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基于最小二乘法leastsq的双孔隙土水曲线拟合

2018.04.20

模型内容:
Dexter AR, Czyż EA, Richard G, Reszkowska A. A user-friendly water retention function that takes account of the textural and structural pore spaces in soil. Geoderma. 2008;143(3):243-53.
基于Python编写了基于最小二乘法的双孔隙土水曲线拟合,记录在此,以便以后查阅。

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# -*- coding: utf-8 -*-
"""
Created on Thu Apr 19 22:40:35 2018
双孔隙土水曲线基于最小二乘法拟合

@author: Think

"""

import numpy as np
from scipy.optimize import leastsq
import matplotlib.pyplot as plt

#from data import x,y  #从外部导入想要拟合的数据,h,theta都是ndarry类型的数据
h=np.array([9.9, 20.5, 42.4,78.47,245.18,526.86,1010,2056,5227,10234,16000])
theta=np.array([0.2051,0.18916,0.18,0.1605,0.1368,0.092,0.079,0.05481,0.04437,0.03278,0.03171])

def fun(p, h):
    """
    定义想要拟合的函数
    """
    c1,A1,h1,A2,h2 = p  #从参数p获得拟合的参数
    return c1+A1*np.exp(-h/h1)+A2*np.exp(-h/h2)

def err(p, h, theta):
    """
    定义误差函数
    """
    return fun(p,h) -theta

#定义初始值
p0 = [0.3,0.233,1000,0.1,100]   

xishu = leastsq(err, p0, args=(h,theta))
c1=xishu[0][0]
A1=xishu[0][1]
h1=xishu[0][2]
A2=xishu[0][3]
h2=xishu[0][4]

print (xishu[0])

# xishu[0],即为获得的参数
plt.semilogx(h,theta,'o')
h=np.logspace(1,4,1000)

#plt.plot(np.log10(h),c1+A1*np.exp(-h/h1)+A2*np.exp(-h/h2))
plt.semilogx(h,c1+A1*np.exp(-h/h1)+A2*np.exp(-h/h2))

最后图形

第二步:求解孔隙分布,即dw/d(logh)
本来想用Sympy符号运算的,但是出现log10,微分出现问题,直接采用最原始的手动微分。

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#from sympy import *
import numpy as np
import matplotlib.pyplot as plt

#手动计算导数
k=np.linspace(0,0,999)
h=np.logspace(1,4,1000)
for i in range(1,999):
    k[i]=-(fun(xishu[0],h[i])-fun(xishu[0],h[i-1]))/(np.log10(h[i])-np.log10(h[i-1]))
plt.semilogx(h[1:],k)

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