Python实现克里金插值法的过程详解
作者:指尖听戏
一、克里金插值法介绍
克里金算法提供的半变异函数模型有高斯、线形、球形、阻尼正弦和指数模型等,在对气象要素场插值时球形模拟比较好。既考虑了储层参数的随机性,有考虑了储层参数的相关性,在满足插值方差最小的条件下,给出最佳线性无偏插值,同时还给出了插值方差。
与传统的插值方法(如最小二乘法、三角剖分法、距离加权平均法)相比,克里金法的优势:
1、在数据网格化的过程中考虑了描述对象的空间相关性质,使插值结果更科学、更接近于实际情况;
2、能给出插值的误差(克里金方差),使插值的可靠程度一目了然
插值方差:就是指实际参数值 zv 与估计值 zv* 两者偏差平方的数学期望:
而插值点的 zv*,通过N个离散点获得;
其中λ与N个离散点指的是加权系数; *变差函数的理论模型*
变差函数与随机变量的距离h存在一定的关系,这种关系可以用理论模型表示。常用的变差函数理论模型包括球状模型、高斯模型与指数模型(还包括:具基台值线性模型、幂函数模型、无基台值线性模型);
1、 球状模型公式:
2、 高斯模型公式:
3、 指数模型公式:
4、 具基台值线性模型:
5、 幂函数模型:
式中: 为幂指数;不存在基台值。两边取对数得ln(γ(h))=αlnh,线性化为γ(hi)=b1X1,i
6、 无基台值线性模型:
式中:k为直线斜率;不存在基台值和变程,当h>0时, γ(hi)=b0+b1X1,i
普通克里格方法的基本步骤如下:
二、算法实现
代码实现:
from gma.algorithm.spmis.interpolate import * class Kriging(IPolate): '''克里金法插值''' # 继承 gma 的标准插值类 IPolate。本处不再做详细说明。 def __init__(self, Points, Values, Boundary = None, Resolution = None, InProjection = 'WGS84', VariogramModel = 'Linear', VariogramParameters = None, **kwargs): IPolate.__init__(self, Points, Values, Boundary, Resolution, InProjection) self.eps = eps # 初始化半变异函数及参数 self.VariogramModel, self.VParametersList = GetVariogramParameters(VariogramModel, VariogramParameters) self.VariogramFUN = getattr(variogram, self.VariogramModel) if self.VParametersList is None: self.VParametersList = self._INITVariogramModel(**kwargs) # 调整输入数据 if self.GType == 'PROJCS': self.Center = (self.Points.min(axis = 0) + self.Points.max(axis = 0)) * 0.5 self.AnisotropyScaling = AnisotropyScaling self.AnisotropyAngle = AnisotropyAngle self.DistanceMethod = cdist else: # 方便后期优化 self.Center = np.array([0,0]) self.AnisotropyScaling = 1.0 self.AnisotropyAngle = 0.0 self.DistanceMethod = GreatCircleDistance self.AdjustPoints = AdjustAnisotropy(self.Points, self.Center, [self.AnisotropyScaling], [self.AnisotropyAngle]) self.XYs = AdjustAnisotropy(self.XYs, self.Center, [self.AnisotropyScaling], [self.AnisotropyAngle]) def _INITVariogramModel(self, **kwargs): '''初始化参数''' if 'NLags' in kwargs: NLags = kwargs['NLags'] initialize.ValueType(NLags, 'pint') else: NLags = 6 if 'Weight' in kwargs: Weight = ToNumericArray(kwargs['Weight']).flatten().astype(bool)[0] else: Weight = False Lags, SEMI = INITVariogramModel(self.Points, self.Values, NLags, self.GType) # 为求解自动参数准备 if self.VariogramModel == "Linear": X0 = [np.ptp(SEMI) / np.ptp(Lags), np.min(SEMI)] BNDS = ([0.0, 0.0], [np.inf, np.max(SEMI)]) elif self.VariogramModel == "Power": X0 = [np.ptp(SEMI) / np.ptp(Lags), 1.1, np.min(SEMI)] BNDS = ([0.0, 0.001, 0.0], [np.inf, 1.999, np.max(SEMI)]) else: X0 = [np.ptp(SEMI), 0.25 * np.max(Lags), np.min(SEMI)] BNDS = ([0.0, 0.0, 0.0], [10.0 * np.max(SEMI), np.max(Lags), np.max(SEMI)]) # 最小二乘法求解默认参数 def _VariogramResiduals(Params, X, Y, Weight): if Weight: Weight = 1.0 / (1.0 + np.exp(-2.1972 / (0.1 * np.ptp(X)) * (0.7 * np.ptp(X) + np.min(X) - x))) + 1 else: Weight = 1 return (self.VariogramFUN(X, *Params) - Y) * Weight