Electromagnetic fields generated by finite‐length wire sources: comparison with point dipole solutions

In present‐day land and marine controlled‐source electromagnetic (CSEM) surveys, electromagnetic fields are commonly generated using wires that are hundreds of metres long. Nevertheless, simulations of CSEM data often approximate these sources as point dipoles. Although this is justified for sufficiently large source‐receiver distances, many real surveys include frequencies and distances at which the dipole approximation is inaccurate. For 1D layered media, electromagnetic (EM) fields for point dipole sources can be computed using well‐known quasi‐analytical solutions and fields for sources of finite length can be synthesized by superposing point dipole fields. However, the calculation of numerous point dipole fields is computationally expensive, requiring a large number of numerical integral evaluations. We combine a more efficient representation of finite‐length sources in terms of components related to the wire and its end points with very general expressions for EM fields in 1D layered media. We thus obtain a formulation that requires fewer numerical integrations than the superposition of dipole fields, permits source and receiver placement at any depth within the layer stack and can also easily be integrated into 3D modelling algorithms. Complex source geometries, such as wires bent due to surface obstructions, can be simulated by segmenting the wire and computing the responses for each segment separately. We first describe our finite‐length wire expressions and then present 1D and 3D examples of EM fields due to finite‐length sources for typical land and marine survey geometries and discuss differences to point dipole fields.     

Insight into the marine controlled‐source electromagnetic signal propagation

In marine controlled‐source electromagnetic (CSEM) surveys the subsurface is explored by emitting low‐frequency signals from an electric dipole source close to the sea‐bed. The main goal is often to detect and describe possible thin resistive layers beneath the sea‐bed. To gain insight into how CSEM signals propagate, it is informative to study a stratified model. The electromagnetic field is then given in terms of integrals over TE‐ and TM‐polarized plane‐wave constituents. An asymptotic evaluation of the field integrals for large propagation distances results in explicit spatial expressions for the field components and the derived expressions can be used to analyse how the CSEM signals propagate. There are two major signal pathways in a standard CSEM model. One of these pathways is via the thin resistive layer and the resulting response is accounted for by a pole in the reflection response for the TM mode. The signal is propagating nearly vertically down to the resistor from the source, then guided while attenuated along the reservoir, before propagating nearly vertically up to the receiver. The response is slightly altered by the sea‐bed interface and further modified in shallow water due to multiple reflections between the sea‐surface and sea‐bed at both the source and receiver sides. The other major signal pathway is via the resistive air half‐space, the so‐called airwave. The airwave is generated by the TE mode and interacts with the subsurface via vertically propagating signals reflected between the sea‐surface and subsurface at both the source and receiver sides.     

一维电阻率各向异性对海洋可控源电磁响应的影响研究

地下介质的电阻率常常表现为各向异性,海底裂隙地层和层状沉积序列可能形成宏观电阻率各向异性.在解释海洋电磁资料时,电阻率各向异性的影响不应该被忽略,否则可能会得到错误的海底地电模型.作者编写了电阻率任意各向异性一维层状介质海洋可控源电磁场计算程序,计算了电阻率各向异性层状模型的海洋可控源电磁响应,讨论了覆盖层和高阻储层分别具有电阻率各向异性时的电磁场响应特征.     

Elimination of the water-layer response from multi-component source and receiver marine electromagnetic data

This paper presents the theory to eliminate from the recorded multi‐component source, multi‐component receiver marine electromagnetic measurements the effect of the physical source radiation pattern and the scattering response of the water‐layer. The multi‐component sources are assumed to be orthogonally aligned above the receivers at the seabottom. Other than the position of the sources, no source characteristics are required. The integral equation method, which for short is denoted by Lorentz water‐layer elimination, follows from Lorentz' reciprocity theorem. It requires information only of the electromagnetic parameters at the receiver level to decompose the electromagnetic measurements into upgoing and downgoing constituents. Lorentz water‐layer elimination replaces the water layer with a homogeneous half‐space with properties equal to those of the sea‐bed. The source is redatumed to the receiver depth. When the subsurface is arbitrary anisotropic but horizontally layered, the Lorentz water‐layer elimination scheme greatly simplifies and can be implemented as deterministic multi‐component source, multi‐component receiver multidimensional deconvolution of common source gathers. The Lorentz deconvolved data can be further decomposed into scattering responses that would be recorded from idealized transverse electric and transverse magnetic mode sources and receivers. This combined electromagnetic field decomposition on the source and receiver side gives data equivalent to data from a hypothetical survey with the water‐layer absent, with idealized single component transverse electric and transverse magnetic mode sources and idealized single component transverse electric and transverse magnetic mode receivers. When the subsurface is isotropic or transverse isotropic and horizontally layered, the Lorentz deconvolution decouples into pure transverse electric and transverse magnetic mode data processing problems, where a scalar field formulation of the multidimensional Lorentz deconvolution is sufficient. In this case single‐component source data are sufficient to eliminate the water‐layer effect. We demonstrate the Lorentz deconvolution by using numerically modeled data over a simple isotropic layered model illustrating controlled‐source electromagnetic hydrocarbon exploration. In shallow water there is a decrease in controlled‐source electromagnetic sensitivity to thin resistors at depth. The Lorentz deconvolution scheme is designed to overcome this effect by eliminating the water‐layer scattering, including the field's interaction with air.     

