This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task,and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four taskspecific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks,i.e.,

ABSTRACT. The study of cognition can provide key insights into the social dimension of coupled social-ecological systems. Values are a fundamental aspect of cognition, which have largely been neglected within the social-ecological systems literature. Values represent the deeply held, emotional aspects of people's cognition and can complement the use of other cognitive constructs, such as knowledge and mental models, which have so far been better represented in this area of study. We provide a review of the different conceptualizations of values that are relevant to the study of human-environment interactions: held, assigned, and relational values. We discuss the important contribution values research can make toward understanding how social-ecological systems function and to improving the management of these systems in a practical sense. In recognizing that values are often poorly defined within the social-ecological systems literature, as in other fields, we aim to guide researchers and practitioners in ensuring clarity when using the term in their research. This can support constructive dialogue and collaboration among researchers who engage in values research to build knowledge of the role and function of values, and hence cognition more broadly, within a social-ecological systems context.

We consider a finite-element-in-space, and quadrature-in-timediscretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinous Galerkin finite element method based on polynomials of degree r-termed DG(r)-and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi-and fully-discrete a priori error estimates are derived without recourse to Gronwall's inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution (e.g. when using the standard P k polynomial basis on simplicies, or tensor product polynomials, Q k , on quadrilaterals). When this is not the case (e.g. using P k on quadrilaterals) the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation.

We consider the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses. A fully discrete approximation is defined, based on a spatially symmetric or non-symmetric interior penalty discontinuous Galerkin finite element method, and a displacement-velocity centred difference time discretisation. An a priori error estimate is given but only the main ideas in the proof of the error estimate are reported here due to the large number of (mostly technical) estimates that are required. The full details are referenced to a technical report.

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