Retrieving climate proxies from an invertible glacier evolution model.

Glacier evolution models (GEMs) are important tools in glaciology to predict future glacier response from climate forcing. However, reconstructing past climates requires inversion tools to infer the climate forcing that explains paleoglacier extents documented through historical records or geomorphic features. 
Such "inverted" GEMs are less common compared to forward GEMs but are important to better constrain climate from the past. 
For instance, Visnjevic et. al proposed a model to estimate the Equilibrium Line Altitude (ELA) from reconstructed paleoglacier extents. 
However, their approach assumes stationary glaciers, neglecting temporal dynamics, and employs a heuristic inversion technique.

Recent implementations of automatic differentiation (AD), coupled with Graphic Processing Unit (GPU) performance improvements, provide a promising pathway to develop fully differentiable and computationally efficient GEMs. Here, we introduce an Invertible Glacier Evolution Model (IGEM), a new framework designed to overcome the limitations of existing inversion methods.
Our IGEM relies on the Shallow Ice Approximation (SIA)  for the ice flow, and surface mass balance is computed with the Positive Degree Day (PDD) .
The model’s tensor-based architecture leverages GPU acceleration and enables efficient computation of gradients with respect to input climate variables, such as temperature and precipitation, which are used for PDD calculations. The gradient-descent inversion scheme employed in our IGEM converges more rapidly, delivers more accurate solutions, and offers greater generality (e.g., it is not constrained by the stationary assumption) compared to heuristic inversion methods.

The main challenge here is due to the fact that one forward GEM simulation requires thousands of iterations to model a glaciation spell.
To compute the gradient of the cost function with respect to climate forcing, a chain derivation of all operations within the forward GEM is necessary, which is memory-challenging, especially on GPUs.
To address this, our IGEM selectively recomputes a subset of intermediate operations during gradient computation. Instead of storing all operations, only those essential for computing gradients are cached, while others are recomputed during the "backward" pass. This approach reduces memory usage at the cost of increased computation time, enabling the methodology to handle large-scale problems effectively.

To illustrate the feasibility of our approach, we apply it to the inference of climate proxies at the Aletsch Glacier for the period 1880–2010. We leverage sequentially dated observations of the glacier geometry during the same timeframe. Given the nonuniqueness of the problem, the method permits the derivation of a set of compatible temperature and precipitation proxies, which are evaluated against weather station data near the glacier.

This proof-of-concept shows that our IGEM approach enables the extraction of compatible climate proxies, such as temperature and precipitation, provided documented glacier former extents. By bridging the gap between glacier dynamics and climate reconstruction, our IGEM has the potential to advance our understanding of past climates in formerly glaciated regions.