# Site Quality Models and Fuel Load Dynamic Equation Systems Disaggregated by Size Fractions and Vegetative States in Gorse and High Heath Shrublands in Galicia (NW Spain)

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Study Area, Shrub Communities and Inventory Plots

#### 2.2. Biomass Sampling

^{2}, depending on shrub height: For shrubs smaller than 1.0 m in height, 4 m

^{2}quadrats were destructively sampled; for shrubs taller than 1.0 m, the quadrat size varied from 3 × 3 m to 6 × 6 m.

_{Shr}) and mean shrub height ($\overline{{h}_{Shr}}$).

#### 2.3. Laboratory Work

_{Shr_G1_dead}= dead fine shrub load, W

_{Shr_G1_live}= live fine shrub load, W

_{Shr_G1}= fine shrub load (dead + live), W

_{Shr_G23}= coarse shrub load and W

_{Shr}= total shrub load = W

_{Shr_G1}+ W

_{Shr_G23}= AGB at stand level. Fractions G2 and G3 were grouped to prevent loss of data, as fraction G3 is infrequent in the juvenile stages of these shrub communities.

#### 2.4. Development of Total Fuel Load Growth Curves

_{Shr}-t curves were constructed on the basis of the information obtained in the sampling plots. This type of relationship has long been used in forestry as a proxy for site productivity, mainly in forest stands, giving rise to site quality or site index curves [44,81,82]. According to Clutter et al. [44], most techniques used to construct site quality curves can be viewed as special cases of three general methods: (1) the guide-curve method, (2) the difference-equation method and (3) the parameter-prediction method. The latter two methods require remeasurement data from the sample plots; however, as we only have one measurement of W

_{Shr}and t for each sample plot, the site quality curves were developed using the guide-curve method with a growth function as the base model.

_{Shr}). Numerous growth functions can be used in forestry, such as the 74 documented by Kiviste et al. [83]. The following are the most important desirable attributes for site quality equations [84,85,86]: (1) polymorphism, (2) sigmoid growth pattern with an inflexion point, (3) horizontal asymptote at old ages, (4) logical behavior (e.g., W

_{Shr}should be zero at age zero), (5) theoretical basis and (6) base-age invariance. Fulfillment of these attributes depends on both the construction method and the base model used to develop the curves, and it cannot always be achieved.

_{shr}is the total shrub load; t is the age considered as the time elapsed since the last natural or anthropic disturbance and a

_{iH}, a

_{iK}and a

_{iB}are parameters to be estimated.

_{i}of each base model are estimated by non-linear regression using the entire W

_{Shr}-t database, obtaining the so-called guide curve, which is the “average” line representing the data used. The same number of families of curves as there are parameters in the base model (two in the case of the Hossfeld model and three in the other two cases) can be obtained from the guide curve. The curves of a given sample plot for each family are obtained from the guide curve equation by varying one site-dependent parameter (W

_{Shr}-t values of the sample plot) and holding the others constant.

_{Shr}-t data of the sampling plots.

_{i}, $\widehat{{Y}_{i}}$ and $\overline{Y}$ are the observed, predicted and mean values of the dependent variable, and n is the number of observations used to fit the equation.

#### 2.5. Compatible System for the Shrub Fuel Complex

^{−2}) at a reference age of 10 years.

_{Shr_G1_dead}, W

_{Shr_G1_live}, W

_{Shr_G1}and W

_{Shr_G23}) for each community. Allometric models ($y={b}_{0\xb7}{X}_{i}^{{b}_{i}}$) for estimating fuel loads were tested for all the biomass fractions by considering the site index (SI) and age (t) as independent variables to be tested.

_{Shr_G1_dead}and W

_{Shr_G1_live}estimates must equal W

_{Shr_G1}estimates or the sum of W

_{Shr_G1}and W

_{Shr_G23}must equal W

_{Shr}estimates). Therefore, in the first step, the equation of each fuel fraction of each shrub community was fitted separately, and the complete system of four equations (one for fraction) was then fitted simultaneously for each shrub community to guarantee additivity:

_{i}= b

_{ig23}− b

_{ig1}; and the equation for estimating the coarse fuel load was as follows:

_{i}= c

_{ig1_live}− c

_{ig1_dead}; and the following equation was fitted to estimate the live fine fuel load:

^{®}[97].

