SUMMARY

The adjoint method is a powerful technique to compute sensitivities (Fréchet derivatives) with respect to model parameters, allowing one to solve inverse problems where analytical solutions are not available or the cost to determine many times the associated forward problem is prohibitive. In Geodynamics it has been applied to the restoration problem of mantle convection—that is, to reconstruct past mantle flow states with dynamic models by finding optimal flow histories relative to the current model state—so that poorly known mantle flow parameters can be tested against observations gleaned from the geological record. By enabling us to construct time dependent earth models the adjoint method has the potential to link observations from seismology, geology, mineral physics and palaeomagnetism in a dynamically consistent way, greatly enhancing our understanding of the solid Earth system. Synthetic experiments demonstrate for the ideal case of no model error and no data error that the adjoint method restores mantle flow over timescales on the order of a transit time (≈100 Myr). But in reality unavoidable limitations enter the inverse problem in the form of poorly known model parameters and uncertain state estimations, which may result in systematic errors of the reconstructed flow history. Here we use high-resolution, 3-D spherical mantle circulation models to perform a systematic study of synthetic adjoint inversions, where we insert on purpose a mismatch between the model used to generate synthetic data and the model used for carrying out the inversion. By considering a mismatch in rheology, final state and history of surface velocities we find that mismatched model parameters do not inhibit misfit reduction: the adjoint method still produces a flow history that fits the estimated final state. However, the recovered initial state can be a poor approximation of the true initial state, where reconstructed and true flow histories diverge exponentially back in time and where for the more divergent cases the reconstructed initial state includes physically implausible structures, especially in and near the thermal boundary layers. Consequently, a complete reduction of the cost function may not be desirable when the goal is a best fit to the initial condition. When the estimated final state is a noisy low-pass version of the true final state choosing an appropriate misfit function can reduce the generation of artefacts in the initial state. While none of the model mismatches considered in this study, taken singularly, results in a complete failure of the recovered flow history, additional work is needed to assess their combined effects.

中文翻译：

---

模型不一致性对伴随法重建过去地幔流的影响

概要

伴随方法是一种针对模型参数计算灵敏度（弗雷谢特导数）的强大技术，允许人们解决无法使用解析解的反问题，或者需要花费很多时间才能确定相关正向问题的问题。在地球动力学中，它已被应用到地幔对流的恢复问题中，也就是说，通过找到相对于当前模型状态的最佳流动历史，通过动态模型重建过去的地幔流状态，从而可以针对观测结果测试未知的地幔流参数从地质记录中收集。通过使我们能够构建时间相关的地球模型，伴随方法可以潜在地以动态一致的方式将地震学，地质学，矿物物理和古磁性的观测结果联系起来，大大增强了我们对固体地球系统的理解。综合实验证明，在没有模型错误和数据错误的理想情况下，伴随方法可以在整个时间范围内以穿越时间（≈100Myr）的量级恢复地幔流。但是实际上，不可避免的局限性以模型参数未知和状态估计不确定的形式进入了反问题，这可能导致重建流量历史的系统误差。在这里，我们使用高分辨率的3-D球形地幔环流模型对合成伴随反演进行系统研究，在此过程中，我们故意在用于生成合成数据的模型与用于进行反演的模型之间插入不匹配项。考虑到流变性不匹配，最终状态和表面速度的历史记录，我们发现不匹配的模型参数不会抑制失配减小：伴随方法仍然会产生适合估计的最终状态的流动历史。但是，恢复的初始状态可能与真实初始状态的近似值很差，在此情况下，重建和真实流动历史在时间上以指数方式发生了分叉，并且在情况更为分散的情况下，重建的初始状态包括物理上不合理的结构，尤其是在热力及其附近边界层。因此，当目标最适合初始条件时，可能不希望完全降低成本函数。当估计的最终状态是真实最终状态的低通低噪版本时，选择合适的失配函数可以减少初始状态下伪像的生成。