0 Introduction

Seismic remigration technique is a kind of seismic imaging technology, which was developed in the end of last century. The main objective of remigration is to realize migration image’s direct mapping among different velocity models in the target space when migration velocity model changes.

人力资源管理信息化开展实现的关键，是要选择一套科学有效的软件管理系统。一步到位的选择固然不错，但如果脱离了企业自身的发展现状和实际需求，加大企业现实的投资成本，拉长了建设周期，也会得不偿失，因此企业要结合自身实际情况，分阶段、分步骤地开展人力资源管理的信息化。

Fomel (1994) was the first who put forward the concept of “velocity continuation” to describe the process of remigration. And he gave the differential equation which was satisfied by time domain migration image field and migration velocity field. On the basis of Fomel (1994)’s achievements, Hubral and Tygel et al. (1996b) put forward the concept of “image wave” to re-explain time domain remigration process, and extended the remigration theory from time domain to depth domain. Hubral et al. (1996a) and Tygel et al. (1996) have established the unified imaging theory. Based on the idea of combination of migration and de-migration, they generalized the basic theory of seismic remigration into the framework of unified imaging theory. Based on the basic theory of Kirchhoff migration, Alder (2002) extended the depth domain remigration method to the data processing in inhomogeneous media for the first time. Fomel (2003a, b), based on the time domain remigration method, achieved the application of remigration to the time domain migration velocity analysis. Schleicher et al. (2004) further extended the application of remigration method to depth domain migration velocity analysis. Subsequently, Schleicher et al. (2006, 2008) extended remigration method from isotropic media to the elliptic anisotropic media. Novais et al. (2008) realized the migration velocity analysis in GPR data processing using remigration method. In recent years, gradual attention has been paid to diffraction waves, and also to the combination of remigration method with diffraction wave information which is used in migration velocity analysis. Based on remigration method, Coimbra et al. (2013) and Decker et al. (2017) have realized depth domain and time domain migration velocity analysis respectively using diffraction wave information.

At present, remigration research mainly concentrates on the differential method, i.e. the differential equation which describes the process of remigration. However, attention should also be paid to the Kirchhoff remigration, which is based on the idea of Kirchhoff migration and de-migration (Sun, 2010; Hubral et al., 1996a; Tygel et al., 1996), and is implemented by stacking the initial migrated image along certain surfaces. Compared with differential remigration, Kirchhoff remigration can be used in inhomogeneous media, and has more adaptability than differential method. Moreover, the calculation speed of Kirchhoff remigration method is obviously faster than differential method, and is suitable for migration image mapping and velocity analysis in 3D.

Based on studies of predecessors, this paper implements Kirchhoff remigration with Kirchhoff migration and Kirchhoff de-migration in depth domain, discusses the basic principle of Kirchhoff remigration operator, and analyses the main parameters that affect the remigration process. Moreover, the remigration method is applied to numerical experiments of groove model and Marmousi model and proved to be effective.

1 Depth domain Kirchhoff remigration

1.1 Theory

The aim of remigration is to transform migrated images corresponding to different velocity model. Without considering the amplitude factors, the basic idea of kinematic Kirchhoff remigration is: when the migration velocity changes, any point in the re-migrated image corresponds to certain curve surface in the initial migrated image (it is curve in 2D), the weighted summation result of all values is taken as the value at this point in remigration image. When the corresponding summation process has been finished for all points in the remigration image, the mapping of initial migrated image to the target image is realized (Hubral et al., 1996a; Tygel et al., 1996).

近一个多月以来，虽然在11月初尿素价格迎来了小幅反弹，但总体行情始终处于震荡下行区间。其原因无非是高价位带来高风险，加之需求疲软，引发下游观望，市场信心不足，厂家适当妥协，在没有上涨动力支撑的情况下，价格逐渐回落。

(1)

Of which, is the Kirchhoff remigration image. V(M) could be an initial migrated image obtained with an arbitrary migration method. KRM is the weighted function of remigration. In order to get more accurate remigration results, the weighting function can be obtained by dynamics ray tracing method. When studying the kinematic characteristics of the remigration, the weighting function can be obtained through a reference velocity model.

Based on the velocity model of Fig.2a, seismic record of 201 shots have been synthesized by Kirchhoff method, and the shot interval is 20 m, 91 traces for each shot, trace interval is 20 m, and minimum offset is 200 m. Fig.2b is the Kirchhoff migration image with the accurate velocity model.

