We propose a novel approach to imaging in diffusive media based on time-resolved reflectance measurements at null source-detector separation. This approach yields better spatial resolution and contrast as compared to the classical approach, which typically employs a separation of 20–40 mm. Results are obtained by an analytical perturbation approach to diffusion theory and on Monte Carlo simulations. Practical implementation with state-of-the-art technology and performance of a complementary approach based on the use of small but not null source-detector separation are also discussed.

The outgoing longwave radiation (OLR) emitted to space is a fundamental component of the Earth’s energy budget. There are numerous, entangled physical processes that contribute to OLR and that are responsible for driving, and responding to, climate change. Spectrally resolved observations can disentangle these processes, but technical limitations have precluded accurate space-based spectral measurements covering the far infrared (FIR) from 100 to 667 cm⁻¹(wavelengths between 15 and 100µm). The Earth’s FIR spectrum is thus essentially unmeasured even though at least half of the OLR arises from this spectral range. The region is strongly influenced by upper-tropospheric–lower-stratospheric water vapor, temperature lapse rate, ice cloud distribution, and microphysics, all critical parameters in the climate system that are highly variable and still poorly observed and understood. To cover this uncharted territory in Earth observations, the Far-Infrared Outgoing Radiation Understanding and Monitoring (FORUM) mission has recently been selected as ESA’s ninth Earth Explorer mission for launch in 2026. The primary goal of FORUM is to measure, with high absolute accuracy, the FIR component of the spectrally resolved OLR for the first time with high spectral resolution and radiometric accuracy. The mission will provide a benchmark dataset of global observations which will significantly enhance our understanding of key forcing and feedback processes of the Earth’s atmosphere to enable more stringent evaluation of climate models. This paper describes the motivation for the mission, highlighting the scientific advances that are expected from the new measurements.

Measurements of optical properties (scattering coefficient μ_s, absorption coefficient μ_a, reduced scattering coefficient μ_s′, and asymmetry factor g) have been carried out up to a volume particle concentration of ρ = 0.227. The results for μ_s and μ_s′ show significant deviations from the linear dependence on ρ as expected when the independent scattering assumption is fulfilled. The asymmetry factor also changed significantly. In contrast, the dependence of μ_a remained linear even at the largest concentration investigated. The simple linear dependence of absorption on the chromophore concentration expected from the independent scattering assumption is thus applicable also to spectroscopic measurements of dense media. A comparison with an approximate theoretical model based on the Foldy-Twersky equation is also reported. The model provides a good description of the dependence of μ_s on particle concentration.

The depth at which photons penetrate into a diffusive medium before being re-emitted has been investigated with reference to a semi-infinite homogeneous medium illuminated by a pencil beam. By using the diffusion equation analytical expressions have been obtained for the probability that photons penetrate at a certain depth before being detected, and for the mean path length they travel inside each layer of the medium. Expressions have been obtained both for the cw and the time domain, and simple approximate scaling relationships describing the dependence on the scattering properties of the medium have been found. For time-resolved measurements both the probability and the mean path length are expected to be independent of the distance from the light beam at which the detector is placed and of the absorption coefficient of the medium. The penetration depth increases as the time of flight increases. In contrast, for cw measurements both the probability and the mean path length strongly depend on the distance and absorption. The penetration depth increases as the distance increases or absorption decreases. The accuracy of the analytical expressions has been demonstrated by comparisons with cw experimental results. The penetration depth and the mean path length provide useful information, for instance, for measurements of tissue oxygenation and for functional imaging of muscle and brain. In particular, the depth reached by received photons provides overall information on the volume of the tissue actually investigated, while the mean path is strictly related to the sensitivity to local variations of absorption.

An exact solution of the time-dependent diffusion equation for the case of a two- and a three-layered finite diffusive medium is proposed. The method is based on the decomposition of the fluence rate in a series of eigenfunctions and upon the solution of the consequent transcendental equation for the eigenvalues obtained from the boundary conditions. Comparisons among the solution of the diffusion equation and the results of Monte Carlo simulations show the correctness of the proposed model.