CN106461802B - 用于多参数全波场反演的有效的线性搜索方法 - Google Patents

用于多参数全波场反演的有效的线性搜索方法 Download PDF

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CN106461802B
CN106461802B CN201580022137.XA CN201580022137A CN106461802B CN 106461802 B CN106461802 B CN 106461802B CN 201580022137 A CN201580022137 A CN 201580022137A CN 106461802 B CN106461802 B CN 106461802B
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唐亚勋
G·阿耶尼亚
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Abstract

用于同时对多个类别的物理特性参数(例如，速度和各向异性)的全波场地震数据进行反演的方法，该方法通过进行以下操作：针对每个类别的参数计算目标函数的梯度(即，搜索方向)；然后将(优选地，穷尽的)第一遍独立线性搜索应用于每个参数类别，以沿着每个参数类别的搜索方向获得对应的步长；然后尚未更新模型，使用步长定义在所有参数类别的梯度之间的相对缩放。接下来，重新结合每个经缩放的搜索方向，以形成新的搜索方向，并且沿着新的搜索方向实行新的第二遍线性搜索，并且用所得的步长同时更新所有的参数。作为前述供选择的两遍实施例的供选择的方案，可以在每个第一遍线性搜索之后更新模型，并且不实行第二遍线性搜索。

Description

用于多参数全波场反演的有效的线性搜索方法
相关申请的交叉引用
本申请要求2014年5月9日提交的标题为“用于多参数全波场反演的有效的线性搜索方法(EFFICIENT LINE SEARCH METHODS FOR MULTI-PARAMETER FULL WAVEFIELDINVERSION)”的美国临时专利申请61/990,860的权益，该专利申请的全部内容以引用方式并入本文。
技术领域
一般来说，本公开涉及用于碳氢化合物的地球物理勘探的领域，并且更具体地，涉及地震数据处理。具体地，本公开涉及用于在地震数据的多参数全波场反演(“FWI”)中实行有效的线性搜索以推断地下物理特性模型的方法。在碳氢化合物的探测或碳氢化合物的生产中这样的模型可以是有用的。
背景技术
全波场反演是非线性反演技术，该非线性反演技术通过使所模拟的地震波场和所观测的地震波场之间的失配最小化，恢复地球模型。由于与FWI相关联的高计算成本，常规的实施方式利用局部优化技术来估计最优模型参数。广泛使用的局部优化技术是基于梯度的一阶方法(例如，最速下降或非线性共轭梯度)，该方法仅利用目标函数的梯度信息定义搜索方向。虽然唯梯度一阶方法是相对有效的(该方法要求仅计算目标函数的梯度)，但是其收敛性一般是缓慢的。通过使用二阶方法，能够显著提高FWI的收敛性。因为二阶方法利用目标函数的梯度和曲率信息两者来确定模型参数空间中的最优搜索方向，所以实现该提高的收敛性。(搜索方向单位矢量s通过m更新的＝m+αs与模型更新过程有关，其中α(标量)是步长。)
一阶方法和二阶方法之间的主要差异是二阶方法用逆矩阵海赛函数(inverseHessian)(例如，高斯牛顿/牛顿方法)或用所投影的海赛函数的逆矩阵(例如，子空间方法)预处理梯度。海赛函数是目标函数关于模型参数的二阶偏导数的矩阵。一般来说，二阶方法具有吸引力，不仅是因为它们相对快速的收敛速率，还因为在多参数反演的情况下平衡不同的参数类别的梯度和为具有不同的数据灵敏度的参数类别(例如，速度、各向异性、衰减等)提供有意义的更新的能力。在二阶方法中，如果同时对此参数类别进行反演，则使用海赛函数的参数类别的最优缩放在多参数反演中是关键的。然而，因为计算海赛函数的逆矩阵非常昂贵，所以这是在实践中广泛采用二阶方法是主要的障碍。二阶方法的另一个缺点是如果目标函数不是二次的或凸性的(convex)(例如，其中初始模型远不是真实模型)，则海赛函数或其近似不能准确地预测目标函数的形状。从此，不能适当地缩放不同参数类别的梯度，从而得到次优搜索方向。