通过以上史料的对比介绍不难发现，《册府元龟》《新唐书》由于借鉴了《旧唐书》和其他资料，因而关于张均流放合浦郡的说法一致；而对于张垍之死的说法，《新唐书》与《旧唐书》虽有歧异，但以《旧唐书》更为准确，似无可疑。宋敏求的《唐大诏令集》为本取材于唐代国史、实录等资料的《旧唐书》对关于二张之死的记载又增添了强有力的证据。

(2)

(3)

From a geometric point of view, the integral surface defined by interpreting Formula (2) and Formula (3) is shown in Fig.1.

In Fig.1, ζR(r) is the imaging strip of the reflection interface in the initial input migrated image, and (ξ,t) defines reflection seismic data space. According to the theory of Kirchhoff migration and de-migration, a point in Kirchhoff migration image corresponds to a Huygens surface in the data space, which is (ξ; Any point N(ξ,τ(ξ; on the Huygens surface τ(ξ; also corresponds to an isochronous surface in the initial migrated image, that is .

Fig.1 Remigration integral surface

In geometry sense, if a point NR on Huygens surface τ(ξ; corresponds to the isochronous surface ζCR(r, ξR; in the initial migrated image, which is tangent to image strip of the reflection interface in the initial migrated image field at point M, then the integral surface of the remigration formula(1)exists, that is ζRM(r; NR is the stationary-phase point related to (Tygel et al., 1993, 1996), if NR is determined, the remigration integral surface can be determined. If there is no tangent point, there will be no effective superposition surface for point

1.2 Travel time computation

Fig.3a is the migration image obtained when migration velocity is. Fig.3c is the migration image when the velocity field is. Fig.3b is the remigration result of Fig.3a using the correct migration velocity, and Fig.3d is the image obtained by re-migrating Fig.3c based on a correct velocity model. It can be seen clearly that obvious error occurred in the migrated image when migration velocity is inaccurate. Based on the true velocity model, Kirchhoff remigration can use inaccurate migrated image to reconstruct a correct one.

|t|=s

(4)

In Formula (4), t is a travel time andis slowness. The upwind difference scheme for fast marching method is

|▽

(5)

(6)

1.3 Remigration aperture

The selected aperture in Kirchhoff remigration is an important parameter that affects the result of the remigration. Similar to Kirchhoff migration and Kirchhoff de-migration, the larger the migration aperture is, the larger the computation, if the aperture is too small, it will cause error in the remigration result. In practice, it is difficult to select a suitable aperture. Usually, remigration aperture selection can refer to the selection methods of Kirchhoff migration and Kirchhoff de-migration, and factors such as embedded depth of imaging points, dip angle of reflection surface, etc. should be taken into account. In the area with larger burial depth or dip angle, the aperture should be properly enlarged.

2 Implementation flow

According to Formula (1), the direct-viewing way of implementing Kirchhoff remigration is to determine the stationary-phase point location in Space (ξ, t) for any pointin the re-migrated image space. If the stationary-phase point is found, we can directly use Formula (1) to get the re-migrated result.

Fig.5a is the imaging result obtained by Kirchhoff pre-stack depth migration when migration velocity is 5% lower than the correct one. Fig.5b is the result obtained by remigration of the image of Fig.5a using correct velocity. Fig.5c is the pre-stack depth migration result obtained when the migration velocity is 5% higher than the correct velocity model, Fig.5d is the remigration image with correct migration velocity using migration result of Fig.5c.

We have adopted the second method, using the direct superposition summation method to implement the Kirchhoff remigration. The implementation flow of the remigration is summarized as follows:

(1) For any point in the target migration image field, calculate the Huygens surface to which pointcorresponds;

每天回家后，我都习惯性地看一眼储存间。因为工作的忙碌，几天没有看储存间，忽地发现鸟儿不知道什么时候飞走了。

1) 通过将破碎的底板围岩视为松散体，依据“朗肯”土压力理论，对巷道底板受力进行力学分析，得出五阳矿76采区2号专用回风巷发生全断面底鼓的破坏形式。

(2) For any pointon the Huygens surface, determine the isochrone in the original migrated image, which corresponds to;

