Terminology & Notation#

This page defines common abbreviations and the mathematical notation used throughout the FIPS documentation and code. Different scientific fields often use different conventions — this guide helps bridge those gaps.

Mathematical Notation#

Notation Framework#

FIPS uses a consistent notation system throughout:

Dimensionality Convention

Lowercase letters (\(x\), \(z\), \(y\)) represent 1-D vectors
Uppercase letters (\(H\), \(S\), \(K\), \(A\)) represent 2-D matrices

Hat Notation

Hat \(\hat{\ }\) = posterior (a posteriori estimate after incorporating observations)
- \(\hat{x}\) = posterior state
- \(\hat{S}\) = posterior covariance
- \(\hat{y}\) = posterior modeled observations

The Forward Model

The fundamental relationship is:

\[z - c = y = Hx\]

where:

\(x\) = state vector (the unknowns we’re solving for)
\(z\) = observations (measured data)
\(c\) = constant (background or offset)
\(y\) = modeled observations (what the forward model predicts)
\(H\) = forward operator (maps state space → observation space)

Subscript Conventions

The subscript system indicates which space or time a variable belongs to:

Subscript \(_0\) = prior information / a priori (before incorporating observations)
- \(x_0\) = prior state vector
- \(S_0\) = prior error covariance matrix
Subscript \(_z\) = observation space (associated with \(z\))
- \(S_z\) = observation/model-data mismatch error covariance matrix

Covariance Matrices

Following the uppercase convention for matrices, covariance matrices are denoted with \(S\):

\(S\) = any covariance matrix (uppercase because 2-D)
\(S_0\) = prior error covariance (subscript _0 for a priori)
\(S_z\) = observation error covariance (subscript _z because it’s in observation space)
\(\hat{S}\) = posterior error covariance (hat for a posteriori)

This framework applies consistently: any variable with subscript \(_0\) refers to the prior, any variable in observation space gets subscript \(_z\), and posterior quantities get a hat.

Quick Reference#

Symbol	Name	Description
\(x\)	State vector	The unknown quantities being estimated (e.g., fluxes, densities)
\(x_0\)	Prior state	A priori estimate before incorporating observations
\(\hat{x}\)	Posterior state	A posteriori optimized state estimate after inversion
\(z\)	Observations	Measured data (e.g., concentrations, gravity anomalies)
\(c\)	Constant / Background	Additive offset or background field
\(y\)	Modeled observations	Forward model output \(y = Hx + c\)
\(y_0\)	Prior observations	\(y_0 = Hx_0 + c\)
\(\hat{y}\)	Posterior observations	\(\hat{y} = H\hat{x} + c\)
\(H\)	Forward operator / Jacobian	Operator mapping state space to observation space
\(S_0\)	Prior error covariance	Uncertainty in the prior state estimate
\(S_z\)	Observation error covariance	Combined measurement error and model representation error
\(\hat{S}\)	Posterior error covariance	Reduced uncertainty after incorporating observations
\(K\)	Kalman gain	Weighting matrix that determines how observations update the prior
\(A\)	Averaging kernel	Shows which states are constrained by observations: \(A = KH\)

Diagnostic Metrics#

Symbol	Name	Description
DOFS	Degrees of Freedom for Signal	Number of independent pieces of information from observations. Equal to \(\text{Tr}(A)\)
\(\chi^2\)	Chi-squared statistic	Goodness-of-fit metric comparing observations to model predictions
\(R^2\)	Coefficient of determination	Fraction of variance explained by the model (0 to 1)

Inverse Problem Terminology#

Prior#: The initial estimate of the state (and its uncertainty) before incorporating observations. Often comes from inventory data, climatology, or a process model.
Posterior#: The updated estimate of the state (and its uncertainty) after incorporating observations through Bayesian inference.
Forward Model / Forward Operator#: The mathematical operator \(H\) that predicts observations from a given state: \(y = Hx + c\). Sometimes called the Jacobian, observation operator, or sensitivity matrix.
Jacobian#: In the linear case, identical to the forward operator \(H\). For nonlinear problems, the Jacobian is the local linearization of the forward model.
Observation Operator#: Another name for the forward operator, emphasizing its role in mapping state space to observation space.
Kalman Gain#: The matrix \(K\) that optimally weights how much each observation updates the prior state. Derived from minimizing posterior uncertainty.
Averaging Kernel#: Matrix \(A = KH\) showing which true state variables are constrained by the observations. Diagonal elements near 1 indicate strong constraint; near 0 indicates weak constraint.
Model-Data Mismatch#: The combined error in observations and forward model representation, captured in the covariance matrix \(S_z\). Includes measurement error, transport error, aggregation error, etc.
Covariance Matrix#: A symmetric positive-definite matrix encoding uncertainties and their correlations. Diagonal elements are variances; off-diagonal elements are covariances.
Posterior Error Reduction#: The decrease in uncertainty from prior to posterior, often expressed as \(1 - \text{diag}(\hat{S}) / \text{diag}(S_0)\).