# Synthesis of Vegetation Indices Using Genetic Programming for Soil Erosion Estimation

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Study Area

^{2}, and its elevation varies between 0 and 1876 m a.s.l. This watershed has two large alluvial valleys at 300 m a.s.l.: Guadalupe and Ojos Negros; and the city of Ensenada is located in a coastal plain at an average of 50 m a.s.l.

## 4. Methodology

#### 4.1. Collecting Field Samples

- Superficial cover percentage (g). This was visually determined from 10 cm around a dropped weight, otherwise known as the micro-plot. Each micro-plot was labeled with one of five different classifications according to the percentage of ground covered by vegetation or rocks: 0 = 0–1%, 1 = 1–25%, 2 = 25–50%, 3 = 50–75%, and 4 = 75–100%.
- Percentage (p) and height (h) of aerial vegetation cover. The percentage, p, is obtained using a similar method as for the superficial cover percentage: labeling a micro-plot according to the same five classifications. However, the center of the micro-plot is not defined by a dropped weight, but by the wire from which it hangs. The height, h, of the aerial cover for the sampling point is defined by the plant or its ramifications that are closest to the ground (without touching it) and touch the wire.
- Roughness (r) and underground biomass (b). Roughness, r, is evaluated according to the empirical method proposed by [41] and Tables 5-5 and 5-6 from the RUSLE manual [7]. Underground biomass, b, is inferred using the primary productivity method defined by [45]. For this study, b was assumed to be uniform.

#### 4.2. Satellite Image Acquisition and Correction

- Radiometric correction diminishes the effects of sensor miscalibration. It also corrects the distortions caused by the angle between the Sun and the satellite over the study area. In order to correct the images produced by the satellite, researchers employ the parameters and methodology published by NASA for instrument recalibration and pixel reflectance values [49]. Figure 4a shows two non-corrected images that were taken at different times; by applying the radiometric correction (Figure 4c), the pixel values are normalized, and the results are comparable.
- Geometric correction adjusts the satellite images to the geographic coordinate system. To perform this adjustment, Ground Control Points (GCP) located in the study area were obtained from the U.S. Geological Survey (USGS) website [46].

#### 4.3. Associating Satellite Images and Field Data

#### 4.4. Applying Conventional VIs

#### 4.5. Applying the Genetic Programming Algorithm

#### 4.5.1. Definition of the Terminal and Function Sets

- Spectral bands represent the median value of the $3\times 3$ pixel window defined in Section 4.3, which was extracted from each satellite image band: Blue (B), Green (G), Red (R), Near-Infrared (NIR), Shortwave Infrared 1 (SWIR1), and Shortwave Infrared 2 (SWIR2).
- Spectral angles are the angles formed by each vertex in the electromagnetic spectrum for the satellite image [68]. For example, ${\beta}_{G}$ corresponds to the combination of pixels from the B, G, and R bands. The formula for this angle is:$${\beta}_{G}=co{s}^{-1}(\frac{{a}^{2}+{b}^{2}-{c}^{2}}{2ab})\phantom{\rule{0.222222em}{0ex}}radians,$$
- Soil line parameters: This is the relationship between the R and NIR bands [70]. When these bands are graphed in a dispersion graph, they tend to group pixels above a numeric threshold, which is called the soil line. For this study, the slope and the intersect of the soil line are included in the primitives’ set as a and b in Table 3.
- Best-performing conventional VIs: The five better performing conventional VIs in Section 4.4 were selected. This performance is based in statistical significance [71], which is the probability that an index has a random correlation value $|{r}_{x,y}|$. For scientific research, the most common statistical significance value is 5%, which means that $|{r}_{x,y}|$ is significant if there is a 5% or less probability that the index has happened by chance. The statistical significance, $SS$, for the correlation coefficient, r, of a sample with N data is [71]:$$SS=\frac{r}{\sqrt{\frac{1-{r}^{2}}{N-2}}}$$The field samples from both watersheds were used in this research, so the sample size for the training dataset was $N=102$. Therefore, correlation coefficients greater than 0.39 will be statistically significant for this experiment.Besides the five indices, it was decided to include the $NDVI$ and $EVI$ indices since these are the most used for scientific applications.
- Arithmetic operators: The function set is formed by the basic arithmetic operators (+, −, and ×) since these are the operators usually employed for VIs. For this work, division (/) is substituted by the compound operator Ratio Spectral Index (RSI) (see below).
- Compound operators: These represent complete arithmetic structures that have been previously defined. This research includes two of the most-employed indices, NDVI and RVI. The first structure is the Normalized Difference Spectral Index (NDSI) [72], which corresponds to NDVI; while the second structure is the Ratio Spectral Index (RSI), which corresponds to RVI. The definitions for these structures are:$$NDS{I}_{[i,j]}=\frac{{R}_{i}-{R}_{j}}{{R}_{i}+{R}_{j}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}},$$$$RS{I}_{[i,j]}=\frac{{R}_{i}}{{R}_{j}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}},$$

