We present a novel method for interpreting time series of multispectral observations of volcanic eruptions. We show how existing models relating to radiance and area emplacement can be generalized into an integration-convolution of a Net Area Emplacement (NAE) function and a cooling function, assuming all surfaces follow the same cooling curve. The NAE describes the variation in the rate of emplacement of hot material with time and temperature, while the cooling function describes the cooling of a hot surface with time. Discretizing the integration-convolution equation yields an underdetermined matrix equation that we solve using second-order Tikhonov regularization to stabilize the solution. We test the inversion by modelling plausible NAE surfaces, calculating the radiances, adding noise and inverting for the original surface. Three or more spectral bands are required to capture the overall shape of the NAE, and recovering specific quantities is difficult. Single wavebands that yield flat kernels recover the total area emplacement curve (rate of increase of hot area – the integral of the NAE with respect to temperature) surprisingly well due to their property of conserving NAE, suggesting novel methods for calculating area emplacement rates (and effusion rates) from time series of satellite images and radiometer measurements.