focusing effects for wavelengths shorter than spherical harmonic degree 9. Selby and Woodhouse (2002) constructed 3-D shear attenuation model of the upper mantle ignored focusing out of concern that it could not be treated accurately, but solved for a frequency-dependent amplitude correction factor for each event to account for uncertainty in the source amplitude. Gung and Romanowicz (2004) used three-component surface wave waveform data that included both overtones and fundamental modes to develop a 3-D model of shear attenuation in the upper mantle, and they inferred from tests with synthetic data that neglecting focusing, source, or instrument effects factors did not significantly bias their degree-8 model.

A large data set of fundamental mode Rayleigh wave amplitudes in the period range 50–250 s for 347 earthquakes observed at 179 seismic stations was analyzed by Dalton and Ekström (2006) to derive global models of surface wave attenuation (Fig. 5).

Dalton et al. (2008) originate a new global three-dimensional model of shear wave attenuation in the upper mantle using a large data set of fundamental mode Rayleigh wave amplitude span periods range 50-250 s (Fig. 6).

B) Seismic frequency shift

The seismic frequency shift method has been used used as crosswell attenuation tomography tool in the geophysical exploration. There are two well-known methods based on frequency shift method to estimate attenuation. The first one known as centroid frequency shift method was introduced by Quan & Harris (1997) and the second known as the peak frequency shift method which introduced by Zhang & Ulrych (2002).

The idea of the centroid frequency shift method based on the assumption that as seismic waves propagate through the media the high‐frequency components of the seismic signal are attenuated more rapidly than the low‐frequency components (Quan & Harris 1997). By the way, the centroid of the signal’s spectrum experiences a downshift along propagation path (Fig. 7). This downshift in frequency is proportional to a path integral through the attenuation distribution and can be used as observed data to reconstruct the attenuation distribution tomographically (Fig. 8). Tomographically, the frequency shift method is relatively insensitive to instrument responses, reflection and transmission effects, geometric spreading, source and receiver coupling and radiation patterns. It is applicable in any seismic survey geometry where the signal bandwidth is broad enough and the attenuation is high enough to cause noticeable losses of high frequencies during propagation.

Zhang & Ulrych (2002) developed an analytical relation between Q-factor and seismic data peak frequency shift both along offset and vertical time direction assuming that the amplitude spectrum of the seismic source signature may be modeled by that of a Ricker wavelet. The Q-factor is estimated from common midpoint CMP gathers using a layer-stripping approach.

A new algorithm was presented by Hermana et al. (2012) to improve the Q-tomography using the frequency shift methods. However, each previous frequency shift method has limitations for definite condition: centroid frequency shift method gives high accuracy in high Q condition for shallow structures only. In the meantime, peak frequency shift method satisfied low Q condition and far target wavelet but dissatisfied high Q condition and near target wavelet. The new proposed technique can improve Q estimation using shift frequency methods in the low and high Q condition, shallow and deep wavelet targets and in the low and high seismic noise ratio conditions of seismic data. It based on the estimation of the corrected factor which derived from the peak frequency shift and the centroid frequency shift methods (Fig. 9).

A modified frequency shift method was presented by Li et al., (2015) to treat the lack of reliable methods for estimating Q from reflection seismic data as a diagnostic exploration tool of hydrocarbon detection. This new technique based on developed an approximate equation and proposed a dominant (peak) and central frequency shift (DCFS) method by combining the quality factor Q, the travel time, and dominant and central frequencies of two successive seismic signals along the wave propagating direction. DCFS method can obtain continuous volumetric Q estimation results. The proposed method was tested using experimental and field data and the results show higher accuracy and robustness compared with the existing methods.

New advances in 3-D attenuation tomography

Although seismic tomography has been very successful in illuminating the Earth’s internal velocity structure, it is insufficient to make direct links with mantle convection, obtain robust estimates of temperature and composition. Consequently, fundamental questions remain unanswered: Do subducting slabs bring water into the transition zone or lower mantle? Are the large low-shear velocity provinces under the Pacific and Africa mainly thermal or compositional? Is there any melt or water near the transition zone or core mantle boundary?

During the last decade, there are dependable advances in the seismic observations and computer power that lead to develop and improve new approaches in tomographic techniques along with the estimation methods of the Q models to retrieve high resolution accurate 3-D Q tomography. These new approaches enlarged the applications of the 3-D attenuation tool in mapping partial melt, water and temperature variations that can answer the above questions and become advantageous tool in hydrocarbon explorations. An outline of new approaches in 3-D attenuation tomography is given along Q model estimation methods and tomographic technique.

Q models has been estimated using various methods: spectral-ratio method (Bath, 1974), the match-technique method (Raikes and White, 1984; Tonn, 1991), analytical signal method (Engelhard, 1996), the centroid frequency-shift method (Quan and Harris, 1997), and the spectrum-modeling method (Janssen et al., 1985; Tonn, 1991; Blias, 2011) where each technique its advantageous and limitations. Tonn (1991) compared between these various methods for Q estimation using vertical seismic profile VSP data and concluded that the optimal method in the noise-free case is the spectral-ratio method. However, the estimation given by spectral-ratio method may deteriorate extremely with increasing noise (Patton, 1988; Tonn, 1991) and the looking for reliable Q estimation remains.

Cheng and Margrave (2008) extended the classic spectral-ratio method and proposed a complex spectral-ratio method. It employs both the phase spectra of signal and the amplitude spectra, in which Q is estimated by solving an inverse problem to minimize the misfit between the measured and modeled complex spectral ratios. On the other hand, Cheng and Margrave (2012a) proposed a time-domain match-filter method for Q estimation, which has been known to be robust to noise and suitable for application to surface reflection data.