#### 4.5.2. Fitness Function

#### 4.5.3. Control Parameters and Stop Criteria

#### 4.6. Results’ Evaluation

#### 4.7. Generating the C Factor Map and the Erosion Map

## 5. Experiments and Results

#### 5.1. Implementing the GP Algorithm

- Initial population: This section generates the initial population and evaluates it according to the fitness function (correlation coefficient) defined in Section 4.5. Population individuals may be generated according to one of three methods: full, grow, and ramped half-and-half. For the full method, the syntactic tree will include all possible nodes at each level (Figure 6a). The grow method generates a random number of nodes at each level, which may generate unbalanced trees (Figure 6b). The ramped Half-and-half method generates half of the tree using the full method and the other half using the grow method. This research employs this last method because it generates trees with a wide variety of sizes and shapes.
- New generation criteria: This section creates a new generation of individuals (children) by either applying genetic mutation operators or crossover (genetic reproduction) to the individuals of the previous population (parents). The following methods may be used to select the next generation’s parents: roulette, stochastic universal sampling, tournament, and lexicographic parsimony pressure tournament [35]. This last method is a special case of the tournament method where competing individuals with the same performance, following the fitness function, are selected according to the one that has the least complexity or the fewer number of nodes. This gives the advantage to simpler VIs that compete against conventional indices and is the reason why lexicographic parsimony pressure tournament was selected as the new generation method.Once the parents have been selected, either mutation or crossover is applied depending on a user-defined probability parameter. Usually, the mutation’s probability is lower than the crossover’s. For this experiment, the crossover probability was 0.7, while the mutation probability was 0.3. The new individuals were evaluated according to the fitness function, and then, these individuals and the best parents became the parents for the next generation. For this research, only the best individual in each generation was preserved for the next generation; this is called the elitist parameter. This procedure was repeated until a stop criteria was met, which for this research was when 50 generations had been produced. This number was employed because the algorithm stopped producing better individuals after 35 iterations.
- Population management: Also called a code bloat in GP parlance, it limits the complexity of individuals. Three parameters are used to define this section: tree depth, maximum dynamic depth, and real maximum depth. If a new individual has a depth greater than the one defined at the beginning, the individual is automatically discarded regardless of the performance provided by the fitness function. This avoids an uncontrolled growth of the syntactic trees that generate the population. The tree depth has two possible values: strict means that the previous rule always applies, while dynamic allows one to preserve individuals that have a better performance than any other previous individual. The dynamic approach verifies that the new individual’s depth is greater than the maximum dynamic depth, but smaller than the real maximum depth. In such a case, the algorithm allows the new individual to survive and sets the maximum dynamic depth to the new individual’s depth value. If later on, there is a better individual with a smaller depth, the previous individual is discarded, and the maximum dynamic depth moves down to the new individual’s depth. In this research, individuals represent a VI in a syntactic tree. Figure 5 shows that the depth of the NDVI syntactic tree was three. The greater the depth of the tree, the greater the complexity; therefore, it is desirable that the GP-synthesized indices have a similar complexity to the conventional indices. Thus, the maximum dynamic depth was set to three, and the real maximum depth to four.

