Uplift histories of landscapes from optimal transport inverse models

地貌的形成是对隆起和侵蚀的反应。一个必然的结论是，景观的几何形状包含了有关其地质演化的宝贵信息。现在存在各种反演方法来提取信息的构造历史（如隆起率）从河流景观。这些模型中的许多都集中在反演选定的纵向河流剖面的几何形状上。例如，阻尼线性最小二乘技术已被用来寻求平滑变化（在空间和时间）隆起的历史，产生理论的河流剖面，有低残差失配观测剖面。这种方法的一个好处是，它们具有相当低的计算成本，从而可以计算出整个大陆的有意义的隆升历史。然而，这种方法使简化的假设（如固定的排水平面），可能会限制计算的隆起率历史的保真度。现在存在有效的景观演化模型，可以预测河流和河间景观的几何形状。此类模型可以包含例如排水流动性、可变基底强度、降水等。我们寻求使用此类模型来反转整个景观以获得隆起率历史。一个挑战是，通常基于L-2范数的方法来评估观测（例如实际景观的海拔）和理论之间的失配可能是不合适的，因为观测和计算的景观不太可能是平滑变化的函数（即我们寻求最小化的目标函数预计会有许多局部极小值）。相反，我们将利用Wasserstein统计和最佳运输技术来评估失配。我们计划联合收割机这种方法来评估计算效率高的景观演变模型反演大陆尺度的景观演变历史的不匹配。

Landscapes form in response to uplift and erosion. A corollary is that the geometries of landscapes contain valuable information about their geological evolution. Various inverse methodologies now exist to extract information about tectonic histories (e.g. uplift rates) from fluvial landscapes. Many of these models have focussed on inverting the geometries of select longitudinal river profiles. For example, damped linear least squares techniques have been used to seek smoothly varying (in space and time) uplift histories which generate theoretical river profiles that have low residual misfits to observed profiles. A benefit to such approaches is that they have reasonably low computational cost such that meaningful uplift histories of entire continents can be calculated. However, such approaches make simplifying assumptions (e.g fixed drainage planforms) that may limit the fidelity of calculated uplift rate histories. Efficient forward models of landscape evolution now exist that can predict fluvial and interfluvial landscape geometries. Such models can incorporate, for example, drainage mobility, variable substrate strength, precipitation, etc. We seek to use such models to invert entire landscapes for uplift rate histories. A challenge is that the usual L-2 norm based approaches for assessing misfit between observations (e.g. the elevation of actual landscapes) and theory are probably not appropriate since observed and calculated landscapes are not likely to be smoothly varying functions (i.e. the objective function we seek to minimise is expected to have many local minima). Instead we will make use of Wasserstein statistics and optimal transport techniques to assess misfit. We plan to combine this approach to assessing misfit with computational efficient landscape evolution models to invert for continental-scale landscape evolution histories.