表面多次波最小二乘逆時偏移成像

劉學建， 劉伊克

1 中國科學院地質(zhì)與地球物理研究所工程地質(zhì)力學重點實驗室， 北京 100029 2 中國科學院大學， 北京 100049

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表面多次波最小二乘逆時偏移成像

劉學建1,2， 劉伊克1

1 中國科學院地質(zhì)與地球物理研究所工程地質(zhì)力學重點實驗室， 北京 100029 2 中國科學院大學， 北京 100049

使用相同的炮記錄，多次波偏移能提供比反射波偏移更廣的地下照明和更多的地下覆蓋但是同時產(chǎn)生很多的串聲噪聲.相比傳統(tǒng)逆時偏移，最小二乘逆時偏移反演的反射波成像結(jié)果具有更高的分辨率和更均衡的振幅.我們主要利用最小二乘逆時偏移壓制多次波偏移產(chǎn)生的串聲噪聲.多次波最小二乘逆時偏移通常需要一定的迭代次數(shù)以較好地消除串聲噪聲.若提前將一階多次波從所有階數(shù)的多次波中過濾出來，使用相同的迭代次數(shù)，一階多次波的最小二乘逆時偏移能夠得到具有更高信噪比的成像剖面，而且能夠提供與多次波最小二乘逆時偏移相似的有效地下結(jié)構(gòu)成像.

最小二乘逆時偏移； 多次波成像； 一階多次波

In order to invert primaries as an image, LSRTM iteratively solves a misfit function that is the L2 norm of the amplitude residual between the modeled and observed primaries. Born modeling is a linear two-step procedure and synthesizes primaries perturbed by an image, which bases the LSRTM. Conventional RTM is the adjoint of Born modeling, whereas the analytical solution of the misfit function is the generalized-inverse of the Born modeling. The analytical solution is hard to be obtained because the Hessian matrix is so large, so a nonlinear optimal scheme, e.g., the steepest-descent method, can be used to iteratively solve the misfit function. Taking the released Sigsbee2b data as an example, we can intuitively conclude that LSRTM provides an image with higher resolution and more balanced amplitude and suppresses the migration artifacts compared with conventional RTM. Different with the misfit function for the conventional LSRTM, the misfit function for the LSRTM of multiples is the L2 norm of the amplitude residual between the modeled multiples and estimated multiples during the regular seismic data processing. The accurate calculation for the modeling of multiples is crucial for the success of this method, where a modified Born modeling procedure and an accurate background velocity are utilized. Instead of a point source, the recorded data including primaries and multiples are forward propagated and stacked as the downgoing wavefield. Each discrete point of the image is seen as a scatter. The two-order time derivative of downgoing wavefield is scattered by the RTM image of multiples, and upgoing wavefield is the stack of scattered waves. Surface-related multiples are modeled by recording the upgoing wavefield at receivers. Similar to the conventional LSRTM, LSRTM of multiples also can iteratively seek the reflectivity model using a nonlinear optimal method. Moreover, to invert first-order multiples as an image, the misfit function based on the L2 norm of the amplitude residual between the observed and Born modeled first-order multiples should be built. Compared with the Born modeling of all-order multiples, instead of total recorded data, only primaries are forward propagated for the Born modeling of first-order multiples. The observed first-order multiples are estimated by a modified SRME, which includes two steps: (1) predicting higher-order multiples by the convolution of primaries and multiples; (2) adaptively subtracting higher-order multiples from all multiples.