#### 2.6. Assessing the Effect of Topographic and Climatic Variables on Site Quality

## 3. Results

#### 3.1. Development of Total Fuel Load Growth Curves

_{Shr}-t growth curves in both communities.

_{0}, a

_{1}or a

_{2}, with the other two held constant). The three families of curves for each community overlaid on the observed W

_{Shr}-t data are shown in Figure 2. The curves represented correspond to W

_{Shr}values of 1.5, 3, 4.5, 6 and 7.5 kg m

^{−2}at a reference age (t

_{ref}) of 10 years for gorse-dominated communities and 1, 2.5, 4, 5.5 and 7 kg m

^{−2}at a reference age (t

_{ref}) of 10 years for high heath-dominated communities.

_{0}varies when considered site-dependent results in anamorphic curves with different asymptotes (Figure 2, left); when parameter a

_{1}is considered site-dependent, the family of curves is polymorphic with a common asymptote (Figure 2, middle), and the same occurs when parameter a

_{2}is considered site-dependent (Figure 2, right).

_{1}depends on the site, resulting in a polymorphic family. The mathematical expression of the curve for a sample plot, in which the total shrub fuel load (W

_{Shr_t}

_{1}) at a given age (t

_{1}) is known, would thus be obtained as follows:

_{ref}) of 10 years, would be obtained as follows:

_{ref}) of 10 years was selected to improve the accuracy of predictions, reducing the prediction bias associated with sample plots, where t differs greatly from the reference t

_{ref}[104].

^{−2}at a reference age of 10 years, with a mean value of 3.79 kg m

^{−2}(sd = 1.15 kg m

^{−2}), while for the high heath-dominated formations, the site index ranged from 1.01 to 8.25 kg m

^{−2}with a mean of 3.68 kg m

^{−2}(sd = 1.61 kg m

^{−2}).

#### 3.2. Effect of Topographic and Climatic Variables on Site Quality

^{−2}, Ps in mm and Tcm in °C; I

_{Ps}is a dummy variable with a value of 1 for Ps greater than 167 mm and 0 otherwise; I

_{Tcm}

_{1}is a dummy variable with a value of 1 for Tcm greater than 8.5 °C and 0 otherwise, and I

_{Tcm}

_{2}is a dummy variable with value 1 for Tcm less than 8.5 °C and 0 otherwise.

^{−2}for every 10 mm increase in precipitation. In the case of Tcm, for equal Ps values, an increase in its value implies a moderate increase in SI until a threshold of 8.5 °C, beyond which SI growth increases considerably. These thresholds and growth rates should be considered with caution as the predictive capacity of the model is limited (ME = 0.11; RMSE% = 25.58%), although the model is useful from an explanatory point of view.

#### 3.3. Compatible System for the Shrub Fuel Complex

_{Shr_G1_dead}, W

_{Shr_G1_live}, W

_{Shr_G1}and W

_{Shr_G23}) was fitted for each shrub community. The mathematical expression of the system (Equations (6) to (9)) fitted to gorse-dominated and high heath-dominated shrub communities and the goodness-of-fit statistics are given in Table 4. All parameters were found to be significant at the 5% level.

_{Shr_G1_dead}) and 94% for coarse shrub load (W

_{Shr_G23}); for the gorse-dominated community, the respective percentages varied between 48% of live fine shrub load (W

_{Shr_G1_live}) and 86% of coarse shrub load (W

_{Shr_G23}).

_{Shr_G1}/W

_{Shr}) and the dead ratio (W

_{Shr_G1_dead}/W

_{Shr_G1}) with the age and the site index of the shrub community or the mean and current fuel load annual increment of each of the five fractions (W

_{Shr_G1_dead}, W

_{Shr_G1_live}, W

_{Shr_G1}, W

_{Shr_G23}and W

_{Shr}). The mathematical expression of the fine and dead ratios would be as follows:

_{1}and b

_{2}are both positive for both shrubland communities under study (Table 3), the fine ratio decreases with age and also with increasing site quality (increasing SI). This behavior is biologically expected, as the fine shrub load increases in absolute terms with age and site quality, but does so at a slower rate than the coarse shrub load (shrub fraction with no upper limit of thickness). On the other hand, as c

_{2}is negative and c

_{1}is zero for both shrub communities analyzed (Table 3), the dead ratio varies exclusively with the age of the community, increasing as the latter increases. In our study, the fine ratio varied in gorse-dominated communities between 0.292 and 0.989 (mean value = 0.673), and in high heath-dominated communities, the range was similar (0.266–0.988) with a mean value slightly higher (0.727).

