southward flow of North Atlantic Deep Water
(NADW) in the North Atlantic and the northward
spreading of Antarctic Bottom Water (AABW) in the
western South Atlantic as well as in the South Pacific
and the Indian Oceans. The quantitative agreement
between simulated and measured D
14 C values is excellent, and the model-data difference is only 1.3%
on average. The root-mean-square (rms) difference
amounts to only 5.2% and is of the same order of
magnitude as the uncertainties of the radiocarbon
data. In the deep North Atlantic and parts of the
Southern Ocean, the measurements are systematically higher than the model simulations, owing to the
contribution of bomb14
C in these waters. Bombradiocarbon is not included in the model, and contaminated data values are not assimilated in the
model.
The model fluxes of particulate organic carbon
(POC) at the base of the euphotic zone (carbon export production) necessary to reproduce the observed oxygen and nutrient fields are shown in
Figure 8. The spatial patterns of carbon export resemble the patterns of primary production in published maps showing high fluxes in coastal upwelling
areas off West Africa, along the West American
coast, in the Arabian Sea and Bay of Bengal, in the
northwest Atlantic and north and tropical Pacific, in
the area of the Indonesian archipelago, and in the
Southern Ocean. Highest values in the productive
areas are on the order of 5–10 mol C m
À 2 yr
À 1
; in
the oligotrophic, open-ocean regions they amount
to between 0.5 and 2 mol C m
À 2 yr
À 1 . The export
of POC in the model predominantly occurs at
mid-latitudes and in the Southern Ocean, and the
contribution of the Northern Hemisphere is comparatively small. Globally integrated, the POC export in the model amounts to about 10 Gt C yr
À 1 .
Error analysis of the solution vector p
à is possible
but computationally very expensive. An eigenvector/
eigenvalue analysis of the inverse Hessian matrix H
À 1
(the Hessian is the square matrix of second partial
derivatives of F with respect to parameters p
à ) reveals
directions in parameter space that are well determined
(eigenvectors associated with large eigenvalues; values
of F increase rapidly when moving away from optimal
point along the direction of the eigenvector) and directions that are only poorly determined by the model
(eigenvectors associated with smallest eigenvalues;
values of F increase slowly when moving away from
optimal point along the direction of the eigenvector).
The ratio of largest and smallest eigenvalues of
H
À 1 is a measure of the anisotropy of F around the
optimal solution p
à . For relatively small problems
with a few hundred parameters in p
à , the Hessian
matrix, its inverse, and the associated eigenvalues and
eigenvectors can actually be calculated and a rigorous
error analysis of the solution is possible. For large
problems with hundred thousands of parameters, this
will remain impossible for the foreseeable future.
Strategies for obtaining sensitivity information at least
for some directions in parameter space or for finding
the largest eigenvalues and associated eigenvectors are
described in the literature.
Conclusion
Recent progress in applications of inverse models to
problems in physical and biogeochemical oceanography clearly shows that inverse methodology is
well advanced and being used by a growing number
of researchers. Mathematical techniques exist that
exploit the available data better than before and
produce new and important scientific results. These
methods successfully cope with problems, such as
sparseness of data and incompleteness of information. The advances were possible because of the
tremendous technological progress in computer
hardware, combined with breakthroughs in the development of efficient and innovative algorithms.
These algorithms finally allowed the numerical application of mathematical principles, such as the
Lagrange multiplier method, whose theory was established for centuries already. Still, the widespread
use of inverse methods would not have been possible,
if there had not been at the same time an increased
awareness among scientists for the need of integrated, global databases and an increased willingness
to contribute to these data sets.
Much more data will become available in the future, enabling inverse-type studies that are still impossible today. While ship observations will continue
to be important, the new autonomous floats, gliders,
or profiling instruments on moorings will provide
data in near real time, even from remote and inaccessible regions. New satellite sensors will complement the water column data with high-resolution
data from the ocean surface, and new geochemical
programs, such as GEOTRACES, will produce highquality data of the ocean’s trace elements and isotopes, including micronutrients such as Fe and Zn, in
unprecedented quality and coverage.
Further Reading
Ganachaud A and Wunsch C (2002) Oceanic nutrient and
oxygen transport and bounds on export production
during the World Ocean Circulation Experiment.
Global Biogeochemical Cycles 16 (doi:10.1029/2000GB
001333).
Gill PE, Murray W, and Wright MH (1981) Practical
Optimization. London: Academic Press.