RTM of all-order multiples, LSRTM of all-order multiples, RTM of first-order multiples and LSRTM of first-order multiples have been tested on a three-layer and the Marmousi2 model. Only 16 shot gathers are used for imaging on the three-layer model. RTM image of multiples provide wider illumination and higher fold for subsurface, whereas there are a lot of artifacts in the image of multiples. After 10 iterations, LSRTM attenuate most of the artifacts in the image of multiples, except the artifacts at bottom. Moreover, LSRTM of first-order multiples provide a more cleaner section than LSRTM of all-order multiples. There are artifacts in the modeled data using the RTM image of multiples, whereas the modeled data using LSRTM image of multiples have a good match with the estimated multiples using SRME and avoid most artifacts. On the Marmousi2 model, there are many artifacts in the RTM image of multiples, which are mostly attenuated by LSRTM after 5 iterations. However, after 5 iterations, there are still residual artifacts in the LSRTM image of multiples, which disappear in LSRTM image of first-order multiples. On above two experiments, LSRTM of multiples and LSRTM of first-order multiples both converge very fast and robust.

Compared with RTM, LSRTM provides image with more balanced amplitude and better resolution and suppresses the migration artifacts. RTM of multiples can provide a wider illumination and higher fold for subsurface. However, there are many crosstalk artifacts in the RTM image of all-order multiples. LSRTM can attenuate most of the crosstalk artifacts in the image of multiples but costs huge computation of many iterations. A modified SRME is proposed to filter first-order multiples. With the same iterations used, LSRTM of first-order multiples provide a much cleaner section, and provider a similar true image of reflectors compared with LSRTM of all-order multiples. Prior to LSRTM of first-order multiples, first-order multiples are needed to be estimated by a modified SRME.

1 引言

多次波通常被認為是一種噪聲，并且在偏移之前的數(shù)據(jù)預處理中盡可能的減掉 (Berkhout and Verschuur, 1997; Verschuur and Berkhout, 1997; Liu et al., 2009, 2010; Dragoset et al., 2010；李鵬等，2007；王維紅和井洪亮，2015).實際上多次波在地下比反射波傳播路徑更長且覆蓋范圍更廣，多次波中含有豐富的小角度信息.在使用相同炮記錄偏移時，多次波能為下地表提供更寬的成像范圍和更多的覆蓋.近年來，很多的學者致力于多次波成像的研究，并且提出了多種多次波成像方法.多次波可以首先被轉(zhuǎn)化為反射波(Berkhout and Verschuur, 2003, 2006; Schuster et al., 2004; Verschuur and Berkhout, 2005； 劉學建等，2015)，并利用傳統(tǒng)的逆時偏移方法成像.更一步的，反射波波動方程偏移方法或者逆時偏移成像方法可以修改為多次波直接波動方程(Guitton, 2002; Muijs et al., 2007; Lu et al., 2011)或者逆時偏移(Liu et al., 2011a, 2011b)成像方法.逆時偏移(Baysal et al., 1983)是一種強有力的成像技術(shù)，能夠利用多種地震波 (包括反射波、回轉(zhuǎn)波以及棱柱波)，從而對速度的橫向變化有良好的適應性并有能力對陡傾角成像.多次波逆時偏移也具有上述傳統(tǒng)逆時偏移成像的優(yōu)勢.然而，因為不同階數(shù)多次波波場之間的互相關(guān)，多次波逆時偏移成像過程中將會產(chǎn)生大量的串聲噪聲.這些串聲噪聲分布在整個成像剖面中，破壞了有效成像的結(jié)構(gòu)和振幅.串聲噪聲很難消除并且大大降低了多次波成像的價值.

相對于傳統(tǒng)的偏移方法，最小二乘逆時偏移(Dong et al., 2012; Dai et al., 2012; Dai and Schuster, 2013; Zhang et al., 2015)能提供振幅更均衡、分辨率更高的反射波成像結(jié)果，并能夠消除偏移噪聲.最小二乘逆時偏移方法也能夠消除多次波成像中的串聲噪聲(Brown and Guitton, 2005; Wong et al., 2014; Zhang and Schuster, 2014)，其目標函數(shù)為波恩模擬的多次波與觀測的多次波之間的差的L2 范數(shù).通過一個最優(yōu)化迭代算法(如最速下降法)求解該目標函數(shù)以得到地下反射率分布的過程，即為多次波的最小二乘逆時偏移反演成像.多次波與反射波的波恩模擬區(qū)別主要在于：不是利用震源子波，而是將包含反射波和多次波的觀測數(shù)據(jù)作為震源正傳.最小二乘逆時偏移每次迭代都消耗大約幾倍逆時偏移的計算量，計算成本非常高.而多次波最小二乘逆時偏移往往需要一定數(shù)量的迭代次數(shù)以較好地壓制串聲噪聲.因此，我們修改SRME方法，只將一階多次波從所有階數(shù)的多次波中過濾出來.基于波恩模擬的一階多次波與記錄的一階多次波差的二范數(shù)最小，一階多次波的最小二乘逆時偏移成像方法能夠以相同的迭代次數(shù)得到更高信噪比的多次波成像剖面.