^{−2}year

^{−1}) in the shrub load of any of the fractions analyzed can be obtained by dividing the expression of the corresponding fitted equation by age; likewise, the current annual increment (CAI, kg m

^{−2}year

^{−1}) can be obtained by calculating the derivative of the same equation relative to age. For example, the MAI and CAI of the total shrub load (W

_{Shr}) can be obtained as follows:

_{Shr_G1}–age curves and the W

_{Shr_G1_dead}–age curves for five different site indices for both shrubland communities analyzed (1.5, 3, 4.5, 6 and 7.5 kg m

^{−2}of W

_{Shr}at a reference age of 10 years for gorse-dominated communities and 1, 2.5, 4, 5.5 and 7 kg m

^{−2}of W

_{Shr}at a reference age of 10 years for high heath-dominated communities).

_{Shr_G1}is reached earlier in the best site qualities for both communities but is more premature in high heath-dominated communities (4 years for a SI of 7 kg m

^{−2}) than in gorse-dominated communities (6 years for a SI of 7.5 kg m

^{−2}). Thereafter, the decrease is more pronounced in the best site qualities, and the curves are even cut at advanced ages (more than 20 and 16 years for gorse- and high heath-dominated communities, respectively).

_{Shr_G1_dead}curves, the pattern is also similar in both shrubland communities, with an initial growth to a higher maximum in the better site qualities and a posterior decrease. The maximum is reached earlier in the best site qualities and, again, is more premature in high heath-dominated communities (5 years for an SI of 7 kg m

^{−2}in these communities vs. 9 years for an SI of 7.5 kg m

^{−2}in gorse-dominated communities). It is important to emphasize that these W

_{Shr_G1_dead}maxima are delayed with respect to the W

_{Shr_G1}maxima, and these delays are longer as site quality becomes worse (from 3 years for an SI of 7.5 kg m

^{−2}to 8 years for an SI of 3 kg m

^{−2}in the gorse-dominated communities and from 1 year for an SI of 7 kg m

^{−2}to 3 years for an SI of 2.5 kg m

^{−2}for the high heath-dominated communities).

## 4. Discussion

#### 4.1. Effect of Topographic and Climatic Variables on Site Quality

#### 4.2. Biomass Accumulation and Mean Annual Increment (MAI)

#### 4.3. Fine Ratio

#### 4.4. Dead Ratio

#### 4.5. Implications for Fuel Hazard Management

_{Shr}and the age of a sample in the target community are required to improve substantially the accuracy in the estimation of different parameters related to shrubland fuel load and predict their change with age. Our results also offer guidance for fuel treatment planning, based on the age at which different fuel fractions reach the maximum fuel load. They suggest conducting fuel reduction treatments in relatively young stands and, due to the fast recovery to high fuel loads, repeating the treatments frequently. However, fuel managers should balance information on the rate of fuel accumulation with the local risk of ignitions to optimize the interval between interventions.

## 5. Conclusions

_{Shr}; therefore, future studies may increase the effort in older stands to better clarify the longer-term fuel load dynamics.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Geographical location of the 316 inventory plots in Galicia encompassing two shrub communities (

**left**). Layout of inventory plot showing transects and location of four destructive sampling subplots (quadrats in green) in each inventory plot (

**right**).

**Figure 2.**Three different families of W

_{Shr}-t curves generated from the guide curve (red line) of the base model proposed by Korf (cited in Ludqvist [90]) for five different site indexes (1.5, 3, 4.5, 6 and 7.5 kg m

^{−2}at a reference age of 10 years for gorse-dominated communities and 1, 2.5, 4, 5.5 and 7 kg m

^{−2}at a reference age of 10 years for high heath-dominated communities).