#### 5.2. Methodology Performance Analysis

- The synthesized indices from one watershed produce a good approximation when applied to the other watershed.
- The combined synthesized indices from both watersheds produce a good approximation when they are later applied to each watershed.

#### 5.2.1. First Hypothesis

#### 5.2.2. Second Hypothesis

#### 5.3. Erosion Rate Analysis for the Todos Santos Watershed

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) Map showing the Rio Martin watershed in Zaragoza, Spain (permission from Matta Trabucchi). (

**b**) Field sites’ location in Rio Martin watershed.

**Figure 3.**Flowchart of the methodology used to define the C factor from the VIs generated by the Genetic Programming (GP) algorithm.

**Figure 4.**Atmospheric effects over the satellite image for the Rio Martin watershed. (

**a**) Non-corrected image, (

**b**) image with atmospheric correction, and (

**c**) image with radiometric correction.

**Figure 6.**Methods to grow a population using GPLab. (

**a**) Full: all possible nodes at each level. (

**b**) Grow: random number of nodes at each level. The ramped half-and-half method means that half the population follows the full method and the other half the grow method.

**Figure 7.**Performance comparison between the conventional and synthetic VIs for the different experiments performed. (

**a**) Todos Santos, (

**b**) Rio Martin, and (

**c**) combined data from both watersheds.

**Figure 8.**Frequency Of Use (FOU) for the elements that form the primitive set for the combined field data for both watersheds using the LandSat-TM sensor. The maximum possible value is 100%.

**Figure 9.**C factor maps produced by the best performing CGPVIs: (

**a**) CGPVI26 and (

**b**) CGPVI1. The pixel histogram for each map is included.

**Figure 10.**C factor maps produced by the best-performing conventional VIs using combined field data: (

**a**) GVI3and (

**b**) NDVI. The pixel histogram for each map is included.

**Figure 11.**Erosion maps generated by the best-performing synthetic indices. (

**a**) Map based on CGPVI1. (

**b**) Map based in the conventional GVI3. For a better visualization, scale has been transformed for a better comparison.

Band | Name | Wavelength |
---|---|---|

1 | Blue light (B) | 0.45–0.52 $\mathsf{\mu}$m |

2 | Green light (G) | 0.53–0.60 $\mathsf{\mu}$m |

3 | Red light (R) | 0.63–0.69 $\mathsf{\mu}$m |

4 | Near-Infrared (NIR) | 0.76–0.90 $\mathsf{\mu}$m |

5 | Shortwave Infrared Channel 1 (SWIR1) | 1.55–1.75 $\mathsf{\mu}$m |

6 | Thermal Infrared (TIR) | 10.4–12.5 $\mathsf{\mu}$m |

7 | Shortwave Infrared Channel 2 (SWIR2) | 2.08–2.35 $\mathsf{\mu}$m |

Indices | Equation | Ref. |
---|---|---|

RVI1 | $\frac{NIR}{R}$ | [50] |

RVI2 | $\frac{NIR}{G}$ | based on [50] |

RVI3 | $\frac{NIR}{SWIR1}$ | based on [50] |

RVI4 | $\frac{SWIR1}{SWIR2}$ | based on [50] |

RVI5 | $\frac{SWIR1}{R}$ | based on [50] |

RVI6 | $\frac{NIR}{SWIR2}$ | based on [50] |

NDVI | $\frac{NIR-R}{NIR+R}$ | [51] |

IPVI | $\frac{NIR}{NIR+R}$ | [52] |

DVI | $NIR-R$ | [53] |

SAVI | $(1+L)\frac{NIR-R}{NIR+R+L}$ where L is a correction factor between 0 and 1 | [54] |

SAVI2 | $\frac{NIR}{R+b/m}$ where m and bare the slope and intercept of the soil line. These parameters are used in the next six indices as well | [55] |