Applying essentially non‐oscillatory interpolation to controlled‐source electromagnetic modelling*

Modelling and inversion of controlled‐source electromagnetic (CSEM) fields requires accurate interpolation of modelled results near strong resistivity contrasts. There, simple linear interpolation may produce large errors, whereas higher‐order interpolation may lead to oscillatory behaviour in the interpolated result. We propose to use the essentially non‐oscillatory, piecewise polynomial interpolation scheme designed for piecewise smooth functions that contains discontinuities in the function itself or in its first or higher derivatives. The scheme uses a non‐linear adaptive algorithm to select a set of interpolation points that represent the smoothest part of the function among the sets of neighbouring points. We present numerical examples to demonstrate the usefulness of the scheme. The first example shows that the essentially non‐oscillatory interpolation (ENO) scheme better captures an isolated discontinuity. In the second example, we consider the case of sampling the electric field computed by a finite‐volume CSEM code at a receiver location. In this example, the ENO interpolation performs quite well. However, the overall error is dominated by the discretization error. The other examples consider the comparison between sampling with essentially non‐oscillatory interpolation and existing interpolation schemes. In these examples, essentially non‐oscillatory interpolation provides more accurate results than standard interpolation, especially near discontinuities.     

Effects of the sea floor topography on the 1D inversion of time‐domain marine controlled source electromagnetic data

Time‐domain marine controlled source electromagnetic methods have been used successfully for the detection of resistive targets such as hydrocarbons, gas hydrate, or marine groundwater aquifers. As the application of time‐domain marine controlled source electromagnetic methods increases, surveys in areas with a strong seabed topography are inevitable. In these cases, an important question is whether bathymetry information should be included in the interpretation of the measured electromagnetic field or not. Since multi‐dimensional inversion is still not common in time‐domain marine controlled source electromagnetic methods, bathymetry effects on the 1D inversion of single‐offset and multi‐offset joint inversions of time‐domain controlled source electromagnetic methods data are investigated. We firstly used an adaptive finite element algorithm to calculate the time‐domain controlled source electromagnetic methods responses of 2D resistivity models with seafloor topography. Then, 1D inversions are applied on the synthetic data derived from marine resistivity models, including the topography in order to study the possible topography effects on the 1D interpretation. To evaluate the effects of topography with various steepness, the slope angle of the seabed topography is varied in the synthetic modelling studies for deep water (air interaction is absent or very weak) and shallow water (air interaction is dominant), respectively. Several different patterns of measuring configurations are considered, such as the systems adopting nodal receivers and the bottom‐towed system. According to the modelling results for deep water when air interaction is absent, the 2D topography can distort the measured electric field. The distortion of the data increases gradually with the enlarging of the topography's slope angle. In our test, depending on the configuration, the seabed topography does not affect the 1D interpretation significantly if the slope angle is less or around 10°. However, if the slope angle increases to 30° or more, it is possible that significant artificial layers occur in inversion results and lead to a wrong interpretation. In a shallow water environment with seabed topography, where the air interaction dominates, it is possible to uncover the true subsurface resistivity structure if the water depth for the 1D inversion is properly chosen. In our synthetic modelling, this scheme can always present a satisfactory data fit in the 1D inversion if only one offset is used in the inversion process. However, the determination of the optimal water depth for a multi‐offset joint inversion is challenging due to the various air interaction for different offsets.     