**Figure 3.**W

_{Shr_G1}–age curves (

**left column**) and W

_{Shr_G1_dead}–age curves (

**right column**) for the SI corresponding to the guide curve (red) and five different site indexes: 1.5, 3, 4.5, 6 and 7.5 kg m

^{−2}of W

_{Shr}at a reference age of 10 years for gorse-dominated communities (

**upper row**) and 1, 2.5, 4, 5.5 and 7 kg m

^{−2}of W

_{Shr}at a reference age of 10 years for high heath-dominated communities (

**lower row**).

**Table 1.**Mean values of the standing shrub fuel strata characteristics. Std. dev. = standard deviation, n = number of plots, $\overline{{h}_{Shr}}$ = shrub height, Cov

_{Shr}= shrub cover, W

_{Shr}= total shrub fuel loadW

_{Shr_G23}= coarse shrub fuel load, W

_{Shr_G1}= fine shrub fuel load, W

_{Shr_G1_dead}= dead fine shrub fuel load, W

_{Shr_G1_live}= live fine shrub fuel load. See definitions in the text. Ea = high heath-dominated communities and Ue = gorse-dominated communities.

Variable | Statistic | Ue | Ea |
---|---|---|---|

n | 165 | 102 | |

$\overline{{h}_{Shr}}$ | Mean | 123.56 | 107.28 |

(cm) | Std. dev. | 67.43 | 69.72 |

Cov_{Shr} | Mean | 8589 | 89.75 |

(%) | Std. dev. | 22.53 | 17.49 |

t | Mean | 9.36 | 8.00 |

(years) | Std. dev. | 4.69 | 5.98 |

W_{Shr} | Mean | 3.49 | 2.40 |

(kg m^{−2}) | Std. dev. | 1.46 | 1.68 |

W_{Shr_G23} | Mean | 1.33 | 0.91 |

(kg m^{−2}) | Std. dev. | 1.06 | 1.11 |

W_{Shr_G1} | Mean | 2.16 | 1.49 |

(kg m^{−2}) | Std. dev. | 0.71 | 0.74 |

WS_{hr_G1_dead} | Mean | 0.85 | 0.41 |

(kg m^{−2}) | Std. dev. | 0.36 | 0.27 |

W_{Shr_G1_live} | Mean | 1.31 | 1.08 |

(kg m^{−2}) | Std. dev. | 0.47 | 0.52 |

**Table 2.**Summary statistics of the physiographical and climate variables extracted from the raster datasets corresponding to the 267 sample plot locations. Standard deviations are shown in parentheses. Different letters indicate significant differences between mean values (α = 5%).

Species | Elevation | Slope | P | Ps | T | Twm | Tcm | P/T | $\overline{\mathit{P}\mathit{s}}/\mathit{T}\mathit{w}\mathit{m}$ | Rd | Rds |
---|---|---|---|---|---|---|---|---|---|---|---|

U. europaeus | 572a | 11a | 1560a | 137a | 11.8a | 18.3a | 6.5a | 132.5a | 2.5a | 10.7a | 4.8a |

n = 156 | (242) | (7) | (239) | (19) | (1.2) | (1.0) | (1.5) | (20.0) | (0.4) | (2.2) | (1.7) |

range | 30–1113 | 0–35 | 1036–1975 | 92–186 | 8.6–14.6 | 15.5–20.7 | 3–9.5 | 89.3–181.7 | 1.6–3.8 | 9.6–11.9 | 3.9–5.9 |

E. australis | 1005b | 18b | 1449b | 144b | 10.0b | 17.9b | 3.5b | 150.0b | 2.7b | 9.8b | 4.8a |

n = 102 | (281) | (9) | (277) | (28) | (1.3) | (1.3) | (1.4) | (43.9) | (0.7) | (2.1) | (1.7) |

range | 497–1710 | 2–43 | 873–2004 | 91–211 | 6.1–12.3 | 14.4–20.8 | −0.6–6.2 | 71.0–272.5 | 1.5–4.1 | 8.6–10.7 | 3.9–5.9 |

**Table 3.**Parameter estimates and goodness-of-fit statistics for the three base models fitted to W

_{Shr}-t data from gorse- and high heath-dominated communities.

Gorse-Dominated Communities | |||||
---|---|---|---|---|---|

Model | a_{0} | a_{1} | a_{2} | ME | RMSE (kg m ^{−2}) |

Hossfeld | 1.3634 | 0.3704 | --- | 0.4792 | 1.0624 |

Korf | 3.0581 | −3.6646 | 0.3291 | 0.5083 | 1.0323 |

Bertalanffy–Richards | 8.1410 | 0.0522 | 0.8504 | 0.5073 | 1.0333 |

High Heath-Dominated Communities | |||||

Model | a_{0} | a_{1} | a_{2} | ME | RMSE(kg m^{−2}) |

Hossfeld | 2.8215 | 0.2965 | --- | 0.7321 | 0.8801 |

Korf | 5.1644 | −6.5425 | 0.2070 | 0.7538 | 0.8436 |

Bertalanffy–Richards | 9.6599 | 0.0498 | 1.2765 | 0.7529 | 0.8453 |

**Table 4.**Mathematical expressions and goodness-of-fit statistics of the system of four equations fitted to estimate the biomass of the shrub fuel complex in gorse- and high heath-dominated communities using site index and age as independent variables.

