Error Propagation & Monte Carlo Simulation
Two methods to quantify uncertainty
VM0042 provides two approaches for calculating the numerical uncertainty deduction: analytical error propagation (straightforward for independent errors) and Monte Carlo simulation (for complex, correlated inputs). Understanding these methods is essential for completing the ERR spreadsheet.
Method 1: Analytical Error Propagation
In plain terms: each measurement you make has some imprecision. When you multiply or add these imprecise numbers together, the imprecision compounds. Error propagation is the mathematical method for tracking how that compounding works so the final uncertainty figure is honest.
The key rules:
Key Rule: Adding/Subtracting values → Add absolute variances
Key Rule: Multiplying/Dividing values → Add relative (%) variances, then take square root
📐 Worked Example: SOC Stock Uncertainty
Calculate SOC stock and its uncertainty:
| Input | Value | Relative Uncertainty |
|---|---|---|
| OC concentration | 1.8% | 8.3% (±0.15%) |
| Bulk density | 1.35 g/cm³ | 5.9% (±0.08 g/cm³) |
| Depth | 30 cm | 0% (assumed exact) |
SOC stock = 1.8% × 1.35 × 30 × 100 = 72.9 tC/ha
Combined uncertainty = √(8.3² + 5.9²) = √(68.9 + 34.8) = √103.7 = 10.2%
Absolute uncertainty = 72.9 × 10.2% = ±7.4 tC/ha
95% confidence interval: 72.9 ± (2 × 7.4) = 58.1 to 87.7 tC/ha
Method 2: Monte Carlo Simulation
🎴 Analogy: Playing Cards 10,000 Times
Error propagation is like using probability formulas to predict card game outcomes. Monte Carlo is like actually playing the game 10,000 times and recording all results. Both give you a distribution of outcomes, but Monte Carlo handles complex, correlated inputs better.
MC simulation is required when using spectroscopy techniques for SOC measurement (because their errors are often non-linear and correlated with carbon content).
📊 How VM0042 Uses the Monte Carlo Distribution
VM0042 does not simply use the "5th percentile." Instead, it uses a probability-of-exceedance construct: the uncertainty deduction percentage is set so that the probability of the true value being lower than (mean − deduction) is only 5%, i.e., a 95% confidence lower bound.
For a symmetric normal distribution:
The 5th percentile is at mean − 1.645σ. So a 10% relative uncertainty (σ/mean = 10%) gives a deduction of 1.645 × 10% = 16.5% of the mean estimate. Because ≥15% triggers a deduction per VM0042's schedule, this project would face a deduction.
For non-symmetric or skewed distributions:
MC simulation directly gives you the 5th percentile value from the simulated distribution, no normal-distribution assumption needed. This is why MC is required for spectroscopy models with non-Gaussian error structures.
Deduction schedule: If 95% confidence interval (CI) half-width <15% of mean → 0% deduction. 15–20% → 5% deduction. 20–30% → 15% deduction. >30% → 35% deduction. Uncertainty is measured as a % of the point estimate.
Model True-Up (Approach 1 only)
🏦 Analogy: Bank Reconciliation
If your personal records say you have $5,000 but the bank statement says $4,700, you adjust your records to match reality. In VM0042, if the model predicted 2.5 tC/ha gain but actual soil samples show 2.0 tC/ha, the model is recalibrated and past credits may be adjusted.
True-up process:
- At each SOC measurement event (≥every 5 years), compare model predictions with actual measurements
- If model systematically over-predicted: recalibrate model, adjust model prediction error estimate
- Credits already issued based on over-predicted values may need buffer pool adjustments
- True-up data is incorporated into the model validation dataset, improving future predictions
Key Takeaways
- 1Analytical error propagation combines uncertainties by adding squared relative uncertainties and taking the square root (for multiplied/divided values)
- 2Monte Carlo simulation runs thousands of random iterations to build a probability distribution - required when spectroscopy is used for SOC measurement
- 3VM0042 uses a 95% confidence lower bound: the deduction ensures only a 5% probability that the true value is lower than claimed
- 4The deduction schedule: less than 15% CI half-width = 0% deduction, 15-20% = 5%, 20-30% = 15%, over 30% = 35%
- 5Model true-up every 5 years compares predictions to actual soil samples - if the model over-predicted, recalibration and buffer adjustments may be required