MSAVI | same that SAVI, but $L=1-2m\xb7NDVI\xb7WDVI$ | [56] |

MSAVI2 | $0.5[(2NIR+1)-\sqrt{{(2NIR+1)}^{2}-8(NIR-R)}]$ | [56] |

TSAVI | $\frac{m(NIR-m\xb7R-b)}{R-m\xb7NIR-m\xb7b+X(1+{m}^{2})}$ | [57] |

OSAVI | $\frac{NIR-R}{NIR+R+\gamma}$ | [58] |

WDVI | $NIR-m\xb7R$ | [59] |

PVI | $\frac{NIR-m\xb7R-b}{\sqrt{{m}^{2}+1}}$ | [60] |

GEMI | $\eta (1-0.25\eta )-(R-0.125)/(1-R)$ where $\eta =[2(NI{R}^{2}-{R}^{2})+1.5NIR+0.5R]/(NIR+R+0.5)$ | [61] |

ARVI | $(NIR-rb)/(NIR+rb)$; where $rb=R-(B-R)$ | [62] |

EVI | $G[(NIR-R)/(NIR+C1\xb7R-C2\xb7B+L)]$ where $G=2.5;C1=6;C2=7.5;L=1$. | [63] |

GVI1 | $-0.2848B-0.2435G-0.5436R+0.7243NIR+0.0840SWIR1-0.1800SWIR2$ | [64] |

GVI2 | $-0.2778B-0.2174G-0.5508R+0.7220NIR+0.0733SWIR1-0.1648SWIR2-0.7310$ | [64] |

GVI3 | $-0.3344B-0.3544G-0.4556R+0.6966NIR+0.0242SWIR1-0.2630SWIR2$ | [64] |

NDWI | $\frac{NIR-SWIR1}{NIR+SWIR1}$ | [65] |

NDII | $\frac{SWIR1-SWIR2}{SWIR1+SWIR2}$ | [66] |

SIWSI | $\frac{NIR-SWIR2}{NIR+SWIR2}$ | [67] |

ANIR | ${\beta}_{NIR}=co{s}^{-1}(\frac{{a}^{2}+{b}^{2}-{c}^{2}}{2ab})$ where a, b, and c are Euclidean distances between R, NIR and SWIR | [68] |

SASI | ${\beta}_{SWIR1}\xb7(SWIR2-NIR)$ where ${\beta}_{SWIR1}$ is defined like ${\beta}_{NIR}$, but a, b, and c are Euclidean distances between NIR, SWIR1 and SWIR2 | [68] |

SANI | ${\beta}_{SWIR1}\xb7\frac{SWIR2-NIR}{SWIR2+NIR}$ ${\beta}_{SWIR1}$ is defined like in SASI | [68] |

**Table 3.**Primitives’ set elements for the Experiments Section. NDSI, Normalized Difference Spectral Index; RSI, Ratio Spectral Index.

Elements | Description |
---|---|

Terminals | |

B, G, R NIR, SWIR1, SWIR2 | Satellite image’s spectral bands |

${\beta}_{G},\phantom{\rule{0.222222em}{0ex}}{\beta}_{R},\phantom{\rule{0.222222em}{0ex}}{\beta}_{NIR},\phantom{\rule{0.222222em}{0ex}}{\beta}_{SWIR1}$ | Calculated spectral angles from the available bands |

RVI1, RVI2, RVI4, RVI5 GEMI, NDVI, EVI | Best-performing conventional indices |

a, b | Slope and the intersect of the soil line |

Functions | |

+, −, × | Arithmetic operators |

NDSI, RSI | Compound operators |

**Table 4.**Confusion matrix showing the measured (correlation) performance $|{r}_{x,y}|$ when applying the Todos Santos synthesized indices to the Rio Martin watershed, and vice versa.

Todos Santos | Rio Martin | |
---|---|---|

Todos Santos | 0.633 ± 0.01 | 0.323 ± 0.010 |

Max. 0.65 | Max. 0.363 | |

Min. 0.614 | Min. 0.290 | |

Rio Martin | 0.329 ± 0.11 | 0.731 ± 0.044 |

Max. 0.401 | Max. 0.829 | |

Min. 0.137 | Min. 0.613 |

**Table 5.**Conventional vegetation indices (Table 2) with the best performance for the combined data from both watersheds. Performance was measured by using the correlation factor ($|{r}_{x,y}|$).