本文首先回顧了反射波最小二乘逆時偏移的基本原理，并通過Sigsbee2b模型來驗證其優(yōu)勢.然后闡述了多次波最小二乘逆時偏移的基本原理；一階多次波的分離方案；一階多次波的最小二乘逆時偏移原理.最后利用一個三層模型及Marmousi2模型，對多次波及一階多次波最小二乘逆時偏移進行數(shù)值實驗.

2 基本原理

2.1 反射波最小二乘逆時偏移

對于二維模型，檢波器記錄到的從震源激發(fā)的地下一次散射波，可以通過波恩近似來模擬，其頻率域的表達式為：

d(xr,xs,ω)= ω2∫G0(xr,x,ω)r(x)G0(x,xs,ω)

×fs(ω)dx.

(1)

其中，ω表示圓頻率，fs(ω)表示震源子波，G0(xr,x,ω)和G0(x,xs,ω)分別表示連接檢波器xr和震源xs與地下散射點x=(x,z)的格林函數(shù)，r(x)表示反射率分布模型，d(xr,xs,ω)為模擬的散射波.波恩模擬的向量表達式為：

(2)

而傳統(tǒng)的偏移方法可以認為是波恩模擬的共軛轉(zhuǎn)置：

(3)

(4)

用算子M(r(x))表示時間域的波恩模擬，其具體實現(xiàn)方法為：

(5)

其中，v0(x)為光滑的背景速度，p0(x,t)為下行的震源波場，pr(x,t)為上行的波場.用算子MT(dobs)表示時間域的逆時偏移，其實現(xiàn)過程簡單概括為：

(6)

(7)

(8)

其中，q(x,t)為檢波器數(shù)據(jù)的逆?zhèn)鞑▓?另外，為滿足成像條件(8)的要求，公式(6)模擬的震源波場需要被重建為時間逆序的波場.

反射波最小二乘逆時偏移最為基本的目標函數(shù)為波恩模擬的反射波d(xr,xs,t)與觀測的反射波dobs(xr,xs,t)之間差的能量：

f(r(x))=∫∫(d(xr,xs,t)-dobs(xr,xs,t))2dtdxr,

(9)

第k次迭代模擬的反射波d(k)(xr,xs,t)和數(shù)據(jù)殘差δd(k)(xr,xs,t)表示為：

(10)

δd(k)(xr,xs,t)=d(k)(xr,xs,t)-dobs(xr,xs,t).

(11)

目標函數(shù)(9)的梯度和基于梯度下降法的迭代解分別為：

(12)

(13)

如公式(10)—(13)所示，最小二乘逆時偏移是一個迭代求解過程.如圖1的對比(使用Sigsbee2b發(fā)布的層速度和帶有鬼波的反射波數(shù)據(jù))，相比傳統(tǒng)逆時偏移，最小二乘逆時偏移成像結(jié)果具有較高分辨率、更均衡的振幅，并能壓制偏移噪聲.