步骤3 根据式(11)～式(13)计算各栖息地的在种群P中随机选择两个比赛栖息地Hα和Hβ，再随机抽取总规模的10%作为子种群S。如果Hα，Hβ中有一个不被子种群S中任何栖息地支配，而另一个被支配，则非支配栖息地获胜，并定义为H0；如果Hα，Hβ都不被支配或都被支配，则比较两者的和更优HSI个数多的栖息地获胜，定义为H0。将栖息地H0加入原本为空集的种群P0，重复上述操作，直到种群P0的规模和种群P的规模相同。用种群P0替换种群P。

For any point in the target migration image field, the basic formula for Kirchhoff remigration is:

(3) Take weighted sum of all points on the isochrone, and the resultis set;

(4) Take weighted sum of all points on the Huygens surface, and the result is placed at the point;

(5) Repeat (1) - (4) until every point in the remigration image has completed the above process.

他们不住地点头，我不忘给他们泼冷水：“挑战才刚刚开始，你们要有心理准备哦！”他们信心满满地说：“老师放心，我们会努力的！”“好样的！”我称赞他们。“你们随时可以来找我，我是你们坚强的后盾。”

3 Numerical examples

3.1 V(z) Model

In order to test the effectiveness of Kirchhoff remigration method, a groove model is presented here, and the velocity of the model varies with depth(Fig.2a).

Compared with the Kirchhoff migration formula, the Kirchhoff remigration expressed in formula (1) is similar to Kirchhoff migration. Considering the kinematic characteristics only, the biggest difference between Kirchhoff remigration and Kirchhoff migration is the definition of the integral surfaces. The definition of integral curve surface of Kirchhoff remigration is:

(a)Velocity Model;(b)the migration image of correct velocity model. Fig.2 Migration velocity and migration image

An important part of the kinematic Kirchhoff remigration method is the travel-time computation. The travel time is applied to the determination of the remigration integral surface. The result of the travel time calculation directly affects the result of the remigration. In addition, the velocity field needs to be changed many times during the remigration process, so the travel-time calculation is very time-consuming in this method. It is of great significance to select the accurate and efficient travel time calculation method for Kirchhoff remigration. We have selected FMM travel time calculation method (Sethian, 1999; Sun et al., 2012), which uses the upwind difference scheme to quickly solve eikonal equation:

溢洪道震损检查信息主要包括进水段、启闭设备、闸门、电源、边墙、桥梁、护坦、泄槽、底板、消力池、渠道、启闭机房等。如果溢洪道和泄水设施由于闸门、阀门或运行设备故障导致不能正常运行时，大坝就有溃坝的危险。闸门、阀门或运行设备的故障可能会导致支撑结构的沉降或移动，及闸门束缚或堆积物阻塞。如果怀疑发生了损伤，应在震后迅速校验闸门运行状况。地震也可能导致闸门自动开启。进水口上方的边坡滑坡会阻塞引水渠，地震可能引起溢洪道进口高程变化。滑坡也会破坏进水建筑物和相关的金属结构，如闸门、启门机和电动机。

3.2 Marmousi model

In order to inspect the adaptability of Kirchhoff remigration to complicated model, we apply it to the Marmousi model. Fig.4a is the Marmousi model. It contains 737 horizontal sampling points with a sampling interval of 12.5 m, and 750 vertical sampling points with a sampling interval of 4 m. The section is seismic records of 240 shot points, 96 traces for each shot, and each trace has 750 sampling points with a sampling interval of 4 ms. Both shot spacing and trace spacing are 25 m, and the minimus offset is 200 m. Fig.4b is the Kirchhoff pre-stack depth migration image using correct migration velocity.

However, there exists another way for remigration, which is not to find the position of the stationary-phase point directly to determine the integral surface. The basic idea of the second way is to compute all values on Huygens surface (ξ; first, to which one point in remigration image field corresponds, then add up all these values, and the final result is taken as the remigration result of the point

(a)The migrated image for model V(z)=1400+0.8 z; (b) the re-migrated image using correct velocity mode; (c) the migrated image for model V(z)=1600+0.8 z; (d) the re-migrated image using correct velocity model. Fig.3 Migration and remigration images for velocity lower and higher than correct velocity

(a) Marmousi velocity model; (b) pre-stack Kirchhoff depth migration image using true velocity model. Fig.4 Maimousi model and migration image

Comparing Fig.4b with Fig.5b, for the inaccurate migrated image obtained when the migration velocity is lower than true one, Kirchhoff remigration can basically reconstruct correct migrated image. Comparing Fig.4b with Fig.5c, it can be seen that the image reconstructed by remigration loss some information in the deep region. This is mainly because when the migration velocity is higher than true velocity, the original migrated image cannot include the deep part of the imaging scope. Besides, Kirchhoff remigration method also has good adaptability to the complicated model.

