Module 2

Crop Model Calibration (MONICA)

MONICA AGRO-ECOSYSTEM MODEL

The selected Agro-ecosystem model to assist us in the DSM of the field is named MONICA „MOdel of Nitrogen and Carbon dynamics in Agro-ecosystems” (Nendel et al., 2011). For a detailed explanation, we suggest the page where MONICA is hosted:

https://github.com/zalf-rpm/monica/wiki

For the software download and detailed information, we also recommend the GitHub page:

https://github.com/zalf-rpm/monica

Important to mention that multiple Agro-ecosystem models are available and could be used here instead of MONICA, strong candidates are WOFOST, DSSAT, APSIM, HERMES (De Wit et al., 2019; Jones et al., 2003; Keating et al., 2003; Kersebaum, 2007).

1. Purpose

Fit cultivar parameters of Winter Rye using observed yield and NDWI data. The

2. Optimization Method

Use the Levenberg–Marquardt Algorithm (LMA) for parameter fitting.

LEVENBERG–MARQUARDT THEORY

The Levenberg–Marquardt algorithm (LMA) is a numerical method designed to solve nonlinear least-squares problems, i.e., situations where a model’s output depends nonlinearly on its parameters and we wish to minimize the sum of squared residuals between observed and predicted data.

It combines two classical optimization strategies:

• Gauss–Newton method: efficient near the solution but unstable if the initial guess is poor.
• Gradient descent: stable but converges slowly.

LMA introduces a damping factor (\(\lambda\)) to transition smoothly between both extremes:

\[ \left( {{J^T}J{\rm{ + }}\lambda I} \right)\Delta p = {J^T}\left( {{y_{obs}} - {y_{sim}}} \right)\]

Where:

• \(J\) is the Jacobian matrix (partial derivatives of model outputs with respect to parameters),
• \(\lambda\) is the damping coefficient,
• \(\Delta p\) is the parameter update step (p as a vector containing the parameter values),
• \(y_{obs}\) and \(y_{sim}\) = observed and simulated values,
• \(I\) is the identity matrix,
• Superscripted \(T\) refers to the transpose of the matrix.

If \(\lambda \rightarrow 0\); then Gauss–Newton behaviour dominance (fast convergence).
If \(\lambda \rightarrow \infty\); then, gradient descent behaviour dominance (safe, small steps).

At each iteration, \(\lambda\) is adjusted:

• If the new parameters reduce the objective function, then \(\lambda\) decreases.
• If the new parameters increase the objective function, then \(\lambda\) increases.

LMA is efficient when derivatives are available or can be approximated numerically, making it particularly suitable for crop model calibration, where each simulation is expensive and convergence speed matters.

LMA is chosen because it converges rapidly when derivatives are available. The limitations of LMA are: (i) it is suitable to find the best parameters locally, which means it is relatively dependent on the initial guess of parameters; (ii) it requires derivatives to calculate the Jacobian matrix.

3. Objective Function (OBF)

The OBF combines:

• Yield error (difference between observed and simulated yield).
• Correlation between NDWI and simulated Leaf Area Index (LAI).

Weights are applied to balance both terms.

BRIEF DISCUSSION ABOUT THE OBF NORMALIZATION

The Objective Function (OBF) aggregates multiple error components with different scales and physical meanings, e.g., yield differences (in kg ha⁻¹) and correlation coefficients (dimensionless).

To ensure balanced influence, each component must be normalized before combination:

\[ OBF = {w_1} \cdot {\left( {{Y_{obs}} - {Y_{sim}}} \right)^2}\,\,\, - \,\,\,{w_2} \cdot {r_p}\left( {{\rm{NDWI}}\,{\rm{,}}\,{\rm{LAI}}} \right)\]

Where:

• \(w_1\) and \(w_2\) are weighting coefficients (typically equal or calibrated),
• \(r_p(\text{NDWI}, \text{LAI})\) is the Pearson correlation between NDWI (from images) and LAI (from MONICA simulations).

The normalization makes each component dimensionless and comparable, avoiding dominance of one variable due to its magnitude or units. Balancing both terms ensures that the calibration process not only minimizes absolute yield errors but also matches the temporal vegetation dynamics captured by NDWI.

4. Output

• Calibrated cultivar parameters.
• Validated MONICA simulation matching yield and NDWI behaviour.