2.2 多次波最小二乘逆時偏移

相對于子波震源，采集的包含反射波dobs(xr,xs,t)和多次波mobs(xr,xs,t)的全波波場記錄Dobs(xr,xs,t)可以看作多次波的二次震源，則波恩模擬的多次波m(xr,xs,t)可以表示為

(14)

相應的，多次波的逆時偏移可以簡單表示為：

(16)

(17)

地震數(shù)據(jù)處理流程中，反射波與多次波將會被分離，分離出的多次波作為觀測的多次波.多次波的最小二乘逆時偏移的目標函數(shù)為波恩模擬的多次波與觀測的多次波之間差的能量：

圖1 將SMAART JV發(fā)布的帶有鬼波的反射波數(shù)據(jù)作為觀測反射波，且將層速度平滑后作為背景速度(a)反射波逆時偏移成像結(jié)果；(b)反射波最小二乘逆時偏移(30次迭代)成像結(jié)果. 圖(b)具有更高的分辨率，如黑色箭頭所示，散射體有更好的聚焦.圖(b)在鹽丘下有更均衡的振幅.如白色箭頭所示，最小二乘逆時偏移能夠壓制偏移噪聲.Fig.1 The released primaries with ghosts by SMAART JV are treated by observed primaries, and the interval velocity is smoothed as to a background velocity(a) RTM image of primaries; (b) LSRTM of primaries with 30 iterations. Figure (b) have a better resolution, as indicated by black arrows, the scatters are better focused. Figure (b) have a more balance amplitude at the subsurface. As indicated by white arrows, LSRTM can suppress migration noises.

f(r(x))=∫∫(m(xr,xs,t)-mobs(xr,xs,t))2dtdxr.

(18)

如公式(10)—(13)所示, 一個相似的迭代過程求解目標函數(shù)，則得到多次波的最小二乘逆時偏移反演成像結(jié)果.

2.3 一階多次波最小二乘逆時偏移

采集的反射波數(shù)據(jù)dobs(xr,xs,t)可以看作一階多次波的二次震源，則波恩模擬的一階多次波m1(xr,xs,t)可以表示為

(19)

(21)

(22)

(23)如公式(10)—(13)所示, 一個相似的迭代過程求解目標函數(shù)，則得到一階多次波的最小二乘逆時偏移反演成像結(jié)果.

3 數(shù)值實驗

如圖2，本次實驗的流程是首先正演帶多次波的數(shù)據(jù)，用SRME分離反射波(含鬼波和層間多次波)和表面多次波，一個修改的SRME流程從所有多次波中分離出一階多次波.反射波，表面多次波和分離出的一階多次波可以應用于最小二乘逆時偏移反演成像中.

3.1 簡單三層模型

如圖3所示為一個簡單三層聲波速度模型，橫向1201網(wǎng)格點，縱向501網(wǎng)格點，網(wǎng)格間距5 m.共有16炮用于偏移成像;震源子波主頻為15 Hz,并等間距的在2.04 km和3.84 km之間激發(fā).中間放炮觀測系統(tǒng)，每個炮記錄有201個檢波器.震源和檢波器的深度為5 m.最大記錄時間長度和采樣間隔分別為3 s和2 ms.

圖2 實驗流程Fig.2 The workflow of the experiments

圖3 三層聲波速度模型Fig.3 Three-layer acoustic velocity model

圖4 (a) SRME估計的反射波的逆時偏移成像結(jié)果；(b)多次波逆時偏移成像結(jié)果多次波偏移為下地表提供了更寬的照明范圍和更多的覆蓋次數(shù)；如箭頭所示，多次波偏移也產(chǎn)生了很多的串聲假象.Fig.4 (a) RTM image of primaries estimated by SRME; (b) RTM image of multiplesMigration of multiples provides wider illumination and more fold for subsurface; however, as indicated by the arrows, migration of multiples also generates many crosstalk artifacts.

圖5 (a)多次波最小二乘逆時偏移成像結(jié)果(10次迭代)； (b)一階多次波最小二乘逆時偏移成像結(jié)果(10次迭代)如(a)中藍色箭頭所示，多次波最小二乘逆時偏移壓制了圖4b中大部分的串聲假象.如黑色箭頭所示，(a)中殘留在深部的串聲假象在(b)中消失.而且(b)與(a)有相似的能對應地下反射位置的有效成像結(jié)果.Fig.5 (a) LSRTM image of all-order multiples (10 iterations); (b) LSRTM image of first-order multiples (10 iterations)As indicated by blue arrows in (a), LSRTM of multiples suppresses most of crosstalk artifacts in Fig.4b. As indicated by black arrows, residual artifacts at deep in (a) disappear in (b). Moreover, panel (b) provides a similar true-image of reflectors to panel (a).