Index | GVI3 | NDVI | GEMI | RVI5 | RVI4 | SASI | NDII | OSAVI | RVI1 | TSAVI |
---|---|---|---|---|---|---|---|---|---|---|

$|{r}_{x,y}|$ | 0.541 | 0.538 | 0.525 | 0.503 | 0.437 | 0.349 | 0.341 | 0.328 | 0.309 | 0.302 |

**Table 6.**Synthesized VIs for the GP method using the combined data from both watersheds. Performance was measured through the correlation factor ($|{r}_{x,y}|$).

Index | Formula | $|{\mathit{r}}_{\mathit{train}}|$ | $|{\mathit{r}}_{\mathit{test}}|$ | Dif. |
---|---|---|---|---|

CGPVI1 | $RSI(RSI(b+R,RSI(NDVI,R)),G+2\xb7NDII)$ | 0.887 | 0.849 | 0.037 |

CGPVI2 | $(NDII+Bl)\times RSI({\beta}_{R},NDII)\times R\times {\beta}_{R}\times {\beta}_{R}$ | 0.884 | 0.787 | 0.097 |

CGPVI3 | $(R-NDII)\times (GEMI-NDII)\times RSI(B,EVI)$ | 0.890 | 0.828 | 0.061 |

CGPVI4 | $RSI(RVI4\times R,NDII)\times {\beta}_{R}\times {\beta}_{R}\times R\times RVI4$ | 0.878 | 0.811 | 0.067 |

CGPVI5 | ${\beta}_{SWIR1}\times {\beta}_{SWIR1}\times R\times G\times (R-G)$ | 0.861 | 0.899 | 0.038 |

CGPVI6 | $RSI(RSI(RSI(R,NDII),GEMI-m),NDVI)$ | 0.869 | 0.768 | 0.102 |

CGPVI7 | $RVI4\times RVI4\times {\beta}_{R}\times R\times R\times RSI({\beta}_{R},NDII)$ | 0.881 | 0.813 | 0.068 |

CGPVI8 | $R\times R\times RSI({\beta}_{R},NDII)\times RVI4\times RVI4$ | 0.875 | 0.809 | 0.066 |

CGPVI9 | $RSI({\beta}_{R},NDII)\times R\times R\times {\beta}_{R}\times {\beta}_{R}\times RVI4$ | 0.881 | 0.805 | 0.075 |

CGPVI10 | $RVI4\times RVI4\times R\times RSI({\beta}_{R},NDII)\times {\beta}_{R}\times {\beta}_{R}$ | 0.880 | 0.807 | 0.073 |

CGPVI11 | $RSI(RSI(R,NDVI),RVI4)\times RSI(RSI(R,RVI4),RVI4)$ | 0.882 | 0.840 | 0.042 |

CGPVI12 | $NDSI(NDII,SWIR1)\times R\times SWIR1\times NDSI(NDII,SWIR1)\times $ $NDSI({\beta}_{SWIR1},RVI5)$ | 0.898 | 0.851 | 0.048 |

CGPVI13 | $NDSI(R,NDII)\times B\times SWIR2$ | 0.879 | 0.829 | 0.050 |

CGPVI14 | $NDSI(RSI(NDII+R,RSI(NDII,R)),{\beta}_{G})$ | 0.878 | 0.840 | 0.038 |

CGPVI15 | $(RVI4-2\xb7NDVI)\times R\times RSI({\beta}_{R},NDII)$ | 0.877 | 0.795 | 0.083 |

CGPVI16 | ${\beta}_{R}\times {\beta}_{R}\times {\beta}_{R}\times RSI(G,NDII)\times R$ | 0.879 | 0.783 | 0.096 |

CGPVI17 | $RSI(R\times R\times RVI4\times {\beta}_{R},NDII)$ | 0.873 | 0.801 | 0.072 |