198 INVERSE MODELING OF TRACERS AND NUTRIENTS
(NADW) in the North Atlantic and the northward
spreading of Antarctic Bottom Water (AABW) in the
western South Atlantic as well as in the South Pacific
and the Indian Oceans. The quantitative agreement
between simulated and measured D
14 C values is excellent, and the model-data difference is only 1.3%
on average. The root-mean-square (rms) difference
amounts to only 5.2% and is of the same order of
magnitude as the uncertainties of the radiocarbon
data. In the deep North Atlantic and parts of the
Southern Ocean, the measurements are systematically higher than the model simulations, owing to the
contribution of bomb14
C in these waters. Bombradiocarbon is not included in the model, and contaminated data values are not assimilated in the
model.
The model fluxes of particulate organic carbon
(POC) at the base of the euphotic zone (carbon export production) necessary to reproduce the observed oxygen and nutrient fields are shown in
Figure 8. The spatial patterns of carbon export resemble the patterns of primary production in published maps showing high fluxes in coastal upwelling
areas off West Africa, along the West American
coast, in the Arabian Sea and Bay of Bengal, in the
northwest Atlantic and north and tropical Pacific, in
the area of the Indonesian archipelago, and in the
Southern Ocean. Highest values in the productive
areas are on the order of 5–10 mol C m
À 2 yr
À 1
; in
the oligotrophic, open-ocean regions they amount
to between 0.5 and 2 mol C m
À 2 yr
À 1 . The export
of POC in the model predominantly occurs at
mid-latitudes and in the Southern Ocean, and the
contribution of the Northern Hemisphere is comparatively small. Globally integrated, the POC export in the model amounts to about 10 Gt C yr
À 1 .
Error analysis of the solution vector p
à is possible
but computationally very expensive. An eigenvector/
eigenvalue analysis of the inverse Hessian matrix H
À 1
(the Hessian is the square matrix of second partial
derivatives of F with respect to parameters p
à ) reveals
directions in parameter space that are well determined
(eigenvectors associated with large eigenvalues; values
of F increase rapidly when moving away from optimal
point along the direction of the eigenvector) and directions that are only poorly determined by the model
(eigenvectors associated with smallest eigenvalues;
values of F increase slowly when moving away from
optimal point along the direction of the eigenvector).
The ratio of largest and smallest eigenvalues of
H
À 1 is a measure of the anisotropy of F around the
optimal solution p
à . For relatively small problems
with a few hundred parameters in p
à , the Hessian
matrix, its inverse, and the associated eigenvalues and
eigenvectors can actually be calculated and a rigorous
error analysis of the solution is possible. For large
problems with hundred thousands of parameters, this
will remain impossible for the foreseeable future.
Strategies for obtaining sensitivity information at least
for some directions in parameter space or for finding
the largest eigenvalues and associated eigenvectors are
described in the literature.
Conclusion
Recent progress in applications of inverse models to
problems in physical and biogeochemical oceanography clearly shows that inverse methodology is
well advanced and being used by a growing number
of researchers. Mathematical techniques exist that
exploit the available data better than before and
produce new and important scientific results. These
methods successfully cope with problems, such as
sparseness of data and incompleteness of information. The advances were possible because of the
tremendous technological progress in computer
hardware, combined with breakthroughs in the development of efficient and innovative algorithms.
These algorithms finally allowed the numerical application of mathematical principles, such as the
Lagrange multiplier method, whose theory was established for centuries already. Still, the widespread
use of inverse methods would not have been possible,
if there had not been at the same time an increased
awareness among scientists for the need of integrated, global databases and an increased willingness
to contribute to these data sets.
Much more data will become available in the future, enabling inverse-type studies that are still impossible today. While ship observations will continue
to be important, the new autonomous floats, gliders,
or profiling instruments on moorings will provide
data in near real time, even from remote and inaccessible regions. New satellite sensors will complement the water column data with high-resolution
data from the ocean surface, and new geochemical
programs, such as GEOTRACES, will produce highquality data of the ocean’s trace elements and isotopes, including micronutrients such as Fe and Zn, in
unprecedented quality and coverage.
Further Reading
Ganachaud A and Wunsch C (2002) Oceanic nutrient and
oxygen transport and bounds on export production
during the World Ocean Circulation Experiment.
Global Biogeochemical Cycles 16 (doi:10.1029/2000GB
001333).
Gill PE, Murray W, and Wright MH (1981) Practical
Optimization. London: Academic Press.
198 INVERSE MODELING OF TRACERS AND NUTRIENTS