如圖4所示為反射波成像與多次波成像的對比.顯而易見，當相同的炮記錄用于偏移時，多次波偏移為下地表提供了更寬的照明范圍和更多的覆蓋次數(shù)；然而，多次波偏移也產(chǎn)生了很多的串聲假象.如圖5為多次波最小二乘逆時偏移與一階多次波最小二乘逆時偏移的對比，它們都用了10次迭代計算.多次波最小二乘逆時偏移壓制了大部分多次波逆時偏移中的串聲假象；然而，在多次波最小二乘偏移剖面的深部，仍有殘留的串聲假象；這些殘留的串聲假象在一階多次波最小二乘逆時偏移剖面中消失.使用同樣的迭代次數(shù)，一階多次波最小二乘逆時偏移能夠提供與多次波最小二乘逆時偏移相似的地下構(gòu)造成像結(jié)果；而一階多次波最小二乘逆時偏移結(jié)果有更高的信噪比.

另外，我們通過數(shù)據(jù)域的對比來說明，多次波最小二乘逆時偏移能夠消除多次波逆時偏移中的串聲假象.如圖6 所示，利用多次波逆時偏移結(jié)果，波恩模擬的多次波中有很多的虛假同相軸；而多次波最小二乘逆時偏移結(jié)果，波恩模擬的多次波沒有虛假的同相軸，與SRME估計的多次波有很好的匹配.圖7中為多次波及一階多次波最小二乘逆時偏移中歸一化的數(shù)據(jù)殘差收斂曲線，它們表現(xiàn)出相似的快速穩(wěn)定收斂性質(zhì).

圖6 (a) SRME估計的所有階數(shù)的多次波；(b)波恩模擬的多次波利用如圖4b所示的多次波偏移結(jié)果；(c)波恩模擬的多次波利用如圖5a所示的多次波最小二乘逆時偏移結(jié)果如箭頭所示，(b)中虛假的同相軸在(c)中消失.(c)中模擬的多次波與(a)中多次波有較好的匹配.Fig.6 (a) Estimated all-order multiples using SRME; (b) Born modeled multiples using the RTM image of all-order multiples in Fig.4b; (c) Born modeled multiples using the LSRTM image of all-order multiples in Fig.5aAs indicated by the arrows, the false events in (b) disappear in (c). The modeled multiples in (c) have a good match with multiples in (a).

圖7 簡單三層模型上多次波(實線)及一階多次波(散點)最小二乘逆時偏移的歸一化數(shù)據(jù)殘差收斂曲線Fig.7 Normalized data residual for LSRTM of multiples (solid line) and first-order multiples (dots) on the simple three-layer model

3.2 Marmousi2模型

如圖8所示為Marmousi2聲波模型的中間部分，橫向1601網(wǎng)格點，縱向561網(wǎng)格點，網(wǎng)格間距6.25 m.共有81炮用于偏移成像;震源子波主頻為20 Hz,并等間距的在2 km和8 km之間激發(fā).中間放炮觀測系統(tǒng)，每個炮記錄有241個檢波器.震源和檢波器的深度為6.25 m.最大記錄時間長度和采樣間隔分別為4 s和2 ms.

圖9對比了多次波逆時偏移成像結(jié)果與多次波最小二乘逆時偏移成像結(jié)果，并對比了多次波最小二乘逆時偏移成像結(jié)果與一階多次波最小二乘逆時偏移成像結(jié)果.在這個例子中，多次波和一階多次波最小二乘逆時偏移都只使用了5次迭代.多次波最小二乘逆時偏移消除了多次波逆時偏移中大部分的串聲假象； 而一階多次波最小二乘逆時偏移提供比多次波最小二乘逆時偏移有更高信噪比的成像結(jié)果.雖然一階多次波最小二乘逆時偏移缺少了高階多次波的信息，依然能提供與多次波最小二乘逆時偏移相似的有效構(gòu)造成像結(jié)果.圖10中, 多次波及一階多次波最小二乘逆時偏移表現(xiàn)出相似的快速穩(wěn)定的收斂性質(zhì).