RES = least_squares(_VariogramResiduals, X0, bounds = BNDS, loss = "soft_l1", args = (Lags, SEMI, Weight)) return RES.x def _GetKrigingMatrix(self): """获取克里金矩阵""" LDs = self.DistanceMethod(self.AdjustPoints, self.AdjustPoints) A = -self.VariogramFUN(LDs, *self.VParametersList) A = np.pad(A, (0, 1), constant_values = 1) # 填充主对角线 np.fill_diagonal(A, 0.0) return A def _UKExec(self, A, LDs, SearchRadius): """泛克里金求解""" Args = LDs.argsort(axis = 1)[:,:SearchRadius] Values = self.Values[Args.T].T # A 的逆矩阵 AInv = inv(A) B = -self.VariogramFUN(LDs, *self.VParametersList) B[np.abs(LDs) <= self.eps] = 0.0 B = np.pad(B, ((0,0),(0,1)), constant_values = 1) X = np.dot(B, AInv) B = B[np.ogrid[:len(B)], Args.T].T X = X[np.ogrid[:len(X)], Args.T].T X = X / X.sum(axis = 1, keepdims = True) UKResults = np.sum(X * Values, axis = 1), np.sum((X * -B), axis = 1) return UKResults def _OKExec(self, A, LDs, SearchRadius): """普通克里金求解""" Args = LDs.argsort(axis = 1)[:,:SearchRadius] LDs = LDs[np.ogrid[:len(LDs)], Args.T].T B = -self.VariogramFUN(LDs, *self.VParametersList) B[np.abs(LDs) <= self.eps] = 0.0 B = np.pad(B, ((0,0),(0,1)), constant_values = 1) OKResults = np.zeros([2, len(LDs)]) for i, b in enumerate(B): BSelector = Args[i] ASelector = np.append(BSelector, len(self.AdjustPoints)) a = A[ASelector[:, None], ASelector] x = solve(a, b) OKResults[:, i] = x[:SearchRadius].dot(self.Values[BSelector]), -x.dot(b) return OKResults def Execute(self, SearchRadius = 12, KMethod = 'Ordinary'): '''克里金插值''' initialize.ValueType(SearchRadius, 'pint') SearchRadius = np.min([SearchRadius, len(self.AdjustPoints)]) A = self._GetKrigingMatrix() LDs = self.DistanceMethod(self.XYs, self.AdjustPoints) if KMethod not in ['Universal', 'Ordinary']: raise ValueError("Undefined Kriging method. Please select 'Universal' or 'Ordinary'!") elif KMethod == 'Universal': KResults = self._UKExec(A, LDs, SearchRadius) else: KResults = self._OKExec(A, LDs, SearchRadius) NT = namedtuple('Kriging', ['Data', 'SigmaSQ', 'Transform']) return NT(KResults[0].reshape(self.YLAT, self.XLON), KResults[1].reshape(self.YLAT, self.XLON), self.Transform)
三、差值应用
示例数据可从:https://gma.luosgeo.com/ 获取
在 gma 1.0.13.15 之后的版本可以直接引用。这里基于 1.0.13.15之后的版本引用做示例。
import gma import pandas as pd Data = pd.read_excel("Interpolate.xlsx") Points = Data.loc[:, ['经度','纬度']].values Values = Data.loc[:, ['值']].values # 普通克里金(球面函数模型)插值 KD = gma.smc.Interpolate.Kriging(Points, Values, Resolution = 0.05, VariogramModel = 'Spherical', VariogramParameters = None, KMethod = 'Ordinary', InProjection = 'EPSG:4326') # 泛克里金类似,这里不做示例 gma.rasp.WriteRaster(r'.\gma_OKriging.tif', KD.Data, Projection = 'WGS84', Transform = KD.Transform, DataType = 'Float32')
四、结果对比
与 ArcGIS Ordinary Kriging 插值结果(重分类后)对比:
与 pykrige 包 Universal Kriging 插值结果(重分类后)对比:
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