Error Theory in CO₂ Retrieval

This section will rigorously describe:
  • Measurement noise propagation
  • Forward model parameter errors
  • Smoothing error
  • Spectroscopic uncertainty
  • Aerosol-induced bias
  • Surface reflectance errors
  • Calibration errors
  • Random vs systematic error decomposition

Mathematical Framework for Error Propagation

We start from the linearized forward model: $$ y = F(x) + \epsilon $$ Linearized around \( x_a \): $$ \delta y = K \delta x + \epsilon $$ Where:
  • \( K = \frac{\partial F}{\partial x} \) (Jacobian matrix)
  • \( \epsilon \sim \mathcal{N}(0, S_\epsilon) \)
Optimal Estimation solution: $$ \hat{x} = x_a + G(y - F(x_a)) $$ Where gain matrix: $$ G = (K^T S_\epsilon^{-1} K + S_a^{-1})^{-1} K^T S_\epsilon^{-1} $$

Total Retrieval Error Decomposition

Define true state \( x_t \). Retrieval error: $$ e = \hat{x} - x_t $$ Insert measurement model: $$ y = F(x_t) + \epsilon $$ Linearized: $$ y - F(x_a) = K(x_t - x_a) + \epsilon $$ Thus: $$ \hat{x} = x_a + G \left[ K(x_t - x_a) + \epsilon \right] $$ Rearrange: $$ \hat{x} = x_a + G K(x_t - x_a) + G\epsilon $$ Subtract \( x_t \): $$ e = (GK - I)(x_t - x_a) + G\epsilon $$ Define averaging kernel: $$ A = GK $$ Thus: $$ e = (A - I)(x_t - x_a) + G\epsilon $$ This equation separates:
  • Smoothing error
  • Measurement noise error

Measurement Noise Error

Noise term: $$ e_{noise} = G\epsilon $$ Covariance: $$ S_{\rm noise} = G S_\epsilon G^T $$ If measurement noise covariance: $$ S_\epsilon = \sigma^2 I $$ Then: $$ S_{\rm noise} = \sigma^2 G G^T $$ Interpretation:
  • Larger Jacobian → smaller noise error
  • Low SNR → large posterior uncertainty
Random error decreases with averaging: $$ \sigma_{\rm mean} = \frac{\sigma}{\sqrt{N}} $$

Smoothing Error

Smoothing term: $$ e_{\rm smooth} = (A - I)(x_t - x_a) $$ Covariance: $$ S_{smooth} = (A - I) S_{\rm true} (A - I)^T $$ Where \( S_{\rm true} \) is true state variability.

Physical meaning:

If retrieval does not fully resolve vertical structure, prior assumptions influence solution.

For XCO₂:

  • DOFS ≈ 1
  • Vertical information limited
  • Smoothing error small for total column but relevant for profile retrievals.

Forward Model Parameter Errors

Suppose forward model depends on additional parameters ( b ): $$ y = F(x, b) $$ If parameter error: $$ b = b_t + \delta b $$ Linearize: $$ \delta y = K_x \delta x + K_b \delta b $$ Where: $$ K_b = \frac{\partial F}{\partial b} $$ Retrieval error due to forward model parameter: $$ e_{\rm model} = G K_b \delta b $$ Covariance: $$ S_{model} = G K_b S_b K_b^T G^T $$ This is key for:
  • Spectroscopy
  • Aerosol properties
  • Surface reflectance
  • Calibration parameters

Spectroscopic Error Propagation

Absorption cross-section: $$ \sigma_\nu = S f(\nu) $$ If line strength uncertainty: $$ S = S_t + \delta S $$ Optical depth: $$ \tau_\nu \propto S XCO_2 $$ Measured radiance fixed.

Thus approximate bias:

$$ \frac{\delta XCO_2}{XCO_2} \approx \frac{\delta S}{S} $$ If: $$ \delta S/S = 0.5% $$ Then: $$ \delta XCO_2 \approx 2 \text{ ppm} $$ Systematic global bias.

Aerosol-Induced Bias

Radiance: $$ I = I_0 e^{-\sigma n L} $$ If effective path length: $$ L = L_t + \delta L $$ Then: $$ I = I_0 e^{-\sigma n (L_t + \delta L)} $$ Retrieval interprets absorption assuming \( L_t \): $$ \delta XCO_2 \approx \frac{\delta L}{L_t} XCO_2 $$ Even: $$ \delta L/L = 0.25% $$

→ 1 ppm bias.

This is dominant systematic contributor.

Surface Reflectance Error

Radiance: $$ I \propto A_\nu e^{-\tau_\nu} $$

If surface albedo error \( \delta A \):

Nonlinear coupling with optical depth. Approximate sensitivity: $$ \delta XCO_2 \approx \frac{\partial XCO_2}{\partial A} \delta A $$ Especially critical over bright desert regions.

Wavelength Calibration Error

Shift in wavelength: $$ \nu \rightarrow \nu + \delta \nu $$ Line misalignment leads to systematic mismatch. Bias magnitude: $$ \delta XCO_2 \propto \frac{\partial I}{\partial \nu} \delta \nu $$ Small spectral shifts can mimic absorption depth changes.

Random vs Systematic Error Decomposition

Total retrieval: $$ X_{\rm retrieved} = X_{\rm true} + \epsilon_{\rm random} + b_{\rm systematic} $$ Random: $$ E[\epsilon_{\rm random}] = 0 $$ Systematic: $$ b = E[X_{\rm retrieved}] - X_{\rm true} $$ Total variance: $$ \sigma_{\rm total}^2 = \sigma_{\rm random}^2 + b^2 $$ Random decreases with averaging. Systematic does not.

Full Error Budget Expression

Combining all components: $$ S_{\rm total} = S_{\rm noise} + S_{\rm smooth} + S_{\rm model} $$ Expanded: $$ S_{\rm total} = G S_\epsilon G^T + (A - I) S_{\rm true} (A - I)^T + G K_b S_b K_b^T G^T $$ This is the complete posterior covariance.

Mission-Level Error Targets

For carbon cycle applications: Precision target: $$ \sigma_{\rm random} \le 1 \text{ppm} $$ Bias target: $$ |b| \le 0.5 \text{ ppm} $$ Because carbon flux inversion amplifies systematic bias.


Reference

Go to Content