4 Conclusions

(1) Based on the Kirchhoff migration and Kirchhoff de-migration theory, Kirchhoff remigration operators are discussed in detail. The travel-time computation method, FMM, is also discussed here. Other key parameters, such as the integral surface and the selection of de-migration aperture, are analyzed in detail, the implementation flow of Kirchhoff remigration is then provided.

(a) Kirchhoff pre-stack depth migration result obtained when migration velocity is 5% lower than the true velocity; (b) Kirchhoff remigration result corresponding to the correct velocity model; (c) Kirchhoff pre-stack depth migration result obtained when migration velocity is 5% higher than true velocity; (d) Kirchhoff remigration result corresponding to the correct velocity model. Fig.5 Migration and remigration images for velocity lower and higher than correct Marmousi velocity model

(2) Kirchhoff remigration can directly map the migrated image corresponding to one migration velocity to the migrated image corresponding to another migration velocity model, this method has broad adaptability. The final results of gradient velocity model and Marmousi model verify the effectiveness of Kirchhoff remigration method.

(3) Kirchhoff remigration, just like Kirchhoff migration, is faster in computation compared with other differential operator methods. This method is suitable for 3D seismic data or the time-lapse seismic data imaging. Besides, this method can also be used in the migration velocity analysis. Migration velocity analysis needs an iterative operation, this can be time-consuming. Kirchhoff remigration offers an alternative way for fast migration velocity analysis.

(4) Although the kinematic characteristics of Kirchhoff remigration have been embodied, the dynamic characteristics of this method needs further study.

References

Adler F. 2002. Kirchhoff image propagation. Geophysics, 67(1): 126-134.

Coimbra T A, de Figueiredo S, Schleicher J, et al. 2013. Migration velocity analysis using residual diffraction moveout in the poststack depth domain. Geophysics, 78(3): S125-S135.

Decker L, Merzlikin D, Fomel S. 2017. Diffraction imaging and time-migration velocity analysis using oriented velocity continuation. Geophysics, 82(2): U25-U35.

Fomel S. 1994. Method of velocity continuation in the problem of temporal seismic migration. Russian Geology and Geophysics, 35(5): 100-111.

Fomel S. 2003a. Time-migration velocity analysis by velocity continuation. Geophysics, 68(5): 1662-1672.

Fomel S. 2003b. Velocity continuation and the anatomy of residual prestack time migration. Geophysics, 68(5): 1650-1661.

Hubral P, Schleicher J, Tygel M. 1996a. A unified approach to 3-D seismic reflection imaging, Part I: basic concepts. Geophysics, 61(3): 742-758.

Hubral P, Tygel M, Schleicher J. 1996b. Seismic image waves. Geophysical Journal International, 125(2): 431-442.

Novais A, Costa J, Schleicher J. 2008. GPR velocity determination by image-wave remigration. Journal of Applied Geophysics, 65(2): 65-72.

Schleicher J, Aleixo R. 2006. Time and depth remigration in elliptically anisotropic media using image-wave propagation. Geophysics, 72(1): S1-S9.

Schleicher J, Novais A, Costa J C. 2008. Vertical image waves in elliptically anisotropic media. Studia Geophysica et Geodaetica, 52(1): 101-122.

Schleicher J, Novais A, Munerato F P. 2004. Migration velocity analysis by depth image wave remigration: first results. Geophysical Prospecting, 52(6): 559-573.

Sethian J A. 1999. Fast marching methods. SIAM Review, 41(2): 199-235.

Sun J G. 2010. The stationary phase analysis of Kirchhoff-type de-migration field. Applied Geophysics: English Edition, 7(1):18-30.

Sun Z Q, Sun J G, Han F X. 2012. The comparison of three schemes for computing seismic wave travel-times in complex topographical conditions. Chinese Journal of Geophysics, 55(2):560-568. (in Chinese with English abstract)

Tygel M, Schleicher J, Hubral P, et al. 1993. Multiple weights in diffraction stack migration. Geophysics, 58(12): 1820-1830.

Tygel M, Schleicher J, Hubral P. 1996. A unified approach to 3-D seismic reflection imaging, part II: theory. Geophysics, 61(3): 759-775.