A hybrid method of debris flow velocity estimation based on empirical equation

2.1 Empirical equation of debris flow velocity

The empirical or semi-empirical formula for the calculation of average flow velocity have been proposed by many researchers. The factors that account for the largest differences of each equation are related to the properties of debris flow materials, the slope of the flow channel and the depth of the debris flow, all of which are used in every equation. The most well known equations for debris flow velocity calculation are as follows:

1) The Manning formula

$V=\frac{1}{n}{{R}^{2/3}}{{S}^{1/2}}$          (1)

where n is the resistance coefficient, R is the hydraulic radius, and S is the slope.

2) Newtonian laminar flow

$V=\frac{1}{3}\frac{\rho g{{h}^{2}}S}{\mu }$         (2)

where $\rho $ is debris flow density, g is gravitational acceleration, h  is debris flow depth, $\mu $ is dynamic viscosity.

3) Bingham fluid

$V=\frac{\rho g{{H}^{2}}S}{2\eta }\left( 1-\frac{H}{3h} \right)$        (3)

where, H is effective shear depth, $\eta $ is Bingham viscosity.

A widely-used method for mean velocity is the Manning-stickler equation, and all aforementioned formulas are based on it. The Manning coefficient n is adjusted through statistical regression analysis based on observation data, and the empirical formula of the research area is obtained. In calculating the velocity of a debris flow, the Manning coefficient n is subjectively determined, but it is closely associated with the accuracy of the results. Generally, the common equations of flow mean velocity calculation can be described as follows:

$V=N{{h}^{b}}{{S}^{c}}$        (4)

where b, c and N are constant values, They can be determined by the research data from the debris flow area.

In this article, debris flow velocity is calculated by the common equation of flow, and the parameters used in the mean flow equation is the integral component in debris flow velocity calculation.

2.2 LS-SVM prediction principle

The least squares support vector machine was developed by Suykens et al. [13]. It is one of the most accessible data mining technologies for small samples based on the structural risk minimization principle. The non-linear prediction problem is resolved by conducting original data into high dimensional space and obtaining the optimal hyperplane. The dimension transformation is conducted by kernel function, so the kernel function is important for prediction accuracy. RBF kernel is selected in this article, and is expressed as follows:

$K({{x}_{i}},{{x}_{j}})=\exp (-\gamma {{({{x}_{i}}-{{x}_{j}})}^{2}})$        (5)

The prediction function of a common fitness curve can be describe as $\ y(x)={{w}^{T}}\varphi (x)+{{w}_{0}}$, where $w$ is the weight vector, ${{w}_{0}}$ is the bias term, $\gamma $ is a constant which should be determined artificially, It is closely associated with the performance of the LSSVM model, and PSO is applied for it's optimization. The structural risk minimization is used to resolve the problem, and it formulates the following optimization problem:

$\left\{ \begin{align}  & \min \ \ J(w,\xi )=\frac{1}{2}{{\left\| w \right\|}^{2}}+C\sum\limits_{i=1}^{l}{\xi _{i}^{2}} \\ & s.t.\ \ \ {{y}_{i}}={{w}^{T}}\varphi ({{x}_{i}})+{{w}_{0}}+{{\xi }_{i}},\ \ i=1,2\cdot \cdot \cdot l \\\end{align} \right.$       (6)

where $K({{x}_{i}},{{x}_{j}})=\varphi ({{x}_{j}})\cdot \varphi ({{x}_{i}});{{R}^{n}}\to {{R}^{{{n}_{h}}}}$ is the kernel function, $w\in {{R}^{{{n}_{h}}}}$ is a weight vector, C is the penalty parameter, ${{\xi }_{i}}\in R$ is an error variant, ${{w}_{0}}$ is a deviation.