CGPVI18 | $RSI(RSI(R,NDVI),NDII)-RVI5-{\beta}_{R}-NDII$ | 0.872 | 0.766 | 0.105 |

CGPVI19 | $RSI(RSI(SWIR2,NDVI),NDVI)+RSI(RSI(R,NDII),NDVI)$ | 0.876 | 0.740 | 0.136 |

CGPVI20 | $RSI(RSI({\beta}_{R},NDII)+RSI({\beta}_{SWIR1},NDVI)+RSI(RSI(NIR,R),R)$ | 0.878 | 0.788 | 0.090 |

CGPVI21 | $NDSI(R,NDII)\times B\times NDSI(NIR,NDII\times R$ | 0.890 | 0.809 | 0.081 |

CGPVI22 | ${\beta}_{R}^{2}\times m\times R\times (RSI(R,NDII)+{\beta}_{R})$ | 0.881 | 0.832 | 0.049 |

CGPVI23 | $RSI(RSI({\beta}_{R},RSI(NDII,R)),RSI(RSI(NDVI,RVI4),{\beta}_{R}))$ | 0.877 | 0.769 | 0.108 |

CGPVI24 | $RSI({\beta}_{R}\times RVI4,NDII)\times {\beta}_{R}^{2}\times R\times R$ | 0.878 | 0.804 | 0.074 |

CGPVI25 | $RSI(R,NDVI)\times (SASI+{\beta}_{R})\times ({\beta}_{R}+RSI(NIR,NDII))$ | 0.881 | 0.785 | 0.096 |

CGPVI26 | $NDSI(NDSI(GVI3\times R,NDII),NDSI(R,RSI(NDII,R)))$ | 0.898 | 0.865 | 0.033 |

CGPVI27 | $RSI(RSI({\beta}_{R},RSI(NIR,R)),RSI(NDII,R))$ | 0.872 | 0.782 | 0.090 |

CGPVI28 | $RSI(RSI(RSI(RVI4,NDVI),RSI(b,R)),RSI(NDTI,$ $RSI(RVI4,{\beta}_{NIR})))$ | 0.873 | 0.751 | 0.122 |

CGPVI29 | $RVI4\times RVI4\times RSI({\beta}_{R},NDII)\times R\times R\times {\beta}_{R}\times {\beta}_{R}$ | 0.881 | 0.813 | 0.068 |

CGPVI30 | $(RVI4-NDVI)\times (RVI4\times R\times {\beta}_{R}\times {\beta}_{R}\times RSI({\beta}_{R},NDII)$ | 0.882 | 0.807 | 0.075 |

**Table 7.**Average and standard deviation for the erosion rate from the sampling points for the training set. Each row shows a different method to calculate C.

Method employed to calculate C | Erosion (Mg·km${}^{-2}$·year${}^{-1}$) |
---|---|

Field data | $76.6\pm 153.6$ |

Indices generated from field data from the Todos Santos and Rio Martin watersheds | |

CGPVI26 | $99.5\pm 216.8$ |

CGPVI1 | $111.1\pm 278.7$ |

GVI3 | $376.5\pm 780.3$ |

Spectral classification method | |

Spectral classification method | $170.6\pm 358.8$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Puente, C.; Olague, G.; Trabucchi, M.; Arjona-Villicaña, P.D.; Soubervielle-Montalvo, C. Synthesis of Vegetation Indices Using Genetic Programming for Soil Erosion Estimation. *Remote Sens.* **2019**, *11*, 156.
https://doi.org/10.3390/rs11020156

**AMA Style**

Puente C, Olague G, Trabucchi M, Arjona-Villicaña PD, Soubervielle-Montalvo C. Synthesis of Vegetation Indices Using Genetic Programming for Soil Erosion Estimation. *Remote Sensing*. 2019; 11(2):156.
https://doi.org/10.3390/rs11020156

**Chicago/Turabian Style**

Puente, Cesar, Gustavo Olague, Mattia Trabucchi, P. David Arjona-Villicaña, and Carlos Soubervielle-Montalvo. 2019. "Synthesis of Vegetation Indices Using Genetic Programming for Soil Erosion Estimation" *Remote Sensing* 11, no. 2: 156.
https://doi.org/10.3390/rs11020156