Accurately quantifying land-atmosphere exchanges is essential at every spatial scale, from aiding a better understanding of climate change globally to informing land management decisions at the smallest scale (e.g. agricultural land management). This quantification may be dealt with relatively easily for homogeneous land surfaces, but in the real world, landscapes are spatially heterogeneous and simple approaches are often inadequate. This thesis uses mathematically advanced methods and/or models to find robust solutions to landatmosphere exchange problems that accommodate spatial heterogeneity. A two-stage sampling strategy (2SS) was developed to reduce the uncertainties in the estimation of chamber-based GHG fluxes when sample size is inadequate to fully capture spatial heterogeneity. A Monte Carlo simulation showed that 2SS improves the estimation of soil GHG fluxes in all but the most homogeneous situations, with the improvement being directly related to the amount of spatial heterogeneity present. EC-based measurements of GHG fluxes invariably contain data gaps that require filling to generate long-term cumulative fluxes, i.e. integrating over a temporally heterogeneous timeseries. Gap-filling methods introduce uncertainty. A robust method based on image inpainting is introduced to fill gaps via a two-dimensional representation of a onedimensional data, i.e. the flux fingerprint. Results show that this unsupervised method, using a more compact and simple form, compares favourably with a widely-used traditional method and can outperform it when applied to de-noised data. The most robust measurements of surface carbon fluxes will be generated when using two independent measurement methods simultaneously. To investigate CO2 and CH4 fluxes from a heterogeneous fen, EC- and chamber-based measurements of surface carbon fluxes were implemented from 2013 to 2015. To implement a direct comparison between these measurements made at differing scales, the chamber-measured data were up-scaled, both temporally by model-based interpolations and spatially by flux footprint modelling. Results show a good linear correlation in CO2 flux and a near zero correlation in CH4 flux between methods. Further analysis on CH4 flux, however, show that the two differed only by a Gaussian distribution, implying the existence of white noise in the signal. The cumulative CO2 flux for the whole season measured by chambers was -376.5 g/m2, 33% higher than the estimated measured by EC (-281.8 g/m2). Similarly, the final cumulative CH4 flux was 4.01 g/m2 by chamber-based estimates, 43% more than EC (2.81 g/m2). The final part of this study investigates the surface flux of momentum in a structured heterogeneous land surface. A logarithmic normal distribution was developed to model the wind speed reduction around a tree-based windbreak. The model showed an excellent fit to field observations made at a real-world windbreak on farm land. A graphical method that describes a 3-d space of wind-chill temperature vs. ambient temperature and wind speed was created to quantify the potential thermal benefits gained by introducing windbreaks and reducing wind speed. The wind-chill thermal tolerance (WTT) of sheep was estimated and compared for a lowland and an upland site. Distinct differences to reduced wind speed were found between the sites, with greater thermal benefits at the upland site. The methods and models generated and developed in this study contribute to an improved quantification of land-atmosphere exchanges, and have potential to be applied to surface fluxes generally, either of mass (GHGs) or energy (heat, momentum), and to landscapes other than those dominated by vegetation. For example, the statistical idea of the two-stage sampling approach provides a generic solution to sample size deficiency in heterogeneous land surfaces; The inpainting-based gap-filling method, as an image processing technique, may be applicable to any signals that can be represented as an image, i.e. a two-dimensional space in which individual locations (pixels) have numerical attributes that can be used as RGB values; The WTT plot/analysis, used here in the context of sheep in upland sites, provides an intuitive and powerful scheme for analysing the thermal tolerance of any animal in any energetically heterogeneous landscape.