# Estimation of Soil Depth Using Bayesian Maximum Entropy Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Material and Methodology

#### 2.1. Research Material

#### 2.1.1. Research District

#### 2.1.2. Soil Depth

#### 2.2. Research Methodology

#### 2.2.1. Kriging Estimation

^{*}$({x}_{0})$. The method first presents the distance function of soil depth measurement (point data) in the semivariogram model and determines the weight coefficient of each point $({\lambda}_{0i})$ with unbiased optimization from semivariogram. The semivariogram measures things nearby tend to be more similar than things that are farther apart, revealing the strength of statistical correlation as a function of distance. The soil depth then can be estimated using a linear combination of the weight coefficient and the measured data. The semivariance (γ) can be calculated by Equation (1), where h is distance between two separated points, N(h) is the set of all pairwise Euclidean distances i − j = h, |N(h)| is the number of pairs in N(h).

- i.
- Calculate the semivariance and distance between two measured points (ranging from meters to kilometers), calculate the average of semivariance and distance of points in the district as the representative value of the district, connect the latter to obtain the experimental semivariogram, overlay it with commonly used theoretical semivariogram models (index, spherical, and Gaussian).
- ii.
- Organize and calculate characteristics of the Best Linear Unbiased Estimator (BLUE) possessed by the ordinary Kriging method. The term of “best” indicates that compared to other unbiased and linear estimators, the lowest variance of the estimate is given
- iii.
- Acquire data Z
_{1}of one point out of n-each soil depth measurements, estimate Z_{1}with Kriging method based on the remaining n − 1 points. Here, n (i.e., 8217) is the data points used in the Kriging model. - iv.
- Replace Z
_{1}with another point Z_{2}and repeat the same steps until all of the n-each points are estimated. - v.
- Compare the soil depth measurements and Kriging method estimates according to the four soil depth gradings.
- vi.
- Calculate the average estimation accuracy of the index, spherical, and Gaussian theoretical semivariogram models in each watershed.

#### 2.2.2. Bayesian Maximum Entropy Method (BME)

_{G}(x

_{map}), given general knowledge G, is calculated via the maximum entropy theory. The variable x

_{map}is a vector of points, x

_{soft}, x

_{hard}, and x

_{k}, representing the information of the soft and hard data points and unknown values at the estimation point, respectively. The expected information is expressed as Equation (2):

_{α}(x

_{map}), a set of functions of x

_{map}such as the mean and covariance moments. To obtain the prior PDF of f

_{G}(x

_{map}), the expectation of Equation (2) is maximized with consideration of g

_{α}(x

_{map}). Equation (3) is the object function if the Lagrange multipliers method (LMM) is adopted for the aforementioned maximization problem, in which μ

_{α}is the Lagrange multiplier and the E[g

_{α}(x

_{map})] is the expected value of g

_{α}(x

_{map}).

_{K}(x

_{k}|x

_{data}) is derived using Bayesian theory, resulting in the following equations:

_{data}is a pointer for a context of knowledge, and (x

_{k}|x

_{data}) stands for the possible values x

_{k}of the map in the context specified by x

_{data}. In this study, the soil depth measurements and the physiographic factor out of 5 m DEM are source of estimates information which can be expressed in formula $S:{X}_{\mathrm{data}}=\left({X}_{\mathrm{hard}},{X}_{\mathrm{soft}}\right)=\left({x}_{1},\dots ,{x}_{n}\right)$ according to information classification by S-KB; here ${X}_{\mathrm{hard}}=\left({x}_{1},\dots ,{x}_{{m}_{h}}\right)$ is the soil depth measurement of ${P}_{i}\left(i=1,2,\dots ,{m}_{h}\right)$; ${X}_{soft}=\left({x}_{{m}_{h}+1},\dots ,{x}_{n}\right)$ is the soil depth estimates on point ${P}_{i}\left(i={m}_{h}+1,\dots ,n\right)$. by the soil depth estimation model and soil depth relevant physiographic factor. The soil depth can be regarded as a space random field with the soil depth of any point in the field expressed by formula ${X}_{P}={X}_{s}$ with $p=\left(s\right)$ where s is the space coordinate. This study takes distribution characteristics of soil depth in space into account and express soil depth of every point in space with ${f}_{KB}$, the PDF; where KB is the knowledge base (KB) used when constructing this PDF.

_{k}is the class; [w

^{T}K(x

_{i}) + b] is the classifier; N is the number of data; and K is the kernel function. In the current study, the Gaussian radial basis function (RBF) kernel is used, as shown in Equation (7):

_{i}are the support vectors. LSSVM [14], instead of solving the QP problem, solves a set of linear equations by modifying the standard SVM, as described in Equation (8):

^{2}, MAPE, and hitting rate of the current soil depth grading, to assess accuracy of estimation by each model. An estimate is defined as “targeted” if both estimate and actual soil depth fall in the same soil depth grade. The formulae for the three indices are shown below:

#### 2.2.3. Method of Accuracy Calculation

## 3. Results and Discussion

#### 3.1. Existing Soil Map in Hsinchu District

#### 3.2. Kriging Method

^{−}

^{3}and 1.186 × 10

^{−}

^{3}, respectively.

#### 3.3. BME

#### 3.3.1. Soft Data and Covariance Model

^{2}. Equation (15) is a good expression of the characteristics of soil depth residual in spatial distribution as its AIC value is −5731.

