Representation Error in
Data Assimilation
Introduction
The fundamental objective of Data Assimilation is to estimate the state of geophysical systems—such as the atmosphere, ocean, and chemical environment—as accurately as possible by comparing and combining observations with prior estimates produced by numerical models. From a Bayesian perspective, this process requires statistical knowledge of both observation errors and background errors.
The challenge arises because numerical models cannot fully represent all spatial and temporal scales, as well as all physical processes, present in the observed geophysical system. As a result, even perfect observations (i.e., observations without measurement error) may differ substantially from model predictions. If this discrepancy is not properly accounted for, the data assimilation results can become significantly distorted.
1. Historical Evolution of Representation Error
1981–1986 — Lorenc
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Emphasized that the estimated state in atmospheric data assimilation is defined by the numerical model.
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Pointed out that observation errors include spatial and temporal variability smaller than the analysis scale.
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Systematically introduced the term representativeness error.
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Proposed the traditional filter, incorporating unresolved-scale representation error into the observation-error covariance matrix.
1991 — Daley
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Used the term representativeness error in the context of sampling error associated with observation grids.
1993 — Daley
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Introduced the term forward interpolation error.
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Linked the concept to the linearized observation operator derived from inverse theory.
1997 — Cohn
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Derived representation error within a Bayesian estimation-theory framework.
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Demonstrated that temporal correlations and cross-scale correlations between background and observation errors must be neglected.
1997 — Mitchell & Daley
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Introduced the term error due to unresolved scales.
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Explicitly distinguished errors arising from subgrid-scale processes.
2001 — Janjić
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Established the theoretical foundation for unresolved-scale error through a PhD dissertation in atmospheric data assimilation.
2006 — Janjić & Cohn
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Proposed a Schmidt–Kalman filtering methodology for unresolved-scale observation error.
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Developed an augmented-state approach treating cross-covariances between resolved and unresolved scales.
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Defined representation error as the combination of unresolved-scale error and observation-operator error.
2007 — Ponte et al.
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Used the term representation error to include both preprocessing error and unresolved-scale error in altimetry data.
2008 — Oke & Sakov
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Published a landmark study on representativeness error in ocean data assimilation.
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Defined representation error as unresolved-scale error plus observation-operator error.
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Quantified differences between 100–200 km SLA maps and 1-second satellite observations.
2014 — Waller et al.
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Introduced the term representativity error.
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Highlighted the component associated with spatial sampling error.
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Further intensified inconsistencies in terminology.
April 2014 — ESA Reading Workshop
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Addressed inconsistencies in terminology across geoscience disciplines.
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Emphasized the need for a unified conceptual framework.
2016 — Schutgens et al.
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Used the term spatial sampling error in aerosol studies.
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Treated it as synonymous with unresolved-scale and process-related errors.
2018 — Janjić et al.
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Unified historically inconsistent terminology.
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Classified representation error into three components:
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Unresolved-scale error
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Observation-operator error
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Preprocessing error
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Presented applications in satellite radiance assimilation, ocean reanalysis, and atmospheric chemistry.
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Summarized theoretical frameworks, diagnostic methods, and assimilation methodologies.
From the 1980s, representation error was developed to explain mismatches between observations and model scales in data assimilation. It was later formalized into components such as unresolved-scale and observation-operator errors. By 2017, these ideas were unified into a standard framework.
2. Formulation and Definition of Representation Error
w(x, t): True (full) state
The true atmospheric state defined on a continuum. It includes all spatial and temporal scales and physical processes, and is represented as a vector-valued function of space x and time t.
wʳ(x, t): Resolved state
The true resolved state that can be represented by the geophysical model. It is assumed to belong to the same function space as w, while corresponding to the model discretization scheme (e.g., a finite sum of spherical harmonics in spectral models). The resolved state is typically obtained by applying the model-resolution filtering or projection operator to the full state.
hᶜ : Continuum observation operator
The (generally nonlinear) continuum observation operator.
It acts on the true atmospheric state w to produce observations y.
Notation convention:
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Lowercase symbols denote nonlinear operators.
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Uppercase symbols denote linear operators.
h : Discrete observation operator
The discrete observation operator used by the numerical model.
It acts on the resolved state wʳ and produces vectors in observation space.
The discrete operator h is determined by the model dynamics and discretization.
y : Observation
The actual observation value: y = hᶜ(w) + εᵐ + ε′′′
where:
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εᵐ: instrument (measurement) error
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ε′′′: preprocessing-related error
R, E, F
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R: observation-error covariance matrix
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E: instrument-error covariance matrix
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F: representation-error covariance matrix
Relations: R = E + F
The framework defines the true state w, the model-resolved state wr, and observations y linked through a continuum observation operator with errors. In data assimilation, a discrete operator h is used, and the observation-error covariance R is split into instrument error E and representation error F (R = E + F).
3. Mathematical Decomposition of Observation Error
εᵒ = y − h(wʳ)
= hᶜ(w) + εᵐ + ε′′′ − h(wʳ)
= ε′′′ + [hᶜ(w) − hᶜ(wʳ)] + [hᶜ(wʳ) − h(wʳ)] + εᵐ
= ε′′′ + ε′ + ε′′ + εᵐ
A key point of this decomposition is that every component of observation error depends on the geophysical model being used. Representation error is therefore not an absolute quantity, but rather a model-dependent quantity defined relative to the chosen model configuration. The paper refers to this as the relative nature of representation error.
4. Three Components of Observation Error
ε′ — Error Due to Unresolved Scales and Processes
This error represents the difference between the perfect observation of the full true state and the perfect observation of the resolved true state. It exists whenever the geophysical model does not fully represent the complete dynamical system.
ε′ ≡ hᶜ(w) − hᶜ(wʳ)
In other words, ε′ is the discrepancy between a perfect (noise-free) observation and the perfect observation of the resolved truth that the data assimilation system aims to estimate.
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For a nonlinear observation operator:
ε′ ≠ hᶜ(w − wʳ) because nonlinear operators do not allow simple separation of unresolved contributions.
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For a linear operator H:
ε′ ≠ hᶜ(w − wʳ) which makes explicit that the error depends on all unresolved scales and processes not
represented by the model.
ε′′ — Observation-Operator Error
This error arises from approximating the continuous observation operator hᶜ with a discrete operator h. It is associated with representing an infinite-dimensional operator in finite dimensions or with computational approximations introduced for efficiency.
ε′′ ≡ hᶜ(wʳ) − h(wʳ)
This error may originate from:
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Representing an infinite-dimensional operator using a finite-dimensional approximation that acts only on resolved scales
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Incomplete knowledge of the physical characteristics required to perfectly describe hᶜ
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Computational simplifications introduced to reduce cost, such as gas absorption parameterizations in RTTOV or the assumption of a single vertical column
ε′′′ — Pre-processing Error
This error originates from imperfections in observation pre-processing and quality-control procedures.
When the pre-processing depends on the data assimilation system itself (e.g., geophysical models or assimilation algorithms), the resulting error is classified as a representation error.