The focus of this paper is observation of the carbon cycle, and in particular its land-atmosphere compo- nents, as one part of an integrated earth observation system.

Introduction

  • model data synthesis: the combination of the information contained in both observations and models through both parameter-estimation and data-assimilation techniques
    • model testing and data quality control
    • interpolation of spatially and temporally sparse data
    • inference from available observations of quantities which are not directly observable
    • forecasting
  • 3 themes:
    • model-data syntheis based on terrestrial biosphere models as essential component of a terrestrial carbon observation ssystem (TCOS)
    • data uncertainties as as important as data values
    • sound uncertain specifications

Purposes and attributes of a TCOS

A succinct statement of the overall purpose of a TCOS might be: to operationally monitor the cycles of carbon and related entities (water, energy, nutrients) in the terrestrial biosphere, in support of comprehensive, sustained earth observation and prediction, and hence sustainable environmental management and socio- economic development.

  • a TCOS needs:
    • scientific credibility
    • respect carbon budgets
    • high spatial resolution
    • high temporal resolution
    • large number of entities
    • sufficient range of processes
    • partitioninong of net fluxes
    • quantification of uncertainty
    • altogether: swiss army knife (eierlegende Wollmilchsau)

Model–data synthesis: methods

Overview

All applications rest on three foundations: a model of the system, data about the system, and a synthesis approach.

Model

  • ODE or difference equation, including a noise term
  • noise accounts for imperfect model formulation and stochastic variability in forcings and parameters

Data

  • two types:
    • observations and measurements
      • $z=h(x, u) + \text{noise}$, where $h$ specifies the deterministic relationship between the measured quantities $z$ and $u$ and the system state $x$; z accounts fot mesurement error and representation error
    • prior estimates for model quantities

Synthesis

  • finding optimal match between data and model
  • 3 kinds of output:
    • optimal estimates for model properties to be adjusted (target variables)
    • uncertainties about these estimates
    • assessment of fitting the data, given the uncertainties
  • 3 basic choices:
    • target variables
    • cost function
    • search strategy for optimal values
      • nonsequential: all data treated at once
      • sequential: data incorporated step by step

Target variables

  • model parameters $p$, forcing variables $u$, initial conditions $x^0$m state vector $x^n$ itself: all collected in vector $y$
  • parameter estimation problems: $y=p$
  • data assimilation problems: target variables can be any model property, with emphasis on state variables

Cost function

  • common choice:

\begin{equation} \label{eqn:cf} J(y) = (z-h(y))^T[\operatorname{Cov}\,z]^{-1}(z-h(y)) + (y-\hat{y})^T[\operatorname{Cov}\,\hat{y}]^{-1}(y-\hat{y}) \end{equation}

  • $\hat{y}$ vector of priors (a priori estimates for target variables)
  • model–data synthesis problem: vary $y$ to minimize $J(y)$, subject to the constraint that $x(t)$ must satisfy the dynamic model
  • $y$ at the minimum is the a posteriori estimate of $y$, including information from the observations as well as the priors
  • Eq. \eqref{eqn:cf} is minimum-variance estimate for $y$
    • for any error distributed it is unbiased
    • minimum error covariance among all in $z$ linear and unbiased estimates
    • if error distributions Gaussian, even a maximum likelihood estimate for $y$, conditional on data and model dynamics
  • other choices for other problems or other error distributions

Search strategies for nonsequential problems

Example

Thus the cost function, and thence the entire minimization, takes a form in which neither the observations nor the prior estimates appear; they are replaced by quantities a and b scaled by the square roots of the inverse covariance matrices, which are measures of confidence. This is no mathematical nicety; rather it demonstrates that the data and the uncertainties are completely inseparable in the formalism. To put the point provocatively, providing data and allowing another researcher to provide the uncertainty is indistinguishable from allowing the second researcher to make up the data in the first place.

Algorithms for nonsequential problems

A high condition number of the Hessian of $J$ indicates that some linear combination(s) of the columns are nearly zero, that is, that the curvature is nearly zero in some direction(s), so that the minimization problem is ill-conditioned, as in the case of a valley with a flat floor.

