Appendix X: Innovation-Space Diagnostics with ufsda-obs-br-check

Overview

The ufsda-obs-br-check utility computes innovation-space diagnostics for radiance observations using the Desroziers (2005) method. These diagnostics assess the statistical consistency between:

  • the background state,

  • the background-error covariance,

  • the analysis state (through OMA),

  • the observations, and

  • the assumed observation-error variance R.

All quantities are derived entirely from innovations and do not depend on the full analysis increment or the Kalman gain.

Innovation Definitions

The tool uses the standard JEDI sign convention:

OMB = y - H(x_b)

Observation minus background.

OMA = y - H(x_a)

Observation minus analysis.

These definitions ensure consistency with the Desroziers identities:

  • Sd     = E[OMB^2]

  • R_est  = E[OMA * OMB]

  • HBH^T  = Sd - R_est

Background State vs Background Covariance Contribution

Two distinct concepts appear in innovation diagnostics:

Background state contribution

The direct mismatch between the background and the observation, represented by the innovation OMB = y - H(x_b). This measures how far the background state is from the observation.

Background covariance contribution

The portion of the innovation variance explained by background error, given by HBH^T = Sd - R_est. This is a covariance-level quantity and reflects how much background-error variance contributes to the innovations.

These two contributions are fundamentally different: one is a state difference, the other is a variance decomposition.

Analysis-State Contribution

Although these diagnostics operate entirely in observation space, the analysis state does appear through the quantity

OMA = y - H(x_a)

which is used in the Desroziers identity

R_est = E[OMA * OMB].

This use of the analysis state does not evaluate the analysis increment or the Kalman gain. Instead, OMA serves only as a statistical probe to estimate the true observation-error variance. In this sense, the analysis contributes to the diagnostics, but only through its projection into observation space, and only for the purpose of variance estimation.

Diagnostic Quantities

For each channel, the following innovation-space quantities are computed:

Sd = E[OMB^2]

Innovation variance. Equal to HBH^T + R_true.

R_est = E[OMA * OMB]

Desroziers estimate of the true observation-error variance.

Sd/R

Innovation chi-square proxy. Values much less than 1 indicate that the assumed R is too large; values much greater than 1 indicate that R is too small.

R_est/R

Ratio of estimated to assumed observation-error variance. Used as a variance scaling indicator.

HBH^T = Sd - R_est

Background-error contribution to the innovation variance.

HBH^T/R

Background-to-observation ratio. Values below ~0.3 are typical for microwave radiances.

scale_R = R_est / R

Recommended variance multiplier for tuning the assumed R.

infl_chi = sqrt((Sd/R) / chi_target)

Standard-deviation inflation factor required to achieve a target chi-square (default chi_target = 0.8).

Interpretation Guidelines

  • Sd/R < 1 The assumed R is too large; observations are under-weighted.

  • Sd/R > 1 The assumed R is too small; observations are over-weighted.

  • HBH^T/R small (0.0–0.3) Background covariance contribution is modest and typical for ATMS.

  • scale_R < 1 Decrease the assumed observation-error variance.

  • scale_R > 1 Increase the assumed observation-error variance.

  • infl_chi Recommended per-channel standard-deviation inflation to achieve the target chi-square.

Example Output

Ch 09: Sd/R=0.158  R_est/R=0.114  HBH^T=0.018  HBH^T/R=0.044
       scale_R=0.114  infl_chi=0.445

Interpretation:

  • Sd/R is well below 1 → assumed R is too large.

  • R_est/R confirms the same.

  • HBH^T/R is small → background covariance contribution is modest.

  • infl_chi suggests multiplying the standard deviation by ~0.45 (a reduction of about 55%).

Usage

Run the tool with:

ufsda-obs-br-check --yaml obs_diag.yaml

The utility reads OMB, OMA, R, and QC from the obs_diag.yaml file and prints per-channel diagnostics.

Notes

These diagnostics operate entirely in innovation space and should be interpreted as statistical consistency checks on R and background error contributions.