To solve the function, we introduce the Lagrannge multiplier, than the function can be defined as follows:

$L(w,{{w}_{0}},\xi ,\alpha )=\frac{1}{2}{{w}^{T}}\cdot w+C\sum\limits_{i=1}^{l}{\xi _{i}^{2}}-\sum\limits_{i=1}^{l}{{{\alpha }_{i}}}({{w}^{T}}\cdot \varphi ({{x}_{i}})+{{w}_{0}}+{{\xi }_{i}}-{{y}_{i}})$        (7)

${{\alpha }_{i}},i=1,2....l$ is the Lagrange multiplier, set $\frac{\partial L}{\partial w},\frac{\partial L}{\partial {{w}_{0}}},\frac{\partial L}{\partial \alpha },\frac{\partial L}{\partial \xi }$ to zero, $\min \ \ J(w,\xi )$ will be optimized. Then we get:

$\left[ \begin{matrix}   0 & 1 & \cdots  & 1  \\   1 & K({{x}_{1}},{{x}_{1}})+1/C & \cdots  & K({{x}_{1}},{{x}_{l}})  \\   \vdots  & \vdots  & \ddots  & \vdots   \\   1 & K({{x}_{l}},{{x}_{1}}) & \ldots  & K({{x}_{l}},{{x}_{l}})+1/C  \\\end{matrix} \right]\left[ \begin{matrix}   b  \\   {{\alpha }_{1}}  \\   \vdots   \\   {{\alpha }_{l}}  \\\end{matrix} \right]=\left[ \begin{matrix}   0  \\   {{y}_{1}}  \\   \vdots   \\   {{y}_{l}}  \\\end{matrix} \right]$       (8)

We calculate ${{w}_{0}}$,  by (6) (7) (8)，Then the prediction model function is as follows:

$f\left( x \right)=\sum\limits_{i=1}^{l}{{{\alpha }_{i}}}K({{x}_{i}},x)+{{w}_{0}}$        (9)

2.3 PSO (particle swarm optimization) method

PSO (particle swarm optimization) is an optimization method for SVM parameters used in this article, which is an improved method of the grid search method. It sets each possible solution as a particle in the optimization process, and gives it an initial speed so that the particle can move in high dimension space. The best choice of one particle and all particles in optimization are recorded as the individual optimal solution ${{p}_{best}}$ and the global optimal solution ${{g}_{best}}$, and the speed changes with ${{p}_{best}}$ and ${{g}_{best}}$ in high dimension space are recorded. After several loops of particles’ changing speeds and positions, the finally optimal solution of parameters are obtained [14].

$+{{c}_{2}}\cdot rand()\times ({{g}_{best}}-{{x}_{i}}(t))$       (10)

${{x}_{i}}(t+1)={{x}_{i}}(t)+{{v}_{i}}(t+1)$        (11)

where ${{v}_{i}}$ is the velocity of the i particle, ${{x}_{i}}$ is the position of the particle, ${{c}_{1}}$ and ${{c}_{2}}$ are learning factors, $\omega $is the inertia factor.

2.4 Improved empirical equation of velocity calculation

According to studies worldwide, the general equation of flow velocity based on the Manning formula is as equation: $V=N{{h}^{b}}{{S}^{c}}$ , where N is a constant, h is flow depth, b and c are coefficients with regional features. Previous studies have yielded the following results. In the Chezy equation, b=c=0.5. In the Manning-Strickler equation, b is 2/3 and c is 1/2. In the debris flow velocity research articles, the b value is varied from 1/3 to 1/2, with 1/2 as the most widely used for debris flow velocity calculation. c is varied from 1/6 to 1, and the c value is generally determined as 1/2 or 1/3 [5].

The values of b and c in the equations are influenced by many factors of debris flow materials. In this article, we propose a PSO-LSSVM model to analysis the coefficients in the empirical equation, and the process of velocity calculation is as follows.

1) The original data of debris flow samples are processed by PCA.

2) The b value is fixed as 0.5, and the c values of each samples are back-calculated through empirical formula.

3) Train the LSSVM model with debris flow samples and the back-calculated c value.

4) Calculate MSE and MAE through five-fold cross-

validation, and obtain the predicted c value based on the LSSVM model.

5) Fix c value as 1, then calculate b as the same as procedures 3) and 4) instead of c parameter.

6) Put the new predicted b and c vectors into the empirical formula, the velocity of debris flow samples will be obtained through the new equation.