Gorse-Dominated Communities | ||
---|---|---|

Model | ME | RMSE (kg m ^{−2}) |

${W}_{Shr\_G1}=\frac{{W}_{Shr}}{1+exp\left[-4.2327+1.0749\xb7log\left(SI\right)+0.9766\xb7log\left(t\right)\right]}$ | 0.6934 | 0.3949 |

${W}_{Shr\_G23}=\frac{{W}_{Shr}\xb7exp\left[-4.2327+1.0749\xb7log\left(SI\right)+0.9766\xb7log\left(t\right)\right]}{1+exp\left[-4.2327+1.0749\xb7log\left(SI\right)+0.9766\xb7log\left(t\right)\right]}$ | 0.8630 | 0.3949 |

${W}_{Shr\_G1\_dead}=\frac{{W}_{Shr\_G1}}{1+exp\left[0.9153-0.2263\xb7log\left(t\right)\right]}$ | 0.5699 | 0.2382 |

${W}_{Shr\_G1\_live}=\frac{{W}_{Shr\_G1}\xb7\left(exp\left[0.9153-0.2263\xb7log\left(t\right)\right]\right)}{1+exp\left[0.9153-0.2263\xb7log\left(t\right)\right]}$ | 0.4803 | 0.3426 |

High Heath-Dominated Communities | ||

Model | ME | RMSE(kg m^{−2}) |

${W}_{Shr\_G1}=\frac{{W}_{Shr}}{1+exp\left[-5.4793+1.0431\xb7log\left(SI\right)+1.5572\xb7log\left(t\right)\right]}$ | 0.8590 | 0.2792 |

${W}_{Shr\_G23}=\frac{{W}_{Shr}\xb7exp\left[-5.4793+1.0431\xb7log\left(SI\right)+1.5572\xb7log\left(t\right)\right]}{1+exp\left[-5.4793+1.0431\xb7log\left(SI\right)+1.5572\xb7log\left(t\right)\right]}$ | 0.9377 | 0.2792 |

${W}_{Shr\_G1\_dead}=\frac{{W}_{Shr\_G1}}{1+exp\left[1.41553-0.2163\xb7log\left(t\right)\right]}$ | 0.7291 | 0.1436 |

${W}_{Shr\_G1\_live}=\frac{{W}_{Shr\_G1}\xb7\left(exp\left[1.4155-0.2163\xb7log\left(t\right)\right]\right)}{1+exp\left[1.4155-0.2163\xb7log\left(t\right)\right]}$ | 0.7715 | 0.2505 |

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## Share and Cite

**MDPI and ACS Style**

Vega, J.A.; Álvarez-González, J.G.; Arellano-Pérez, S.; Fernández, C.; Ruiz-González, A.D.
Site Quality Models and Fuel Load Dynamic Equation Systems Disaggregated by Size Fractions and Vegetative States in Gorse and High Heath Shrublands in Galicia (NW Spain). *Fire* **2024**, *7*, 126.
https://doi.org/10.3390/fire7040126

**AMA Style**

Vega JA, Álvarez-González JG, Arellano-Pérez S, Fernández C, Ruiz-González AD.
Site Quality Models and Fuel Load Dynamic Equation Systems Disaggregated by Size Fractions and Vegetative States in Gorse and High Heath Shrublands in Galicia (NW Spain). *Fire*. 2024; 7(4):126.
https://doi.org/10.3390/fire7040126

**Chicago/Turabian Style**

Vega, José A., Juan Gabriel Álvarez-González, Stéfano Arellano-Pérez, Cristina Fernández, and Ana Daría Ruiz-González.
2024. "Site Quality Models and Fuel Load Dynamic Equation Systems Disaggregated by Size Fractions and Vegetative States in Gorse and High Heath Shrublands in Galicia (NW Spain)" *Fire* 7, no. 4: 126.
https://doi.org/10.3390/fire7040126