Two triaxial induction tools: sensitivity to radial invasion profile‡

I study the responses of two different triaxial induction tools to invaded dipping anisotropic formations. I show that the triaxial measurements have generally higher sensitivity to the radial invasion profile, compared to the conventional induction measurements. This enables accurate interpretation of both the anisotropic formation properties and the invasion parameters. Multi‐spacing and single‐spacing multi‐frequency triaxial induction tools can both be used for this purpose. Failure to take the invasion properties into account may lead to misinterpretation of the vertical formation resistivity. Symmetrization of the apparent conductivity matrix opens ways for a visual interpretation of triaxial induction logs for the formation and the invasion zone properties. This technique enables simpler and faster inversion algorithms. I study how the effect of a conductive annulus forming around the invasion zone couples with effects of the dipping anisotropy and the dipping boundaries and show when these effects are additive. Thus, a visual detection of log parts affected by a conductive annulus becomes possible. The key tool for interpretation in complex 3D scenarios is efficient modelling software. I use a 3D finite‐difference modelling approach to simulate responses of induction logging tools of the new generation. Its high efficiency enables simultaneous multi‐spacing and multi‐frequency computing of the tool responses to arbitrary 3D anisotropic formations that made the study possible.     

1D joint multi‐offset inversion of time‐domain marine controlled source electromagnetic data

The accurate estimation of sub‐seafloor resistivity features from marine controlled source electromagnetic data using inverse modelling is hindered due to the limitations of the inversion routines. The most commonly used one‐dimensional inversion techniques for resolving subsurface resistivity structures are gradient‐based methods, namely Occam and Marquardt. The first approach relies on the smoothness of the model and is recommended when there are no sharp resistivity boundaries. The Marquardt routine is relevant for many electromagnetic applications with sharp resistivity contrasts but subject to the appropriate choice of a starting model. In this paper, we explore the ability of different 1D inversion schemes to derive sub‐seafloor resistivity structures from time domain marine controlled source electromagnetic data measured along an 8‐km‐long profile in the German North Sea. Seismic reflection data reveal a dipping shallow amplitude anomaly that was the target of the controleld source electromagnetic survey. We tested four inversion schemes to find suitable starting models for the final Marquardt inversion. In this respect, as a first scenario, Occam inversion results are considered a starting model for the subsequent Marquardt inversion (Occam–Marquardt). As a second scenario, we employ a global method called Differential Evolution Adaptive Metropolis and sequentially incorporate it with Marquardt inversion. The third approach corresponds to Marquardt inversion introducing lateral constraints. Finally, we include the lateral constraints in Differential Evolution Adaptive Metropolis optimization, and the results are sequentially utilized by Marquardt inversion. Occam–Marquardt may provide accurate estimation of the subsurface features, but it is dependent on the appropriate conversion of different multi‐layered Occam model to an acceptable starting model for Marquardt inversion, which is not straightforward. Employing parameter spaces, the Differential Evolution Adaptive Metropolis approach can be pertinent to determine Marquardt a priori information; nevertheless, the uncertainties in Differential Evolution Adaptive Metropolis optimization will introduce some inaccuracies in Marquardt inversion results. Laterally constrained Marquardt may be promising to resolve sub‐seafloor features, but it is not stable if there are significant lateral changes of the sub‐seafloor structure due to the dependence of the method to the starting model. Including the lateral constraints in Differential Evolution Adaptive Metropolis approach allows for faster convergence of the routine with consistent results, furnishing more accurate estimation of a priori models for the subsequent Marquardt inversion.     

Controlled‐source electromagnetic monitoring of reservoir oil saturation using a novel borehole‐to‐surface configuration

To advance and optimize secondary and tertiary oil recovery techniques, it is essential to know the areal propagation and distribution of the injected fluids in the subsurface. We investigate the applicability of controlled‐source electromagnetic methods to monitor fluid movements in a German oilfield (Bockstedt, onshore Northwest Germany) as injected brines (highly saline formation water) have much lower electrical resistivity than the oil within the reservoir. The main focus of this study is on controlled‐source electromagnetic simulations to test the sensitivity of various source–receiver configurations. The background model for the simulations is based on two‐dimensional inversion of magnetotelluric data gathered across the oil field and calibrated with resistivity logs. Three‐dimensional modelling results suggest that controlled‐source electromagnetic methods are sensitive to resistivity changes at reservoir depths, but the effect is difficult to resolve with surface measurements only. Resolution increases significantly if sensors or transmitters can be placed in observation wells closer to the reservoir. In particular, observation of the vertical electric field component in shallow boreholes and/or use of source configurations consisting of combinations of vertical and horizontal dipoles are promising. Preliminary results from a borehole‐to‐surface controlled‐source electromagnetic field survey carried out in spring 2014 are in good agreement with the modelling studies.     