发明内容
在一个实施例中，本发明是用于对地震数据进行迭代反演以同时推断地下的至少两个物理特性的模型的方法，所述方法包括：
(a)对于每个物理特性，针对物理特性的参数，计算目标函数的梯度，所述目标函数测量地震数据中的全部或一部分和对应模型所模拟的地震数据之间的不匹配；
(b)对于每个物理特性，根据梯度计算模型空间中的搜索方向；
(c)针对至少两个物理特性在搜索方向之间或之中进行交替线性搜索，以确定沿着搜索方向中的每个搜索方向的最优步长；以及
(d)使用最优步长更新模型。
附图说明
通过参考以下详细描述和附图可以更好地理解本发明及其优点，在附图中：
图1例示在本发明的交替的一遍线性搜索实施例中的最优搜索方向；
图2例示可以由本发明的交替的一遍线性搜索实施例得到的次优搜索方向；
图3例示使用本发明的交替的两遍线性搜索实施例的最优搜索方向；
图4A至图4F示出，对于本发明的方法的测试示例，用于两个参数(速度和各向异性)的初始模型和最终反演模型，并且使用初始模型和最终模型比较在迁移之后的数据；
图5是示出在本发明的方法的交替的一遍线性搜索中的基本步骤的流程图；
图6是示出在本发明的方法的交替的两遍线性搜索中的基本步骤的流程图；以及
图7是示出在本文中所公开的扩展的交替的两遍线性搜索中的基本步骤的流程图，其中级联反演方法与交替的两遍线性搜索结合。
由于对附图中颜色使用的专利规则限制，所以图4A至图4F中的一些是原始彩色生成的附图的黑白再现。
将结合示例实施例描述本发明。然而，在一定程度上，以下详细描述针对本发明的特定实施例或特定用途，这旨在仅是说明性的，并且不被解释为限制本发明的范围。相反，旨在覆盖可以包含在如由随附权利要求所限定的本发明的范围内的所有的供选择的方案、修改和等价物。
具体实施方式
通过使用可以被称为交替的一遍/两遍线性搜索方法(该方法不要求来自海赛函数矩阵的明示信息，而是通过连续的线性搜索近似二阶信息)，本发明弥合一阶和二阶最优方法之间的差距。将示出本发明的方法能够适当地缩放具有不同的数据灵敏度的参数类别的梯度，并且能够同时为多个参数类别提供有意义的更新。在实践中，因为本发明的方法没有假设目标函数是二次的，并且如果能够有效地实施每个线性搜索，则本发明的方法还能够显著地更廉价，所以与基于海赛函数的二阶方法相比，本发明的方法能够更加强健。
虽然为了简单起见，使用两个参数类别描述了理论，但是本发明适用于任何数量的参数类别的同时反演，并且该方法扩展到超过两个参数类别是简单明了的。对于其中对两个参数类别进行反演的情况，模型能够表示为包含两个不同子模型的矢量，即，m＝(m1m2)T，其中m1和m2分别是第一模型参数类别和第二模型参数类别，其中T代表转置。当前迭代s的搜索方向是这两个参数类别的搜索方向的级联，并且能够被写为：
其中，s1和s2分别是第一参数类别和第二参数类别的搜索方向。基于一阶的方法通常沿着方向s更新模型m。该方法的一个主要问题是如果模型参数类别是具有非常不同的单位和对FWI目标函数的非常不同的灵敏度的物理量(例如，速度、各向异性、衰减等)，则从FWI梯度导出的得到的搜索方向将通常具有非常不同的量值，即，s1的量值可以与s2的量值显著不同。这常常导致FWI选取收敛路径，FWI沿着该收敛路径主要更新对目标函数更加灵敏的参数类别，同时使对目标函数不太灵敏的参数类别保持基本不更新。因此，FWI可以朝次优解收敛。下面是所描述的本发明的基于交替的线性搜索的方法的两个供选择的实施例，所述实施例解决了该问题，其目标是为所有的参数类别同时提供最优更新，同时不会由于不同的单位或数据灵敏度而偏向某些参数类别。
