Glacial rebound of the British Isles .2. Constraints on mantle viscosity

Observations of sea-level change since the time of the last glacial maximum provide important constraints on the response of the Earth to changes in surface loading on time-scales of 10(3)-10(4) years. This response is conveniently described by an effective elastic lithospheric thickness and effective viscosities for one or more mantle layers. Considerable trade-off between the parameters describing these layers can occur, and different combinations can give rise to comparable predictions of sea-level change. In particular, the trade-off between lithospheric thickness and upper-mantle viscosity can be important, and for any reasonable value for the lithospheric thickness a corresponding mantle viscosity structure can be found that gives a plausible comparison of sealevel predictions with observations. In particular, thin-lithosphere models will lead to low estimates for the upper-mantle viscosity, while thick-lithosphere models lead to high viscosity values. However, either solution may represent only a local minimum in the model parameter space, and may not correspond to the optimum solution. It becomes important, therefore, that in the inversion of observational data, a comprehensive search is conducted throughout the entire model-parameter space, to ensure that the solution identified does indeed correspond to the optimum solution. The sea-level data for the British Isles lend themselves well to such an inversion because of the relatively high quality of the data, the good geographic distribution of the data relative to the former ice sheet, and reasonable observational constraints on the dimensions of the former ice sheet and on its retreat. Furthermore, because of the contribution to the sea-level signal from the distant ice sheets, as well as from the melt-water load, the observational data base for the region also has some resolving power for the viscosity of the deeper mantle. The parameter space explored is defined by up to five mantle layers, the lithosphere of effective elastic thickness D-1, and a series of upper-mantle layers, i = 2-4, extending down to depths of 200, 400 and 670 km, respectively, each of viscosity eta(i), and a lower-mantle layer of viscosity eta(im) extending down to the core-mantle boundary. The range of parameters explored is 30 less than or equal to D-1 less than or equal to 120 km, 3 x 10(19) less than or equal to eta(i) (i = 2, 3, 4) less than or equal to 5 x 10(21) Pa s, 10(21) less than or equal to eta(lm) less than or equal to 10(23) pa s with eta(2) less than or equal to eta(3) less than or equal to eta(4) less than or equal to eta(lm) g. Simple models comprising three layers with D-1 similar to 70 km, D-2 similar to 670 km, eta(2) similar to (4-5) 10(20) Pa s, and eta(3) > 10(22) Pa s describe the sea-level response to the glacial unloading well. Earth models with low-viscosity channels immediately beneath the lithosphere are not required, but if a thin lithosphere (<50 km) is imposed in the inversion then the solution for the mantle viscosity leads to a low-viscosity (<10(20) Pa s) channel. Such a model does not, however, represent the overall least variance solution that would be obtained if D-1 were also introduced as an unknown. Likewise, if a thick lithosphere (>120 km) is imposed, then the solution points to a considerably higher value for the upper-mantle viscosity (similar to 10(21) Pa s). But this also represents only a local minimum solution. The observational data do point to some stratification in the viscosity of the upper mantle, and the optimum solution is for a five-layer model with the following effective parameters: 55 < D-1 < 60 km (2 < eta(2) < 4) x 10(20) Pa s for (D-1 < D less than or equal to 200) km (4 < eta(3) < 6) x 10(20) Pa s for (200 < D less than or equal to 400) km eta(4) similar to 2 x 10(21) Pa s for (400 < D less than or equal to 670) km eta(lm) greater than or similar to 10(22) Pa s for (670 < D < D-cmb) km