层状垂直各向异性介质海洋CSEM发射源及接收站姿态和位置对电磁响应影响研究

众所周知,电阻率各向异性对海洋可控源电磁(CSEM)响应有显著影响.在模拟和解释海洋可控源电磁资料时,通常假设接收站布放于平稳的海底,发射源以理想的水平电偶极源形式按预设路径拖曳前进.然而,在实际海洋作业中,自由下沉的接收站可能沉降在局部倾斜的海底面上,在海底的方位可能是任意取向的.同时,由于洋流等因素的影响,发射源偶极子会发生旋转、倾斜,其位置也会偏离预设的拖曳路径,所以发射源和接收站姿态的变化都将对电磁资料产生影响.本文计算了电阻率各向异性介质中,发射源任意姿态任意位置激发、接收站任意姿态任意位置接收的海洋CSEM响应,分析观测系统引起数据误差的机制,分别在电阻率各向同性、围岩电阻率各向异性和高阻储层电阻率各向异性三种模型中,讨论发射源和接收站的姿态及位置变化对响应的影响.计算结果表明,电磁场振幅误差主要来源于发射源和接收站的倾斜角,而相位的变化为发射源位置和接收站倾斜角、位置共同影响的结果.     

Joint inversion of long‐offset and central‐loop transient electromagnetic data: Application to a mud volcano exploration in Perekishkul,Azerbaijan

Mud volcanism is commonly observed in Azerbaijan and the surrounding South Caspian Basin. This natural phenomenon is very similar to magmatic volcanoes but differs in one considerable aspect: Magmatic volcanoes are generally the result of ascending molten rock within the Earth's crust, whereas mud volcanoes are characterised by expelling mixtures of water, mud, and gas. The majority of mud volcanoes have been observed on ocean floors or in deep sedimentary basins, such as those found in Azerbaijan. Furthermore, their occurrences in Azerbaijan are generally closely associated with hydrocarbon reservoirs and are therefore of immense economic and geological interest. The broadside long‐offset transient electromagnetic method and the central‐loop transient electromagnetic method were applied to study the inner structure of such mud volcanoes and to determine the depth of a resistive geological formation that is predicted to contain the majority of the hydrocarbon reservoirs in the survey area. One‐dimensional joint inversion of central‐loop and long‐offset transient electromagnetic data was performed using the inversion schemes of Occam and Marquardt. By using the joint inversion models, a subsurface resistivity structure ranging from the surface to a depth of approximately 7 km was determined. Along a profile running perpendicular to the assumed strike direction, lateral resistivity variations could only be determined in the shallow depth range using the transient electromagnetic data. An attempt to resolve further two‐dimensional/three‐dimensional resistivity structures, representing possible mud migration paths at large depths using the long‐offset transient electromagnetic data, failed. Moreover, the joint inversion models led to ambiguous results regarding the depth and resistivity of the hydrocarbon target formation due to poor resolution at great depths (>5 km). Thus, 1D/2D modelling studies were subsequently performed to investigate the influence of the resistive terminating half‐space on the measured long‐offset transient electromagnetic data. The 1D joint inversion models were utilised as starting models for both the 1D and 2D modelling studies. The results tend to show that a resistive terminating half‐space, implying the presence of the target formation, is the favourable geological setting. Furthermore, the 2D modelling study aimed to fit all measured long‐offset transient electromagnetic Ex transients along the profile simultaneously. Consequently, 3125 2D forward calculations were necessary to determine the best‐fit resistivity model. The results are consistent with the 1D inversion, indicating that the data are best described by a resistive terminating half‐space, although the resistivity and depth cannot be determined clearly.     

基于改进的接收点插值算法的频率域海洋可控源电磁法2.5维正演

在海洋可控源电磁法勘探中,接收站常置于海底.在进行海洋电磁场模拟时,由于海水和海底介质存在显著电性差异,这给海底接收点处场值的求取带来困难.本文提出一种新的接收点插值算法,该算法考虑到海底电场法向分量不连续性问题,用法向电流分量进行插值以准确求取海底任意接收点处电磁场值.本文利用交错网格有限差分法实现了二维介质中频率域海洋可控源法(CSEM)正演.对构造走向做傅里叶变换,将三维电磁模拟问题转换为波数域2.5维问题,即三维场源激励下针对二维地电模型的电磁模拟问题.使用交错网格有限差分法,基于一次场/二次场分离方法导出波数域二次电场离散形式,并进一步求得波数域电磁场.采用本文提出的改进的插值算法可求得海底任意接收点处波数域电磁场,采用傅里叶逆变换对波数域电磁场进行积分可得到接收点处空间域电磁场.模型算例表明,与常规的线性插值和严格插值算法相比,本文提出的改进的插值算法具有更高的精度.     