#### 3.3.2. Estimates

#### 3.4. Comparison of Estimation Models

## 4. Conclusions

- The Kriging method is commonly used for space estimation. It suffers in two aspects: first, the statistical assumption of normally distributed estimation data may be not the case of actual soil depth data; secondly, the Kriging method focuses on space distribution characteristics and does not take physiographic factors into account. In spite of being better than existing soil depth distribution diagrams available now in Taiwan, its accuracy rate is a mere 40%.
- The BME method incorporates both hard data of soil depth measurement and physiographic factor (soft data)-based soil depth estimations to take both natural environmental factors and space distribution characteristics into account at the same time. The BME method not only performs without the normal distribution assumption, but also comes out with much better estimation (80%) than that of the Kriging method.
- The soil depth distribution map of Hsinchu district in Taiwan produced by the BME method in this study gives soil depth estimation at grid points with an error range of a mere ±5.62 cm. This is acceptable for current soil depth grading standards and may be adopted for districts without soil depth data. This is of great help for disaster prevention and land management of slopeland in Taiwan.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Soil Depth Grade | Ratings |
---|---|

Very deep | >90 cm |

Deep | 50–90 cm |

Shallow | 20–50 cm |

Very shallow | <20 cm |

Physiographic Factor | Correlation with Soil Depth | Distribution Diagram |
---|---|---|

Slope | Steep slopes tend to have thinner soils which are hard to retain, while gradual slopes may have more pedogenesis and pileup | |

Aspect | Different aspects may feature large differences in solar radiation, water evaporation, and soil moisture. For example, a sunny slope may have greater solar radiation and water evaporation and thinner soil due to more erosion by wind and rainfall than the sunless side | |

Plan curvature | Perpendicular to the direction of the maximum slope which is related to the convergence and dispersion of water flowing through the surface | |

Profile curvature | Parallel to the direction of the maximum slope, that is related to acceleration and deceleration of the water flowing through ground surface and also erosion and accumulation of soil on the slope | |

Topographic wetness index | A function of both the slope and the upstream contributing area per unit width orthogonal to the flow direction |

Soil Depth Measured (cm) | |||||
---|---|---|---|---|---|

<20 | 20–50 | 50–90 | >90 | ||

Soil Depth Estimated (cm) | <20 | (1) | (2) | (3) | (4) |

20–50 | (5) | (6) | (7) | (8) | |

50–90 | (9) | (10) | (11) | (12) | |

>90 | (13) | (14) | (15) | (16) |

**Table 4.**Soil depth grading by measurements and slopeland soil map (bold number: correct prediction).

Soil Depth Measured (cm) | |||||
---|---|---|---|---|---|

<20 | 20–50 | 50–90 | >90 | ||

Soil Depth Estimated by Soil Depth Map (cm) | <20 | 586 | 199 | 111 | 11 |

20–50 | 500 | 233 | 183 | 16 | |

50–90 | 808 | 473 | 461 | 91 | |

>90 | 426 | 276 | 320 | 74 | |

Accuracy = 28.4% |

**Table 5.**Estimates by the Kriging method on three semivariogram models (bold number: correct prediction).

Soil Depth Measured (cm) | ||||||
---|---|---|---|---|---|---|

<20 | 20–50 | 50–90 | >90 | |||

Soil Depth Estimated by the Kriging (cm) | Exponential model | <20 | 280 | 54 | 1 | 0 |

20–50 | 1450 | 1807 | 469 | 174 | ||

50–90 | 481 | 1151 | 811 | 789 | ||

>90 | 42 | 143 | 143 | 422 | ||

Accuracy = 40.4% | ||||||

Spherical model | <20 | 273 | 49 | 1 | ||

20–50 | 1449 | 1804 | 477 | 171 | ||

50–90 | 490 | 1162 | 804 | 805 | ||

>90 | 41 | 140 | 142 | 409 | ||

Accuracy = 40.04% | ||||||

Gaussian model | <20 | 270 | 50 | 1 | ||

20–50 | 1451 | 1792 | 485 | 177 | ||

50–90 | 493 | 1176 | 796 | 801 | ||

>90 | 39 | 137 | 142 | 407 | ||

Accuracy = 39.74% |

Model | R^{2} | MAPE (%) | Hitting Rate |
---|---|---|---|

LSSVM | 0.12 | 98.5 | 0.30 |

SVR | 0.01 | 70.6 | 0.37 |

KNN | 0.30 | 74.9 | 0.43 |

LSSVM+KNN | 0.11 | 94.8 | 0.31 |

SVR+KNN | 0.27 | 63.7 | 0.44 |

**Table 7.**Estimation outcomes of the Bayesian maximum entropy (BME) model (bold number: correct prediction).

Soil Depth Measured (cm) | |||||
---|---|---|---|---|---|

<20 | 20–50 | 50–90 | >90 | ||

Soil Depth Estimated by BME (cm) | <20 | 1632 | 143 | 11 | |

20–50 | 618 | 2651 | 80 | 3 | |

50–90 | 3 | 359 | 1201 | 51 | |

>90 | 2 | 132 | 1331 | ||

Accuracy = 82.94% |

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**MDPI and ACS Style**

Liao, K.-W.; Guo, J.-J.; Fan, J.-C.; Huang, C.L.; Chang, S.-H.
Estimation of Soil Depth Using Bayesian Maximum Entropy Method. *Entropy* **2019**, *21*, 69.
https://doi.org/10.3390/e21010069

**AMA Style**

Liao K-W, Guo J-J, Fan J-C, Huang CL, Chang S-H.
Estimation of Soil Depth Using Bayesian Maximum Entropy Method. *Entropy*. 2019; 21(1):69.
https://doi.org/10.3390/e21010069

**Chicago/Turabian Style**

Liao, Kuo-Wei, Jia-Jun Guo, Jen-Chen Fan, Chien Lin Huang, and Shao-Hua Chang.
2019. "Estimation of Soil Depth Using Bayesian Maximum Entropy Method" *Entropy* 21, no. 1: 69.
https://doi.org/10.3390/e21010069