We study the disturbance decoupled estimation problem for linear differential-algebraic systems which are not necessarily regular. We introduce the notion of partial state observers following a recent approach to observer design motivated by considerations for behavioural systems. In our framework, a partial state observer is itself a differential-algebraic system. We derive a characterisation for the existence of (asymptotic) partial state observers. Exploiting the freedom in the proposed observer design, we derive a solution of the disturbance decoupled estimation problem. The characterisation of solvability is obtained via geometric conditions in terms of the generalised Wong sequences.