Lloyd, Andrew J.Crawford, OpheliaAl-Attar, DavidAustermann, JacquelineHoggard, Mark J.Richards, Fred D.Syvret, Frank2025-05-302025-05-300956-540XWOS:001133623000002http://www.scopus.com/inward/record.url?scp=85183080152&partnerID=8YFLogxKhttps://hdl.handle.net/1885/733754874A key initial step in geophysical imaging is to devise an effective means of mapping the sensitivity of an observation to the model parameters, that is to compute its Frechet derivatives or sensitivity kernel. In the absence of any simplifying assumptions and when faced with a large number of free parameters, the adjoint method can be an effective and efficient approach to calculating Frechet derivatives and requires just two numerical simulations. In the Glacial Isostatic Adjustment problem, these consist of a forward simulation driven by changes in ice mass and an adjoint simulation driven by fictitious loads that are applied at the observation sites. The theoretical basis for this approach has seen considerable development over the last decade. Here, we present the final elements needed to image 3-D mantle viscosity using a dataset of palaeo sea-level observations. Developments include the calculation of viscosity Frechet derivatives (i.e. sensitivity kernels) for relative sea-level observations, a modification to the numerical implementation of the forward and adjoint problem that permits application to 3-D viscosity structure, and a recalibration of initial sea level that ensures the forward simulation honours present-day topography. In the process of addressing these items, we build intuition concerning how absolute sea-level and relative sea-level observations sense Earth's viscosity structure and the physical processes involved. We discuss examples for potential observations located in the near field (Andenes, Norway), far field (Seychelles), and edge of the forebulge of the Laurentide ice sheet (Barbados). Examination of these kernels: (1) reveals why 1-D estimates of mantle viscosity from far-field relative sea-level observations can be biased; (2) hints at why an appropriate differential relative sea-level observation can provide a better constraint on local mantle viscosity and (3) demonstrates that sea-level observations have non-negligible 3-D sensitivity to deep mantle viscosity structure, which is counter to the intuition gained from 1-D radial viscosity Frechet derivatives. Finally, we explore the influence of lateral variations in viscosity on relative sea-level observations in the Amundsen Sea Embayment and at Barbados. These predictions are based on a new global 3-D viscosity inference derived from the shear-wave speeds of GLAD-M25 and an inverse calibration scheme that ensures compatibility with certain fundamental geophysical observations. Use of the 3-D viscosity inference leads to: (1) generally greater complexity within the kernel; (2) an increase in sensitivity and presence of shorter length-scale features within lower viscosity regions; (3) a zeroing out of the sensitivity kernel within high-viscosity regions where elastic deformation dominates and (4) shifting of sensitivity at a given depth towards distal regions of weaker viscosity. The tools and intuition built here provide the necessary framework to explore inversions for 3-D mantle viscosity based on palaeo sea-level data.This work has been supported by the National Science Foundation under grants: NSF-EAR-2002352 and OPP-2142592. Support was also provided through the Natural Environment Research Council grant number NE/V010433/1. We acknowledge computing resources from Columbia University's Shared Research Computing Facility project, which is supported by NIH Research Facility Improvement Grant 1G20RR030893-01, and associated funds from the New York State Empire State Development, Division of Science Technology and Innovation (NYSTAR) Contract C090171, both awarded 15 April 2010. MH acknowledges support from Geoscience Australia and the Australian Research Council's Discovery Early Career Researcher Award DE220101519. FDR acknowledges support from the Imperial College Research Fellowship scheme. Figures were created using the Generic Mapping Tools (GMT; Wessel et al.2019). Finally, we thank Jerry Mitrovica for input on the paper draft and two anonymous reviewers for their critiques of this paper.33enPublisher Copyright: © 2023 The Author(s). Published by Oxford University Press on behalf of The Royal Astronomical Society.Inverse theoryRheology: mantleSea level change2023-12-0510.1093/gji/ggad45585183080152