Two‐dimensional controlled‐source electromagnetic interferometry by multidimensional deconvolution: spatial sampling aspects

We use numerically modelled data sets to investigate the sensitivity of electromagnetic interferometry by multidimensional deconvolution to spatial receiver sampling. Interferometry by multidimensional deconvolution retrieves the reflection response below the receivers after decomposition of the fields into upward and downward decaying fields and deconvolving the upward decaying field by the downward decaying field. Thereby the medium above the receiver level is replaced with a homogeneous half‐space, the sources are redatumed to the receiver level and the direct field is removed. Consequently, in a marine setting the retrieved reflection response is independent of any effect of the water layer and the air above. A drawback of interferometry by multidimensional deconvolution is a possibly unstable matrix inversion, which is necessary to retrieve the reflection response. Additionally, in order to correctly separate the upward and the downward decaying fields, the electromagnetic fields need to be sampled properly. We show that the largest possible receiver spacing depends on two parameters: the vertical distance between the source and the receivers and the length of the source. The receiver spacing should not exceed the larger of these two parameters. Besides these two parameters, the presence of inhomogeneities close to the receivers may also require a dense receiver sampling. We show that by using the synthetic aperture concept, an elongated source can be created from conventionally acquired data in order to overcome these strict sampling criteria. Finally, we show that interferometry may work under real‐world conditions with random noise and receiver orientation and positioning errors.     

浅水区的瞬变电磁法：一维数值模拟结果分析

在浅水环境中,利用频率域海洋可控源电磁法探测油气藏薄层面临着挑战,这是因为空气波支配着电磁响应而它只含有少量海底电阻率结构的信息.本文研究了空气波对时间域可控源电磁(CSEM)响应的影 响. 空气波到达时间取决于海水深度.在浅水区空气波到达早,而在深海水区空气波到达晚,空气波与来自深部电阻体的电磁波出现在不同时间段.虽然在中等深度的海水区空气波和来自深部电阻体的电磁信号几乎同时到达,但与不含有油气藏的背景模型相比,仍可以看到明显的异常.浅水区勘探的好处在于可以使用表面拖曳系统.     

A new marine controlled‐source electromagnetic receiver with an acoustic telemetry modem and arm‐folding mechanism

We developed a new marine controlled‐source electromagnetic receiver for detecting methane hydrate zones and oil and gas reservoirs on the seafloor, which is not imaged well by seismic reflection surveys. To determine the seafloor structure, the electromagnetic receiver should have low noise, power consumption, clock drift error, and operating costs while being highly reliable. Because no suitable receiver was available in our laboratory, we developed a new marine controlled‐source electromagnetic receiver with these characteristics; the receiver is equipped with acoustic telemetry modem and an arm‐folding mechanism to facilitate deployment and recovering operations. To demonstrate the applicability of our new receiver, we carried out a field experiment offshore of Guangzhou in the South China Sea, where methane hydrates have been discovered. We successfully obtained controlled‐source electromagnetic data along a profile about 13 km long. All six new receivers were recovered, and high‐quality electromagnetic data were obtained. Relatively high apparent resistivity values were detected. The results of the offshore field experiment support the claim that the electromagnetic data obtained using the new receiver are of sufficient quality for the survey target.     

A land controlled‐source electromagnetic experiment using a deep vertical electric dipole: experimental settings,processing, and first data interpretation

A multichannel borehole‐to‐surface controlled‐source electromagnetic experiment was carried out at the onshore CO₂ storage site of Hontomín (Spain). The electromagnetic source consisted of a vertical electric dipole located 1.5 km deep, and the electric field was measured at the surface. The subsurface response has been obtained by calculating the transfer function between the transmitted signal and the electric field at the receiver positions. The dataset has been processed using a fast processing methodology, appropriate to be applied on controlled‐source electromagnetics (CSEM) data with a large signal‐to‐noise ratio. The dataset has been analysed in terms of data quality and repeatability errors, showing data with low experimental errors and good repeatability. We evaluate if the induction of current along the casing of the injection well can reproduce the behaviour of the experimental data.     

