Interval Data Assimilation: A Data Assimilation Scheme Where All Observations Are Intervals

Abstract

With the proliferation of non‐conventional observations from low‐cost sensors, citizen science networks, and binary detection systems, data assimilation increasingly confronts observations that take the form of inequalities or intervals rather than precise point values. Yet most existing methods assume point‐valued observations and cannot fully utilize inequality‐constrained data. This paper introduces Interval Data Assimilation (IDA), a unified Bayesian framework that seamlessly incorporates both equality and inequality observations within a single algorithmic structure. The key insight is that all observations can be represented as intervals: a point observation is a special case where the interval collapses to a single value, while an inequality observation that a variable is below a threshold corresponds to a one‐sided interval. We show that the logistic likelihood function uniquely satisfies the requirements for a well‐posed formulation, including convexity of the cost function and automatic quality control for outliers. Numerical experiments with the Lorenz‐96 and quasi‐geostrophic models demonstrate that IDA matches the Ensemble Transform Kalman Filter for conventional observations while achieving substantial improvements when inequality observations are incorporated. We further show that IDA handles observation biases effectively through interval representation and can be interpreted as a soft solver for quadratic programming problems, enabling applications in model predictive control. We also demonstrate that logistic regression emerges as a special case of IDA, offering a more principled justification for the logistic function and the binary cross‐entropy loss. These results make IDA valuable for atmospheric modeling and environmental monitoring while opening a bridge between data assimilation and machine learning.