Although the match-filter method which is a time-domain alternative to spectrum-modeling method is theoretically sophisticated wavelet modeling technique (Janssen et al., 1985; Tonn, 1991; Blias, 2011), it has superior performance in comparing with the other methods (Cheng and Margrave, 2012b). Cheng and Margrave (2013) used real and synthetic data in terms of accuracy and robustness to noise to evaluate the performance of the three methods: time-domain match-filter method, centroid frequency-shift method and the complex spectral ratio method for Q estimation. They concluded that the match-filter method and centroid frequency-shift method are suitable for application to reflection data in which all three methods are robust to noise.

Li et al. (2016) improved the approach of the centroid frequency shift (CFS) for Q measurement to the weighted centroid frequency shift method (WCFS), which integrates a Gaussian weighting coefficient into the calculation procedure of the conventional CFS method. The basic idea is to enhance the proportion of advantageous frequencies in the amplitude spectrum and reduce the weight of disadvantageous frequencies.

Yuan and Simons (2014) established a new algorithm that regularizes the inversion via the use of wavelet-based constructive approximations applied to elastic waveform data, synthetic and observed, in a model that evolves as part of a gradient-based iterative scheme relying on forward and adjoint modeling that carried out with a spectral-element method. Its idea is based on that the full-waveform seismic inversions based on minimizing the distance between observed and synthetic seismograms are able to yield better-resolved earth models than those minimizing misfits derived from traveltimes alone.

Métivier et al. (2016b) improved an optimal transport for seismic tomography to 3D full waveform inversion (FWI) approach. This approach proposed further analyze the mathematical foundation of the modified dual Kantorovich problem that proposed by Métivier et al. (2016a) and a novel numerical strategy for the computation of the Kantorovich–Rubinstein norm. this approach can apply to realistic size 3D data sets. Figure (10) shows a performed 3-D experiment on the overthrust model that verify the optimal transport distance technique that looks an interesting tool for 3-D full waveform inversion FWI.

Paste youTomographically, there are new trends takes the attention of seismologists to overcome the uneven observations and disadvantages of the common methods. These trends are known as seismic noise tomography and finite frequency tomography. Recent studies (Shapiro et al., 2005; Sabra et al., 2005; Pollitz and Fletcher, 2005) have caught wide attention as an approach that can take advantage of the growing archive of continuous seismic data (ambient noise observations). Apparently, seismic data that regarded as noise may include extremely useful signal that can benefit in improving seismic tomography.

Ambient seismic noise tomography has advantages of the continuous and global distributed sources additions to the high energy in short-period surface wave, which has useful applications to constrain crustal structure (Yang et al.2007). It has been widely used in the past few years for exploring shallow Earth structure (e.g. Sabra et al.2005; Shapiro et al.2005; Brenguier et al.2007; Liang & Langston 2008; Lin et al.2008).

Li and Lin (2014) presented a new adaptive tomography method using ambient seismic noise with irregular grids that provides a few advantages over the traditional methods. First, irregular grids with different sizes and shapes can fit the ray distribution better and the traditionally ill-posed problem can become more stable owing to the different parametrizations. Secondly, the data in the area with dense ray sampling will be sufficiently utilized so that the model resolution can be greatly improved.

In the seminal study of Shapiro et al. (2005), only one month of data from the US Array stations was required to produce high resolution images of the California crust, which clearly discriminates between regions of thick sedimentary cover and crystalline basement. More recent efforts have been directed towards recovering phase velocity in addition to group velocity (Benson et al., 2008), and attempting to resolve 3D shear wave velocity structure from the inversion of Rayleigh and Love wave dispersion maps (Benson et al., 2009).

The finite frequency tomography beginning was accompanied by some discussion as to its validity in general heterogeneous media and the degree of improvement it brought to conventional ray-based tomography (de Hoop and van der Hilst, 2005a,b; Dahlen and Nolet, 2005; Montelli et al., 2006; Trampert and Spetzler, 2006). Besides the study of Montelli et al. (2004), others to have used finite frequency tomography include Hung et al. (2004), who report increased resolution in the upper mantle transition zone beneath Iceland; Chevrot and Zhao (2007), who use finite frequency Rayleigh wave tomography to image the Kaapval craton; and Sigloch et al. (2008), who exploit teleseismic P-waves to elucidate the structure of subducted plates beneath western North America. Compared to seismic traveltime tomography based on geometric ray theory, the advantage of finite frequency traveltime tomography is that a larger range of phase information is used to constrain structure and has the potential to improve seismic imaging on many fronts (Rawlinson et al., 2010).

Future approaches are expected to realize some dreams about the interior of the earth. Arwen Deuss (http://www.geo.uu.nl/~deuss/grants/attenuation-tomography-using-novel-observations-earths-free-oscillations/) planned a project to study the Attenuation Tomography Using Novel observations of Earth’s free oscillations (ATUNE) which granted by the European Research Council from June 2016 to June 2021. The aim of ATUNE is to develop novel full-spectrum techniques and apply them to Earth’s long period free oscillations to observe global-scale regional variations in seismic attenuation – distinguished between scattering (redistribution of energy) versus intrinsic attenuation- from the lithosphere to the core mantle boundary. ATUNE will deliver the first ever full-waveform global tomographic model of 3D attenuation variations in the lower mantle, providing essential constraints on melt, water and temperature for understanding the complex dynamics of our planet.

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