We introduce a general framework for quantifying the information content and representation quality of complex geophysical datasets based on the intrinsic dimension and differentiable information imbalance of data manifolds. We use it to derive and compare optimal representations of surface ocean carbon in the SOCAT database of observations and in global ocean biogeochemistry models (GOBMs) and to assess the robustness of the information we can extract from existing data. We find that within the most widely used feature set, the complexity of the data space of SOCAT observations is not fully captured by GOBMs, but the ranking and relative importance of variables learned through GOBMs are substantially correct. We observe that the learned representation of ocean carbon is less accurate in some regions, including the Southern Ocean, but doesn't appear to have evolved significantly over the last two decades. Finally, we show how the optimal representations can be used to improve the skill of distance-based machine learning models and demonstrate it for ocean carbon, and we propose two new metrics to compare models and observations that can be used to build more accurate weighted ensembles of estimates.
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