We present an adaptation of the Backus–Gilbert method that enables (i) the incorporation of arbitrary prior knowledge and (ii) the solution of multiparameter inverse problems, providing a tunable balance between spatial resolution, inference errors and interparameter trade-offs. This yields a powerful approach for solving a class of inverse problems where the forward relation is linear or weakly nonlinear. The method rests on a probabilistic reformulation of Backus–Gilbert inversion and the solution of an optimization problem that maximizes deltaness while minimizing interparameter trade-offs. Applying the theory to multimode surface wave dispersion data collected by distributed acoustic sensing on the Northeast Greenland Ice Stream, we show that density in the firn layer may be constrained directly and without the need for scaling relations to depths of around ten metres, provided that dispersion data up to at least the third overtone of Rayleigh waves are available in the 10–50 Hz frequency band. The limiting factor that prevents the resolution of density at greater depth is data quality. Hence, progress on the direct inference of density could be made by repeated experiments or higher signal-to-noise ratios that would require better coupling and shielding of fibre-optic cables from wind and temperature fluctuations.