方法I：交替的一遍线性搜索
为了开始，定义两个基矢量，两个基矢量中的每个基矢量包含具体模型参数类别的搜索方向：
其中，0指代包含零的矢量。然后，原始搜索方向能够被写为上面的两个基矢量的和，上面的两个基矢量彼此正交(即，)：
在该第一方法中，用图5中所示的基本步骤以交替的方式更新两个模型参数类别。当前模型52被用于模拟预测的数据，并且其与所测量的数据51结合，以计算目标函数，并且然后在步骤53计算关于第一模型参数的目标函数的梯度，以在步骤54生成搜索方向在步骤55，沿着方向实行线性搜索，并且更新模型。由于仅对于第一参数类别来说，是非零的，所以将仅更新参数类别m1。在已经更新模型之后，然后，沿着方向实行线性搜索，并且再次更新模型(步骤56)。在该情况下，由于仅对于第二参数类别来说，是非零的，所以将仅更新参数m2，如图1所示。在该方法中，一个FWI迭代是指以该交替的方式更新两个参数类别的过程，并且在步骤57，针对另一个迭代重复该过程，除非根据一些预定收敛准则另一个迭代已经收敛或已经达到另一个停止点。因为以独立的方式更新m1和m2，所以克服了在上面所描述的搜索方向(或梯度)中的缩放失衡。然后，所有的参数类别以交替的方式接收显著的更新。该方法可以被称为交替的一遍线性搜索。该方法的一个潜在的缺陷是连续的线性搜索的次序能够影响收敛路径，并且如果目标函数具有相对复杂的空间，则这可以导致之字形收敛路径(图2)。
更详细地，线性搜索涉及在高度多维模型参数空间中沿着搜索方向针对各种不同的“步”长的模型模拟的地震数据。选择使模型模拟的数据和所测量的数据之间的不匹配最小化的步长。在上面所描述的方法I中，在模型中更新第一物理特性之后，然后所更新的模型被用于模拟用于线性搜索的数据，以更新第二物理特性。尽管第二线性搜索仅更新第二物理特性，但是根据第一线性搜索更新第一物理特性将影响用于第二线性搜索的模型模拟的数据，并且因而将影响对第二物理特性的所得的更新。
为了进一步解释图1，由在目标函数轮廓的中心中的星号表示解，因为与模型参数m2相比，模型参数m1对目标函数更加灵敏，所以目标函数轮廓是椭圆而不是圆。图1示出在这种情形下通过使用本发明的方法的交替的一遍线性搜索实施例进行两个独立步骤能够在单个迭代过程周期中获得该解。图1还示出传统的线性搜索s由不同的灵敏度偏离右边方向，并且将要求更多的迭代以获得该解。图2示出其中交替的一遍线性搜索可能是有问题的，即，当m1和m2耦合时，如由是倾斜的目标函数轮廓所示的，意味着这两个参数类别之间的强相关性。在此类状况下，交替的一遍线性搜索可能要求额外的迭代以找到最优搜索方向。图3示出本发明的交替的两遍线性搜索实施例(接下来所解释的)如何能够解决该问题。
方法II：交替的两遍线性搜索
通过对方法I的修改，能够减轻上面所识别的方法I的缺陷中的一些。如在图6中所指示的，所观测的数据61和当前地下模型62被用于针对每个参数类别计算目标函数的梯度(步骤63)，得到每个参数类别的搜索方向(64)。在步骤65，针对每个参数类别，使用在步骤64针对此类参数类别所计算的搜索方向，实行第一遍独立线性搜索。保存所计算的步长，但尚不实行模型更新。在步骤66，该修改的方法在其方面中的一个方面上类似于所谓的子空间方法(Kennett等人，1998)，但不是以交替的方式更新所有模型类别，该修改的方法首先使用两个缩放因子α和β，即，步长，重新结合两个基矢量和以形成如下的新的搜索方向：
一旦获得该新的结合的搜索方向，就实行第二遍线性搜索(利用新的搜索方向)，以同时更新两个参数类别(步骤67)。该方法和已知的子空间方法(Kennett等人，1998)之间的主要差异是如何估计缩放因子α和β。在子空间方法中，通过对以下所投影的海赛函数矩阵求逆矩阵，确定缩放因子：