4 討論

利用SRME方法將反射波和多次波分離是針對海上采集數(shù)據(jù)的常規(guī)處理流程之一.SRME方法需要較密的炮檢排列和近偏移距數(shù)據(jù)，因此在使用之前需要做數(shù)據(jù)規(guī)則化；尤其是對于三維數(shù)據(jù)，橫向上的采集數(shù)據(jù)較為稀疏，增加了數(shù)據(jù)規(guī)則化的難度(Dragoset et al., 2010).另外，在三維數(shù)據(jù)上使用SRME方法，需要存儲大規(guī)模的共道集數(shù)據(jù).

圖8 Marmousi2 聲波速度模型Fig.8 Marmousi2 acoustic velocity model

圖9 (a)多次波逆時偏移成像結(jié)果；(b)多次波最小二乘逆時偏移成像結(jié)果(5次迭代)；(c)一階多次波最小二乘逆時偏移成像結(jié)果(5次迭代)

As indicated by white labels in (a) and (b), LSRTM of multiples suppresses most of the crosstalk artifacts in the RTM of multiples. Black labels in (b) and (c) highlight that LSRTM of first-order multiples provides an image with a higher signal to noise ratio than LSRTM of multiples; and they provide similar true images.

圖10 Marmousi2模型上多次波(實線)及一階多次波(散點)最小二乘逆時偏移的歸一化數(shù)據(jù)殘差收斂曲線Fig.10 Normalized data residual for LSRTM of multiples (solid line) and first-order multiples (dots) on the Marmousi2 model

常規(guī)的數(shù)據(jù)處理提供了分離的反射波和多次波; 多次波成像利用了傳統(tǒng)處理流程中被認為是噪聲而丟掉的多次波，提供了除反射波外的額外地下照明.而從所有多次波中分離一階多次波，無需額外的數(shù)據(jù)規(guī)則化，增加的計算量主要為：通過反射波與多次波的一次褶積來預測除了一階外的所有高階多次波，將高階多次波從所有多次波中減去.一階多次波的分離方法很容易拓展到三維算法，難點在于增加的存儲量和計算量.

圖11 在R2位置記錄到的反射波偏移成像時，總的傳播路徑為SX1和R2X1；在R2位置記錄到的多次波偏移成像時(Liu et al., 2011a, 2011b)，總的傳播路徑為R1X2和R2X2Fig.11 When the primary recorded at R2 is migrated, the total propagation path is SX1 and R2X1. When the multiple recorded at R2 is migrated, the total propagation path is R1X2and R2X2

反射波最小二乘逆時偏移需要較好的偏移速度(Dai and Schuster, 2013; Huang et al., 2014).而多次波或者一階多次波最小二乘逆時偏移，受偏移速度不準確的影響相對較小.因為，在相同的偏移距處，反射波成像比多次波成像的傳播路徑要更長(如圖11).

多次波成像與反射波成像的主要區(qū)別在于震源項的不同，而兩者的正演算法是相同的.因此多次波或者一階多次波最小二乘逆時偏移也可以拓展到三維模型上.三維的算法能夠使實際資料的偏移歸位更加準確，因此，拓展到三維算法能提高多次波或者一階多次波最小二乘逆時偏移在實際資料應用時的收斂性.

