production, for instance, can be inferred from the
observed strength of the vertical nutrient gradient,
and the depth range of particle remineralization can
be determined from the observed vertical position and
spatial extent of the nutrient maxima and oxygen
minima. Differences in the vertical structure of different nutrients (e.g., phosphate, silicate, carbon, and
alkalinity) reveal differences in the remineralization
depths of organic material, CaCO 3 , and opal.
Basin-wide observations of a variety of oceanic
tracers have been conducted since the 1950s. The
first coordinated and global tracer program, GEOSECS, produced tracer data of unprecedented quality
and coverage during the 1970s. More recently, the
World Ocean Circulation Experiment (WOCE) has
provided an even more detailed tracer data set describing the distributions during the 1990s. These
data are publicly available in electronic form over the
Internet or as colored distribution plots in printed
atlases. Availability of original tracer data is essential
for the inverse methods described below.
Inverse Model Concepts
Deriving quantitative results about the underlying
biogeochemical processes from oceanic nutrient
and tracer data, and separating the effects of biogeochemistry from the effects of circulation is a
challenging task, and requires the use of coupled
physical/biogeochemical numerical models. There
are a variety of possible approaches, which can
broadly be divided into two categories. The so-called
‘forward models’ assume rates of physical and biogeochemical processes to be known a priori, and
require ocean currents (or the physical forcing at the
ocean surface), as well as biological production and
particle remineralization rates to be specified as
input. Oceanic tracer concentrations are treated as
unknowns, and the tracer fields that would evolve
under the assumed flows and biogeochemical parameters are then simulated. Forward models, in
general, lead to mathematical systems that are relatively easy to solve. The conceptual disadvantage of
forward models is that physical and biogeochemical
rate information that is supposed to be determined
from the tracer data and, in fact, only becomes
available after the data evaluation, is required
a priori to enable the model run.
‘Inverse models’, in contrast, follow a seemingly
more intuitive approach, treating the measured tracer concentrations and other auxiliary knowledge
formally as knowns, whereas the physical and biogeochemical parameters, to be determined on the
basis of the tracer data, are treated as unknowns.
As described in more detail below, the inverse
approach generally leads to underdetermined mathematical systems that are much harder to solve than
the systems encountered in forward models. Error
and resolution analysis are two issues of particular
importance when solving underdetermined inverse
problems. First, the solved-for physical and biogeochemical parameters depend directly on the tracer
data, and errors in the data propagate into errors in
the solution. Second, owing to the incompleteness of
information in underdetermined systems, the unknowns are usually not fully resolved. Instead, only
specific linear combinations of unknowns may be
well constrained by the data, while individual unknowns or other combinations of unknowns may
remain poorly determined. Both, error and resolution analysis are essential for a quality assessment
of the solution of underdetermined systems.
Two widely used practical inverse approaches are
presented in the next two sections (‘Estimating absolute velocities and nutrient fluxes across sections’
and ‘Estimating carbon export fluxes with the
adjoint method’). These serve as examples to describe details of the mathematical methods and to
list achievable results. The first method, the section
inverse approach, infers nutrient, carbon, and tracer
fluxes across sets of sections, based on hydrographic, tracer, and nutrient data along these sections. The second example describes an application
of the adjoint method for the determination of
ocean currents, biological productivity, and downward particle fluxes. This method is specifically
adapted for the utilization of many different tracers
and can handle problems with heterogeneous and
sparse data coverage.
Estimating Absolute Velocities and
Nutrient Fluxes across Sections
The section inverse approach developed by Wunsch
exploits hydrographic data along sets of oceanographic sections, and allows estimation of absolute
flow velocities perpendicular to the sections. The
method was later extended to include nutrient and
oxygen data allowing to estimate nutrient and oxygen fluxes across the sections.
The section inverse method is based on the geostrophic principle, which for a given pair of hydrographic stations allows calculating the geostrophic
velocity v g (z) perpendicular to the connecting line
between the two stations as function of depth z.