基于二次场电导率分块连续变化的三维可控源电磁有限元数值模拟

从电偶源三维地电断面可控源电磁法的二次电场边值问题及其变分问题出发,采用任意六面体单元对研究区域进行剖分,并且在单元分析中同时对电导率及二次电场进行三线性插值,实现电导率分块连续变化情况下,基于二次场的可控源电磁三维有限元数值模拟.这个新的可控源电磁三维正演方法可以模拟实际勘探中地下任意形状及电性参数连续变化的复杂模型.理论模型的计算结果表明,均匀大地计算的视电阻率误差和相位误差分别为0.002%和0.0005°.分层连续变化模型的有限元计算结果表明,其与对应的分层均匀模型解析结果有明显差异.三维异常体组合模型以及倾斜异常体等复杂模型的有限元计算结果也有效地反映了异常形态.     

Analytical solution for the electric potential in arbitrary anisotropic layered media applying the set of Hankel transforms of integer order

The analytical solution and algorithm for simulating the electric potential in an arbitrarily anisotropic multilayered medium produced by a point DC source is here proposed. The solution is presented as a combination of Hankel transforms of integer order and Fourier transforms based on the analytical recurrent equations obtained for the potential spectrum. For the conversion of the potential spectrum into the space domain, we have applied the algorithm of the Fast Fourier Transform for logarithmically spaced points. A comparison of the modelling results with the power‐series solution for two‐layered anisotropic structures demonstrated the high accuracy and computing‐time efficiency of the method proposed. The results of the apparent‐resistivity calculation for both traditional pole‐pole and tensor arrays above three‐layered sequence with an azimuthally anisotropic second layer are presented. The numerical simulations show that both arrays have the same sensitivity to the anisotropy parameters. This sensitivity depends significantly on the resistivity ratio between anisotropic and adjacent layers and increases for the models with a conductive second layer.     

Using an airgun array in a land reservoir as the seismic source for seismotectonic studies in northern China: experiments and preliminary results

This paper reports the field setup and preliminary results of experiments utilizing an airgun array in a reservoir in north China for a seismotectonic study. Commonly used in offshore petroleum resource exploration, the airgun source was found to be more useful than a traditional explosive source for large‐scale and long offset land seismic surveys. The airgun array, formed by four 1,500 in³ airguns (a total of 6,000 in³ in volume) was placed at a depth of 6–9 m into the reservoir to generate the pressure impulse. No direct evidence was found that the airgun source adversely affected the fish in the reservoir. The peak ground acceleration recorded on the top of the reservoir dam 100 m away was 17.8 gal in the horizontal direction; this is much less than the designed earthquake‐resistance threshold of 125 gal for this dam. The energy for one shot of this airgun array is about 6.68 MJ, equivalent to firing a 1.7 kg explosive. The seismic waves generated by the airgun source were recorded by receivers of the regional seismic networks and a temporary wide‐angle reflection and refraction profile formed by 100 short‐period seismometers with the maximum source‐receiver offset of 206 km. The seismic wave signature at these long‐offset stations is equivalent to that generated by a traditional blast source in a borehole with a 1,000–2,000 kg explosive. Preliminary results showed clear seismic phases from refractions from the multi‐layer crustal structures in the north China region. Forward modelling using numerical simulation confirms that the seismic arrivals are indeed from lower crustal interfaces. The airgun source is efficient, economical, environmentally friendly and suitable for being used in urbanized areas. It has many advantages over an explosive source for seismotectonic studies such as the high repeatability that is supreme for stacking to improve signal qualities. The disadvantage is that the source is limited to existing lakes or reservoirs, which may restrict experimental geometry.     

Source–receiver two‐way wave extrapolation for prestack exploding‐reflector modelling and migration

Most modern seismic imaging methods separate input data into parts (shot gathers). We develop a formulation that is able to incorporate all available data at once while numerically propagating the recorded multidimensional wavefield forward or backward in time. This approach has the potential for generating accurate images free of artiefacts associated with conventional approaches. We derive novel high‐order partial differential equations in the source–receiver time domain. The fourth‐order nature of the extrapolation in time leads to four solutions, two of which correspond to the incoming and outgoing P‐waves and reduce to the zero‐offset exploding‐reflector solutions when the source coincides with the receiver. A challenge for implementing two‐way time extrapolation is an essential singularity for horizontally travelling waves. This singularity can be avoided by limiting the range of wavenumbers treated in a spectral‐based extrapolation. Using spectral methods based on the low‐rank approximation of the propagation symbol, we extrapolate only the desired solutions in an accurate and efficient manner with reduced dispersion artiefacts. Applications to synthetic data demonstrate the accuracy of the new prestack modelling and migration approach.