5 結(jié)論

多次波逆時偏移成像能夠?qū)ο碌乇硖峁╊~外的照明，但是卻產(chǎn)生了很多串聲噪聲.我們在數(shù)據(jù)和成像域驗證了最小二乘逆時偏移能夠消除多次波逆時偏移產(chǎn)生的串聲假象.利用多次波的最小二乘逆時偏移的成像剖面，波恩模擬的多次波與觀測的多次波有很好的匹配.然而， 在多次波最小二乘逆時偏移成像剖面中，往往會有殘余的噪聲.我們利用修改的SRME流程將一階多次波從所有多次波中分離出后，使用同樣的迭代次數(shù)，一階多次波最小二乘逆時偏移能夠提供與多次波最小二乘逆時偏移相似的有效構(gòu)造成像結(jié)果；而一階多次波最小二乘逆時偏移結(jié)果中有更少的噪聲.總之，多次波或者一階多次波最小二乘逆時偏移，能夠以較高的信噪比為下地表提供額外的照明，或許可以為復雜結(jié)構(gòu)的成像做出貢獻.

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(本文編輯 何燕)

Least-squares reverse-time migration of surface-related multiples

LIU Xue-Jian1,2, LIU Yi-Ke1

1KeyLaboratoryofEngineeringGeomechanics,InstituteofGeologyandGeophysics,ChineseAcademyofSciences,Beijing100029,China2UniversityofChineseAcademyofSciences,Beijing100049,China

Surface-related multiples are traditionally treated as noise and are attenuated using surface-related multiples elimination (SRME) and/or radon-based multiple-elimination methods. Multiples penetrate into the subsurface several times and contain abundant reflection information of small angles. Compared with migrating of primaries, migrating of multiples extends all the receivers as second sources and sometimes provides additional subsurface illumination. For reverse-time migration (RTM) of all-order multiples, however, the main challenge is that undesired crosscorrelations between forward and backward propagated seismic waves generate so many crosstalk artifacts. The crosstalks may distribute in the whole image profile, which can destruct the true image of reflectors and mislead the interpreting result of a migrated image. Compared with conventional RTM, least-squares reverse-time migration (LSRTM) can invert recorded primaries as an image with more balanced amplitude and higher resolution. Moreover, we develop the conventional LSRTM to invert multiples as an image while iteratively suppressing crosstalk artifacts. However, LSRTM of multiples can′t totally attenuate the artifacts in the image of multiples, and usually many iterations are required to invert a well-accepted image. Alternatively, if first-order multiples can be separated from all-order multiples in advance, LSRTM of first-order multiples can be developed to reduce the iteration number. With the same iterations used, compared with LSRTM of multiples, LSRTM of first-order multiples can provide a much cleaner image section and a similar true image of reflectors. The motivation to develop LSRTM of first-order multiples can be further summarized as: (1) conventional migration of first-order multiples can avoid the most undesired crosscorrelations between forward and backward propagated wavefields and can maintain some advantages of imaging multiples at the same time; although the subsurface information contributed by higher-order multiples is neglected, RTM of first-order multiples have already avoided most artifacts. (2) There are still some crosstalk artifacts in the RTM image of first-order multiples; then, compared with RTM of first-order multiples, LSRTM of first-order multiples can further enhance the image in detail by suppressing the crosstalk artifacts, balancing the amplitude, and improving the resolution.

Least-Squares Reverse-Time Migration (LSRTM); Migration of multiples; First-order multiples

10.6038/cjg20160919.

國家自然科學基金項目(41430321，41374138)和中國科學院戰(zhàn)略性先導科技專項(B類) (XDB01020300)聯(lián)合資助.

劉學建，男，1987年生，在讀博士研究生，從事表面多次波消除、多次波成像方法、最小二乘逆時偏移以及逆時偏移角道集等方面的研究. E-mail:liuxuejian10@mails.ucas.ac.cn

10.6038/cjg20160919

P631

2015-09-01，2016-07-28收修定稿

劉學建，劉伊克. 2016. 表面多次波最小二乘逆時偏移成像. 地球物理學報，59(9):3354-3365,

Liu X J, Liu Y K. 2016. Least-squares reverse-time migration of surface-related multiples.ChineseJ.Geophys. (in Chinese),59(9):3354-3365,doi:10.6038/cjg20160919.