Conditioning of the weak-constraint variational data assimilation problem for numerical weather prediction

4-Dimensional Variational Data Assimilation (4DVAR) assimilates observations through the minimisation of a least-squares objective function, which is constrained by the model flow. We refer to 4DVAR as strong-constraint 4DVAR (sc4DVAR) in this thesis as it assumes the model is perfect. Relaxing this assumption gives rise to weak-constraint 4DVAR (wc4DVAR), leading to a different minimisation problem with more degrees of freedom. We consider two wc4DVAR formulations in this thesis, the model error formulation and state estimation formulation. The 4DVAR objective function is traditionally solved using gradient-based iterative methods. The principle method used in Numerical Weather Prediction today is the Gauss-Newton approach. This method introduces a linearised `inner-loop' objective function, which upon convergence, updates the solution of the non-linear `outer-loop' objective function. This requires many evaluations of the objective function and its gradient, which emphasises the importance of the Hessian. The eigenvalues and eigenvectors of the Hessian provide insight into the degree of convexity of the objective function, while also indicating the difficulty one may encounter while iterative solving 4DVAR. The condition number of the Hessian is an appropriate measure for the sensitivity of the problem to input data. The condition number can also indicate the rate of convergence and solution accuracy of the minimisation algorithm. This thesis investigates the sensitivity of the solution process minimising both wc4DVAR objective functions to the internal assimilation parameters composing the problem. We gain insight into these sensitivities by bounding the condition number of the Hessians of both objective functions. We also precondition the model error objective function and show improved convergence. We show that both formulations' sensitivities are related to error variance balance, assimilation window length and correlation length-scales using the bounds. We further demonstrate this through numerical experiments on the condition number and data assimilation experiments using linear and non-linear chaotic toy models.

Biogeochemical cycles and biodiversity as key drivers of ecosystem services provided by soils

Soils play a pivotal role in major global biogeochemical cycles (carbon, nutrient, and water), while hosting the largest diversity of organisms on land. Because of this, soils deliver fundamental ecosystem services, and management to change a soil process in support of one ecosystem service can either provide co-benefits to other services or result in trade-offs. In this critical review, we report the state-of-the-art understanding concerning the biogeochemical cycles and biodiversity in soil, and relate these to the provisioning, regulating, supporting, and cultural ecosystem services which they underpin. We then outline key knowledge gaps and research challenges, before providing recommendations for management activities to support the continued delivery of ecosystem services from soils. We conclude that, although soils are complex, there are still knowledge gaps, and fundamental research is still needed to better understand the relationships between different facets of soils and the array of ecosystem services they underpin, enough is known to implement best practices now. There is a tendency among soil scientists to dwell on the complexity and knowledge gaps rather than to focus on what we do know and how this knowledge can be put to use to improve the delivery of ecosystem services. A significant challenge is to find effective ways to share knowledge with soil managers and policy makers so that best management can be implemented. A key element of this knowledge exchange must be to raise awareness of the ecosystems services underpinned by soils and thus the natural capital they provide. We know enough to start moving in the right direction while we conduct research to fill in our knowledge gaps. The lasting legacy of the International Year of Soils in 2015 should be for soil scientists to work together with policy makers and land managers to put soils at the centre of environmental policy making and land management decisions.

Biodiversity and resilience of ecosystem functions

Accelerating rates of environmental change and the continued loss of global biodiversity threaten functions and services delivered by ecosystems. Much ecosystem monitoring and management is focused on the provision of ecosystem functions and services under current environmental conditions, yet this could lead to inappropriate management guidance and undervaluation of the importance of biodiversity. The maintenance of ecosystem functions and services under substantial predicted future environmental change (i.e., their ‘resilience’) is crucial. Here we identify a range of mechanisms underpinning the resilience of ecosystem functions across three ecological scales. Although potentially less important in the short term, biodiversity, encompassing variation from within species to across landscapes, may be crucial for the longer-term resilience of ecosystem functions and the services that they underpin.

Functional identity and diversity of animals predict ecosystem functioning better than species-based indices

Drastic biodiversity declines have raised concerns about the deterioration of ecosystem functions and have motivated much recent research on the relationship between species diversity and ecosystem functioning. A functional trait framework has been proposed to improve the mechanistic understanding of this relationship, but this has rarely been tested for organisms other than plants. We analysed eight datasets, including five animal groups, to examine how well a trait-based approach, compared with a more traditional taxonomic approach, predicts seven ecosystem functions below- and above-ground. Trait-based indices consistently provided greater explanatory power than species richness or abundance. The frequency distributions of single or multiple traits in the community were the best predictors of ecosystem functioning. This implies that the ecosystem functions we investigated were underpinned by the combination of trait identities (i.e. single-trait indices) and trait complementarity (i.e. multi-trait indices) in the communities. Our study provides new insights into the general mechanisms that link biodiversity to ecosystem functioning in natural animal communities and suggests that the observed responses were due to the identity and dominance patterns of the trait composition rather than the number or abundance of species per se.