The forward sensitivity and adjoint-state methods of glacial isostatic adjustment
The forward sensitivity and adjoint-state methods of glacial isostatic adjustment
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冰川均衡调节的前向灵敏度和伴随态方法
DOI：
10.1093/gji/ggu378
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发表时间：
2015
影响因子：
2.8
通讯作者：
Velímský
中科院分区：
文献类型：
--
作者：
Martinec;Sasgen;Velímský
在这项研究中，一个新的方法计算的冰川均衡调整（GIA）的正演解决方案的灵敏度相对于地球的地幔粘度，所谓的前向灵敏度方法（FSM），和一种方法计算的梯度数据失配相对于粘度参数，所谓的伴随状态方法（ASM），提出。这些先进的形式化方法在GIA相关观测的逆建模中相互补充。当解决这个逆问题时，第一步是通过FSM计算前向灵敏度，并使用它们来固定不影响前向模型解的模型参数，以及识别和去除推断的粘度结构的冗余部分。一旦根据前向灵敏度优化了粘度模型，就可以通过利用ASM的梯度技术来最小化相对于粘度参数的数据失配。本文的目的是推导出与Martinec开发的GIA正向求解器密切相关的形式的FSM和ASM。由于这种方法是基于一个连续形式的正演模型方程，然后离散的谱和有限元，我们首先推导出的FSM和ASM的连续形式，然后离散它们的谱和有限元离散的正演模型方程。这种方法的优点是，所有三种方法（向前，FSM和ASM）具有相同的方程矩阵，并使用相同的方法来实现应力的时间演变。正演方法与FSM和ASM的唯一区别在于，在各自的方法中，对麦克斯韦和广义麦克斯韦粘性应力的时间演化采用了不同的数值差分格式。然而，它只需要一点额外的计算时间来执行FSM和ASM数值。计算数据失配的梯度的直接方法是蛮力法，由此失配相对于模型参数的偏导数由两个正向模型运行的中心差近似。虽然蛮力法对于计算相对于少量模型参数的数据失配的梯度是有用的，但是对于具有大量参数的粘度模型，蛮力法变得昂贵。由于ASM的计算时间与粘度参数的数量无关，因此ASM提供了用于计算失配梯度的有效替代方案。因此，ASM对于计算具有大量参数的模型的失配的梯度是高效的。然而，每个时间步的正演模型解必须被存储，因此存储器需求与时间步的数量成线性比例。这是ASM的主要缺点。
In this study, a new method for computing the sensitivity of the glacial isostatic adjustment (GIA) forward solution with respect to the Earth's mantle viscosity, the so-called the forward sensitivity method (FSM), and a method for computing the gradient of data misfit with respect to viscosity parameters, the so-called adjoint-state method (ASM), are presented. These advanced formal methods complement each other in the inverse modelling of GIA-related observations. When solving this inverse problem, the first step is to calculate the forward sensitivities by the FSM and use them to fix the model parameters that do not affect the forward model solution, as well as identifying and removing redundant parts of the inferred viscosity structure. Once the viscosity model is optimized in view of the forward sensitivities, the minimization of the data misfit with respect to the viscosity parameters can be carried out by a gradient technique which makes use of the ASM. The aim is this paper is to derive the FSM and ASM in the forms that are closely associated with the forward solver of GIA developed by Martinec. Since this method is based on a continuous form of the forward model equations, which are then discretized by spectral and finite elements, we first derive the continuous forms of the FSM and ASM and then discretize them by the spectral and finite elements used in the discretization of the forward model equations. The advantage of this approach is that all three methods (forward, FSM and ASM) have the same matrix of equations and use the same methodology for the implementation of the time evolution of stresses. The only difference between the forward method and the FSM and ASM is that the different numerical differencing schemes for the time evolution of the Maxwell and generalized Maxwell viscous stresses are applied in the respective methods. However, it requires only a little extra computational time for carrying out the FSM and ASM numerically. An straightforward approach to compute the gradient of the data misfit is the brute-force method, whereby the partial derivatives of the misfit with respect to model parameters are approximated by the centred difference of two forward model runs. Although the brute-force method is useful for computing the gradient of the data misfit with respect to a small number of model parameters, it becomes expensive for a viscosity model with a large number of parameters. The ASM offers an efficient alternative for computing the gradient of the misfit since the computational time of the ASM is independent of the number of viscosity parameters. The ASM is thus highly efficient for calculating the gradient of the misfit for models with large numbers of parameters. However, the forward-model solution for each time step must be stored, hence the memory demands scale linearly with the number of time steps. This is the main drawback of the ASM.
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根据具有岩石圈根的轴对称粘度分布反演芬诺斯坎迪亚弛豫时间谱
DOI：
--
发表时间：
2005
期刊：
影响因子：
--
作者：
Z. Martinec;D. Wolf
通讯作者：
D. Wolf
DOI：
10.5194/tc-7-1499-2013
发表时间：
2013-09
期刊：
The Cryosphere
影响因子：
--
作者：
I. Sasgen;H. Konrad;E. Ivins;M. Broeke;J. Bamber;Z. Martinec;V. Klemann
通讯作者：
I. Sasgen;H. Konrad;E. Ivins;M. Broeke;J. Bamber;Z. Martinec;V. Klemann
影响因子：
2.8
作者：
M. Křížek;P. Neittaanmäki
通讯作者：
P. Neittaanmäki
影响因子：
2.8
作者：
D. Wolf
通讯作者：
D. Wolf
表面环形牵引引起的偏心嵌套球体粘弹性松弛的半解析解
影响因子：
2
作者：
Z. Martinec
通讯作者：
Z. Martinec