其中，H是海赛函数矩阵。代替使用海赛函数，方法II的优选的实施例通过线性搜索估计最优缩放因子(步骤65)。在本公开的方法II的该两个参数示例中，如下首先实行两个独立的线性搜索：
(i)首先，确定最优步长α，使得该最优步长α沿着由定义的搜索方向，使目标函数最小化，但实际上不更新模型；
(ii)然后，确定最优步长β，使得最优步长β沿着由定义的搜索方向，使目标函数最小化，实际上也不更新模型。
注意，该方法不用所估计的步长更新模型参数的事实表示该两遍线性搜索方法和先前所描述的一遍线性搜索方法之间的主要差异中的一个。一旦确定了缩放因子，则使用等式4形成新的搜索方向。然后，使用新的搜索方向s新的实行第二遍线性搜索，以更新两个参数类别，即，我们尝试找到步距λ，使得沿着方向s新的，使目标函数最小化。该方法可以被称为交替的两遍线性搜索方法：第一遍线性搜索确定不同的搜索分量之中的相对缩放，并且所估计的缩放有效地使原始搜索方向旋转，以获得新的搜索方向；然后，第二遍线性搜索使用新的搜索方向更新模型参数。由于通过独立的线性搜索确定缩放因子，所以然后能够使用所估计的缩放因子，校正原始搜索方向(或梯度)的缩放。主要通过沿着基矢量每个参数类别有多灵敏(即，沿着s1，m1的灵敏度和沿着s2，m2的灵敏度)，确定所结合的搜索方向的每个分量的相对量值。
实践考虑和扩展
通过穷尽的线性搜索确定缩放
在本文中所公开的交替的两遍线性搜索方法中，一个目标是第一遍线性搜索为每个参数类别找到适当的缩放因子(步长)。为了适当地这样做，当实行第一遍线性搜索时，可以实行穷尽的线性搜索，以找到拐点。采取两个参数类别的反演，例如，意味着通过在扰动强度的范围内沿着搜索方向使级联模型m扰动，迭代地扫描目标函数值(实质上，这意味着沿着s1扫描m1，因为的第二分量是零)。当目标函数值大于先前的目标函数值时，终止该线性搜索。返回的α被用作缩放因子，以为第一参数类别的搜索方向进行加权。接下来描述了例示穷尽的线性搜索的算法：
do while
αi←αi+Δα
计算
如果则停止且返回α＝αi
i←i+1
end
其中，Δα是用户提供的增量值，并且J是目标函数值。类似地，能够通过沿着搜索方向使m扰动，找到第二参数类别的最优缩放因子β。
扩展的方法-合并数据集的级联分解
还能够通过并入在发明人Ayeni等人的母案专利申请“一种用于通过波场分量的级联反演估计多个地下参数的方法(A Method for Estimating Multiple SubsurfaceParameters by Cascaded Inversion of Wavefield Components)”中所描述的级联反演方法来扩展本发明的方法，该母案专利申请以允许其的所有司法权以引用方式并入本文。该扩展允许使用对各个参数类别最灵敏的数据的部分，容易地调节所估计的梯度(并且因此调节搜索方向)。在图7的流程图中概括了本发明的方法的扩展的实施例。本文中所公开的该扩展的方法与Ayeni等人的文章中所描述的不同，至少因为：在本发明中，在每次迭代中同时更新所有的参数类别，而在由Ayeni等人教导的反演中，在每次迭代时仅更新一个参数类别。
多个线性搜索的成本考虑
与常规的基于梯度的一阶方法相比较，交替的一/两遍方法要求实行更多的线性搜索，这可能是高成本的。为了减轻该问题，可以仅使用震源激发的(一个或更多个)小的子集，可以优选地从整个调查随机选择震源激发的(一个或更多个)小的子集，以实行线性搜索。这些随机选择的激发能够被用于第一遍和/或第二遍线性搜索。
示例
