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| Titel |
The forward and adjoint sensitivity methods of glacial isostatic adjustment: Existence, uniqueness and time-differencing scheme |
| VerfasserIn |
Zdeněk Martinec, Ingo Sasgen, Jakub Velimsky |
| Konferenz |
EGU General Assembly 2014
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 16 (2014) |
| Datensatznummer |
250088332
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| Publikation (Nr.) |
 EGU/EGU2014-2429.pdf |
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| Zusammenfassung |
| In this study, two new methods for computing the sensitivity of the glacial isostatic
adjustment (GIA) forward solution with respect to the Earth’s mantle viscosity are
presented: the forward sensitivity method (FSM) and the adjoint sensitivity method
(ASM). These advanced formal methods are based on the time-domain,spectral-finite
element method for modelling the GIA response of laterally heterogeneous earth
models developed by Martinec (2000). There are many similarities between the
forward method and the FSM and ASM for a general physical system. However,
in the case of GIA, there are also important differences between the forward and
sensitivity methods. The analysis carried out in this study results in the following
findings.
First, the forward method of GIA is unconditionally solvable, regardless of whether or not
a combined ice and ocean-water load contains the first-degree spherical harmonics. This is
also the case for the FSM, however, the ASM must in addition be supplemented by nine
conditions on the misfit between the given GIA-related data and the forward model
predictions to guarantee the existence of a solution. This constrains the definition of data
least-squares misfit.
Second, the forward method of GIA implements an ocean load as a free boundary-value
function over an ocean area with a free geometry. That is, an ocean load and the shape of
ocean, the so-called ocean function, are being sought, in addition to deformation and
gravity-increment fields, by solving the forward method. The FSM and ASM also apply the
adjoint ocean load as a free boundary-value function, but instead over an ocean area with the
fixed geometry given by the ocean function determined by the forward method. In other
words, a boundary-value problem for the forward method of GIA is free with respect to
determining (i) the boundary-value data over an ocean area and (ii) the ocean function itself,
while the boundary-value problems for the FSM and ASM are free only with respect to
(i).
Third, the forward method of GIA traditionally uses an explicit time-differencing scheme
to calculate the evolution of the viscous stress tensor Ï over time. We show that this scheme
is not sufficiently accurate for calculating the time evolution of the viscous gradient of the
viscous stress tensor, --ăÎ·Ï , underlying the FSM and ASM. This is particularly evident in the
case where the size of time step is comparable to the shortest Maxwell relaxation time of the
studied earth model, although the explicit time-differencing scheme of the forward method is
still sufficiently accurate. The inaccuracy in --ăÎ·Ï which is unacceptable for computing
the gradient of least-squares misfit, --ăη Ï2, can only be reduced by making the
time steps shorter. This subsequently increases the demands on computation time
and storage space, since the forward solution must be stored for solving the FSM
and ASM. We propose and numerically test a new, semi-explicit time-differencing
scheme for --ăÎ·Ï with a numerical accuracy comparable to the explicit scheme for
Ï .
Fourth, the FSM determines, among other forward sensitivities, the viscosity gradient of
the ocean function which describes the sensitivity of coastal regions to a particular Earth’s
viscosity structure and shows how this sensitivity varies with time. This relation is
particularly important for interpreting sea-level indicator data. The ASM does not, however,
provide, this kind of sensitivity. On the other hand, the ASM is more efficient for the
sensitivity analysis of models with a large number of viscosity parameters. In this sense, the
FSM and ASM complement each other. |
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