[Paper Review] Commentary on 'Modified MMF (Morgan-Morgan-Finney) model for evaluating effects of crops and vegetation cover on soil erosion' by Morgan and Duzant (2008)

Kwanghun Choi, Bernd Huwe+1 more|arXiv (Cornell University)|Dec 28, 2016

Soil erosion and sediment transport16 references4 citations

TL;DR

This paper identifies and corrects three critical errors in the modified Morgan–Morgan–Finney (MMMF) soil erosion model: an incorrect trigonometric factor in effective rainfall, an erroneous interflow calculation, and an improperly normalized C-factor in the transport capacity equation. The authors propose revised formulations that restore physical consistency and improve accuracy in runoff and soil erosion predictions, particularly under steep slopes and complex surface conditions.

ABSTRACT

The Morgan-Morgan-Finney (MMF) model is a widely used semi-physically based soil erosion model that has been tested and validated in various land use types and climatic regions. The latest version of the model, the modified MMF (MMMF) model, improved its conceptual physical representations through several modifications of the original model. However, the MMMF model has three problematic parts to be corrected: 1) the effective rainfall equation, 2) the interflow equation, and 3) the improperly normalized C-factor of the transport capacity equation. In this commentary, we identify and correct the problematic parts of the MMMF model, which should result in more accurate estimations of runoff and soil erosion rates.

Motivation & Objective

• To identify and correct three key errors in the modified Morgan–Morgan–Finney (MMMF) model that compromise runoff and soil erosion predictions.
• To address a trigonometric error in the effective rainfall calculation where 1/cos(S) is incorrectly used instead of cos(S) as the slope adjustment factor.
• To resolve a quantity estimation error in the interflow calculation that leads to overestimation of subsurface flow.
• To correct the improper normalization of the C-factor in the transport capacity equation, which causes inconsistent units and erroneous erosion rate predictions when multiple surface conditions are combined.
• To improve the physical consistency and predictive accuracy of the MMMF model for use in diverse climatic and land use conditions.

Proposed method

• Reformulates effective rainfall using cos(S) instead of 1/cos(S) based on geometric and trigonometric principles, deriving Rf_corrected = R·(1−PI)·cos(S).
• Revises the interflow equation by correcting the volume estimation method, ensuring mass balance and accurate subsurface flow representation.
• Replaces the original C-factor formulation (C_MMMF = v_a·v_v·v_t / v_b) with a properly normalized version: C_corrected = (v_a/v_b)·(v_v/v_b)·(v_t/v_b) = (v_a·v_v·v_t)/v_b³.
• Validates the corrected formulations using physical principles and mathematical derivation, ensuring unit consistency and correct scaling of flow resistance and acceleration effects.
• Applies the corrected equations to standard MMMF model components, ensuring compatibility with existing model structure and parameterization.
• Demonstrates that the corrected C-factor maintains unitless consistency regardless of the number of surface condition types included.

Experimental results

Research questions

• RQ1 Why does the original MMMF model overestimate effective rainfall due to the use of 1/cos(S) instead of cos(S) as the slope adjustment factor?
• RQ2 How does the incorrect interflow calculation in the MMMF model affect subsurface water flow estimation and overall hydrological balance?
• RQ3 What is the root cause of the C-factor inconsistency in the transport capacity equation, and how does it lead to erroneous erosion rate predictions?
• RQ4 How does the improper normalization of the C-factor affect model outputs when multiple surface conditions (e.g., rills, vegetation, roughness) are combined?
• RQ5 To what extent do the corrected formulations improve the physical consistency and predictive accuracy of the MMMF model under varying slope and land cover conditions?

Key findings

• The effective rainfall equation in the MMMF model incorrectly uses 1/cos(S) as the slope adjustment factor, leading to overestimation of rainfall on sloping surfaces; the correct factor is cos(S), reducing effective rainfall on steeper slopes.
• The interflow calculation in the MMMF model contains a quantity estimation error that results in overestimated subsurface flow; the corrected formulation ensures proper mass balance and realistic flow volume estimation.
• The C-factor in the MMMF model is improperly normalized, as the original formulation C_MMMF = v_a·v_v·v_t / v_b introduces inconsistent units due to the lack of proper normalization by v_b³.
• The corrected C-factor is C_corrected = (v_a·v_v·v_t)/v_b³, which is unitless and physically consistent, ensuring accurate scaling of flow resistance and acceleration effects.
• The error in the C-factor leads to significant overestimation of erosion rates when runoff velocities exceed 1 m/s, especially on steep slopes, due to unbalanced velocity normalization.
• The corrected model formulations improve physical consistency, reduce prediction bias, and enhance reliability for runoff and soil erosion estimation across diverse land use and slope conditions.