  • analytical solution: only possible of $h(y)=H\,y+\text{noise}$ linear- gradient descent: simple and low cost, but tend to find local minima near starting value rather than global minimum
  • global search: find global minimum by searching through the whole $y$ space: overcome local minimum problem, but high costs
    • for example, simulated annealing finds the vicinity of a global minimum
    • then apply gradient descent from there

Search strategies for sequential problems

  • Kalman filter, genetic methods
  • adjoint methods (backward integration)

Discussion of model–data synthesis methods

Differences between nonsequential and sequential strategies.
  • advantages sequential:
    • optimal state can differ from that embodied in model equations
    • $x^n$ required to be included in $y$ (leads to intractable dimensionality in nonsequential models
    • size does not grow with length of model integration
    • can easily handle incremental extensions to time series observations
  • advantages nonsequential:
    • treat all data at once (see impacts of data at different points in time)
Model and data error structures
  • often assumend Gaussian, no temporal correlations
  • generalizations active area of research
Nonsequential and sequential parameter estimation
  • usually done nonsequantially (LS)
  • but: one can incorporate $p$ as part of $y$ and do sequential analysis
    • allows change of parameters by data, caused by catastrophic events

Model–data synthesis: examples

  • parameter estimation
  • atmospheric inversion methods to infer surface-atmosphere fluxes from atmospheric composition observations and atmospheric transport models
  • combination, advantages:
    • different observations constrain different processes
    • different observations have different resolutions in space and time (also a problem)
  • weather forecast by atmospheric and ocean circulation models

Data characteristics: uncertainty in measurement and representation

We have emphasized that data uncertainties affect not only the predicted uncertainty of the eventual result of a model–data synthesis process, but also the predicted best estimate.

  • scale mismatches between measurement and models is part of the representation error

An analogous temporal representation error arises when flask measurements (actually grab samples in time) are interpreted as longer-term means…

A further contribution to representation errors for most atmospheric inversion studies to date has been the projection of possible source distributions to a restricted subspace, usually by dividing the earth into a number of large regions. This is done both for computational reasons and to reduce the error amplification arising from under-determined problems. Errors in the prescription of flux distributions within these regions give rise to a so-called aggregation error, described and quantified by Kaminski et al. (2001). This error can be avoided by using adjoint representations of atmospheric transport that do not require aggregation (Rodenbeck et al., 2003a,b).

There are few experiments where representation errors can be evaluated, since this requires simultaneous knowledge of sources and atmospheric transport. However, one can use the range of model simulations as a guide (e.g. Law et al., 1996; Gurney et al., 2003).

  • GPP = [net assimilation]
  • net primary productivity (NPP) = [GPP - autotrophic respiration]
  • net ecosystem productivity (NEP) = [NPPheterotrophic respiration]
  • net biome productivity (NBP) = [NEP - disturbance flux]
  • disturbance flux: grazing, harvest, and catastrophic events (fire, windthrow, clearing)

Mistmatches of measurement and model scale

Observational issue

There are several options to relate fine-scaled measurements to a coarse-scaled model:

  • $z_{\text{fine}}$ is a noisy sample of $z_{\text{coarse}}$, variability in $z_{\text{fine}}$ (covariance $R_{\text{fine}}$) treated as contribution to representation error
  • direct aggregation: $z_{\text{coarse}}$ is weighted average of $z_{\text{fine}}$
  • $z_{\text{fine}} = g(x_{\text{coarse}}, a_{\text{fine}})$: relate fine-scale observations to coarse-scale state variables and additional fine-scale sncillary data such as topography

Scaling of dynamic model

Translate $\mathrm{d}x/\mathrm{d}t=f(x, u, p)$ between scales:

fine-scale and coarse-scale equations are different (for instance, biased with respect to each other) because of interactions between fine-scale variability and nonlinearity in the fine-scale function f(x, u, p)

Summary and conclusions

Critical error properties include:

  • the diagonal elements $[\operatorname{Cov}(z)]_{mm}=\sigma_m^2$ of the measurement error covariance matrix (where $\sigma_m^2$ is the error magnitude for an observation $z_m$)
  • the correlations between different observations, quantified by the off-diagonal elements of the covariance matrix
  • the temporal and the spatial structure of errors
  • the error distribution
  • possible scale mismatches between measurements and models
  • the representation of the observations in the model