A method for infering bed topography beneath glaciers from surface measurements (elevation from altimetry and velocity from InSAR) and sparse thickness measurements is developed and evaluated. The method is based on an original non-isothermal Reduced Uncertainty (RU) version of the Shallow Ice Approximation (SIA) equation that natively incorporates the surface measurements. The flow model has a single dimensionless multi-physics parameter γ. This parameter takes into account the basal slipperiness and the variable vertical rate factor profiles, thus the vertical thermal variations. The inversions are based on three steps involving: an Artificial Neural Network (ANN) and two Variational Data Assimilation (VDA) processes. The ANN-based stage aims at estimating the multi-physics number γ from the thickness measurements; the resulting estimator is remarkably robust. The full inversion method is valid for half-sheared flows (presenting a moderate basal slipperiness): it can be applied to inland ice-sheets areas. Also these estimates connect continuously with estimates from mass conservation only, i.e. with areas of sliding flows. Numerical results are presented for areas of the East Antarctica Ice Sheet where bed elevation can be very uncertain (Bedmap2 values). Estimates are valid for wavelengths longer than \(\sim 10 \bar h\) (due to the long wave assumption, shallow flow model) with resolution at \(\sim \bar h\) (\(\bar h\) a characteristic thickness value).

开发并评估了一种从表面测量（高度测量的高程和 InSAR 速度）和稀疏厚度测量推断冰川下床地形的方法。该方法基于浅冰近似 (SIA) 方程的原始非等温降低不确定性 (RU) 版本，该方程本身结合了表面测量。流动模型具有单个无量纲多物理参数γ。该参数考虑了基础滑度和可变垂直速率因子分布，因此考虑了垂直热变化。反演基于三个步骤：人工神经网络 (ANN) 和两个变分数据同化 (VDA) 过程。基于 ANN 的阶段旨在估计多物理场数γ从厚度测量；由此产生的估计量非常稳健。全反演方法适用于半剪切流（呈现中等基础滑度）：它可以应用于内陆冰盖地区。此外，这些估计仅与质量守恒的估计，即滑流区域的估计连续连接。给出了南极洲东部冰盖区域的数值结果，这些区域的床高度可能非常不确定（Bedmap2 值）。估计值对于长于\(\sim 10 \bar h\) 的波长有效（由于长波假设，浅流模型），分辨率为\(\sim \bar h\) ( \(\bar h\) a特征厚度值）。