A reference velocity b has to be added to the geostrophic velocity to obtain the absolute flow velocity
vðzÞ ¼ v g ðzÞ þ b
½1
190 INVERSE MODELING OF TRACERS AND NUTRIENTS
observed strength of the vertical nutrient gradient,
and the depth range of particle remineralization can
be determined from the observed vertical position and
spatial extent of the nutrient maxima and oxygen
minima. Differences in the vertical structure of different nutrients (e.g., phosphate, silicate, carbon, and
alkalinity) reveal differences in the remineralization
depths of organic material, CaCO 3 , and opal.
Basin-wide observations of a variety of oceanic
tracers have been conducted since the 1950s. The
first coordinated and global tracer program, GEOSECS, produced tracer data of unprecedented quality
and coverage during the 1970s. More recently, the
World Ocean Circulation Experiment (WOCE) has
provided an even more detailed tracer data set describing the distributions during the 1990s. These
data are publicly available in electronic form over the
Internet or as colored distribution plots in printed
atlases. Availability of original tracer data is essential
for the inverse methods described below.
Inverse Model Concepts
Deriving quantitative results about the underlying
biogeochemical processes from oceanic nutrient
and tracer data, and separating the effects of biogeochemistry from the effects of circulation is a
challenging task, and requires the use of coupled
physical/biogeochemical numerical models. There
are a variety of possible approaches, which can
broadly be divided into two categories. The so-called
‘forward models’ assume rates of physical and biogeochemical processes to be known a priori, and
require ocean currents (or the physical forcing at the
ocean surface), as well as biological production and
particle remineralization rates to be specified as
input. Oceanic tracer concentrations are treated as
unknowns, and the tracer fields that would evolve
under the assumed flows and biogeochemical parameters are then simulated. Forward models, in
general, lead to mathematical systems that are relatively easy to solve. The conceptual disadvantage of
forward models is that physical and biogeochemical
rate information that is supposed to be determined
from the tracer data and, in fact, only becomes
available after the data evaluation, is required
a priori to enable the model run.
‘Inverse models’, in contrast, follow a seemingly
more intuitive approach, treating the measured tracer concentrations and other auxiliary knowledge
formally as knowns, whereas the physical and biogeochemical parameters, to be determined on the
basis of the tracer data, are treated as unknowns.
As described in more detail below, the inverse
approach generally leads to underdetermined mathematical systems that are much harder to solve than
the systems encountered in forward models. Error
and resolution analysis are two issues of particular
importance when solving underdetermined inverse
problems. First, the solved-for physical and biogeochemical parameters depend directly on the tracer
data, and errors in the data propagate into errors in
the solution. Second, owing to the incompleteness of
information in underdetermined systems, the unknowns are usually not fully resolved. Instead, only
specific linear combinations of unknowns may be
well constrained by the data, while individual unknowns or other combinations of unknowns may
remain poorly determined. Both, error and resolution analysis are essential for a quality assessment
of the solution of underdetermined systems.
Two widely used practical inverse approaches are
presented in the next two sections (‘Estimating absolute velocities and nutrient fluxes across sections’
and ‘Estimating carbon export fluxes with the
adjoint method’). These serve as examples to describe details of the mathematical methods and to
list achievable results. The first method, the section
inverse approach, infers nutrient, carbon, and tracer
fluxes across sets of sections, based on hydrographic, tracer, and nutrient data along these sections. The second example describes an application
of the adjoint method for the determination of
ocean currents, biological productivity, and downward particle fluxes. This method is specifically
adapted for the utilization of many different tracers
and can handle problems with heterogeneous and
sparse data coverage.
Estimating Absolute Velocities and
Nutrient Fluxes across Sections
The section inverse approach developed by Wunsch
exploits hydrographic data along sets of oceanographic sections, and allows estimation of absolute
flow velocities perpendicular to the sections. The
method was later extended to include nutrient and
oxygen data allowing to estimate nutrient and oxygen fluxes across the sections.
The section inverse method is based on the geostrophic principle, which for a given pair of hydrographic stations allows calculating the geostrophic
velocity v g (z) perpendicular to the connecting line
between the two stations as function of depth z.
A reference velocity b has to be added to the geostrophic velocity to obtain the absolute flow velocity
vðzÞ ¼ v g ðzÞ þ b
½1
190 INVERSE MODELING OF TRACERS AND NUTRIENTS