在测试示例中，本文中所公开的交替的两遍线性搜索方法被应用于3D场数据示例的各向异性VTI(垂直横向各向同性)反演。在该示例中，同时对两个参数类别进行反演：Thompson的各向异性参数η和正常时差(Normal Moveout)速度VNMO。在反演期间，其它Thompson各向异性参数δ被设置为等于零，并在整个反演中，其它Thompson各向异性参数δ是固定的。图4A-图4F将模型与从初始模型和反演模型导出的迁移集合进行比较。图4A和图4D示出从初始(4A)模型和最终(4D)模型导出的基尔霍夫深度迁移集合。图4B和图4E分别示出初始模型和最终η模型，而图4C和图4F示出初始模型和反演VNMO模型。注意，相对于使用指示所导出的模型的准确性的开始模型迁移的图4A中所示的集合，使用从交替的两遍线性搜索方法(图4D中所示的迁移的集合)导出的VNMO和η的FWI反演模型迁移的所迁移的集合是扁平的。
上述描述针对本发明的特定的实施例，是出于例示其的目的。然而，对于本领域中的技术人员将明显的是，对本文中所描述的实施例进行很多修改和变化是可能的。所有此类修改和变化旨在在如由随附权利要求书限定的本发明的范围内。如对于在该技术领域中工作的人员来说将明显的是，使用根据本文中的本公开编程的计算机实行本发明的方法的所有实践应用。
参考文献
Kennett、B.L.N.、M.S.Sambridge和P.R.Williamson，“用于多个参数类别的大型逆矩阵问题的子空间方法(Subspace methods for large inverse problems withmultiple parameter classes)”，地理物理学杂志94，237-247(1988)。

Claims (8)

1.一种用于对地震数据进行迭代反演以同时推断地下的至少两个物理特性的模型的方法，所述方法包括：
(a)对于每个物理特性，针对所述物理特性的参数计算目标函数的梯度，所述目标函数测量所述地震数据中的全部或部分和对应模型模拟的地震数据之间的不匹配；
(b)对于每个物理特性，根据所述梯度计算模型空间中的搜索方向；
(c)针对所述至少两个物理特性，在所述搜索方向之间或之中进行交替线性搜索，以确定沿着所述搜索方向中的每个搜索方向的最优步长；
(d)使用所述最优步长更新所述模型；以及
(e)根据所更新的模型产生地下的图像，该图像显示所述至少两个物理特性；
其中(c)-(d)包括：
(i)使用初始模型针对第一物理特性实行第一线性搜索，并且保存来自所述第一线性搜索的所述最优步长，但尚不更新所述模型；
(ii)使用所述初始模型针对第二物理特性实行第二线性搜索，并且保存来自所述第二线性搜索的所述最优步长，但尚不更新所述模型；
(iii)针对正被建模的任何额外的物理特性，如同在(ii)中的使用所述初始模型实行线性搜索并保存所述最优步长；以及
(iv)使用来自(i)-(iii)的所保存的步长计算新的搜索方向，然后沿着所述新的搜索方向实行第二遍线性搜索，确定第二遍最优步长，并且使用所述第二遍最优步长为所述反演的当前迭代生成更新的模型。
2.根据权利要求1所述的方法，其中在(a)中，所述地震数据中的一个或更多个部分被选择和被用于针对每个物理特性计算所述梯度，所述一个或更多个部分是基于所述物理特性对所述数据的灵敏度，根据所选择的准则确定的。
3.根据权利要求2所述的方法，进一步包括，在所述迭代反演的下一个迭代中，针对所述至少两个物理特性中的至少一个物理特性，扩展所述地震数据的所选择的一个或更多个部分，以包含所述地震数据的更多部分。
4.根据权利要求1所述的方法，其中所述第一线性搜索和所述第二线性搜索以及在(iii)中的任何额外的线性搜索是穷尽的线性搜索，其被实行以根据所述步长在所述目标函数中找到拐点。
5.根据权利要求1所述的方法，其中所述地震数据中的一些震源激发被丢弃并且不在所述方法中使用，以便减少计算成本和时间。
6.根据权利要求1所述的方法，其中数据反演是全波场反演。
7.根据权利要求1所述的方法，其中所述至少两个物理特性包括P波速、S波速、正常时差速度、一个或更多个各向异性参数和衰减中的至少两个。
8.根据权利要求1所述的方法，其中计算所述搜索方向而不用所述目标函数的海赛函数预处理所述梯度。
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