Universal Kriging and heatmap speed flow of river section

[ChironeX1976]

Hi,
We measured the flow speed in the cross section of a river at certain points.
This results in a really ugly heatmap.
I was hoping to do spatial interpolation, to get a better representation.
What kriging-setttings do i have to use?
Below my code.
Thanks for helping me out.

import plotly.graph_objs as go
from plotly.subplots import make_subplots
import numpy as np
from pykrige.uk import UniversalKriging

# x y z values are a cross section of the speed of riverflow
xyz = np.array([[ 2.,0.,0.],[ 1.9, 0.07,0.002], [1.9,0.21,-0.002],[ 1.9,0.28,-0.006],[ 1.9,0.35,0.],[ 1.8,0.066,0.052],
       [ 1.8,0.198,0.02 ],[ 1.8,0.264,0.012],[ 1.8,0.33,0.],[ 1.7,0.07,0.138],[ 1.7,0.21,0.11 ],[ 1.7,0.28,0.097],
       [ 1.7,0.35,0.], [ 1.6,0.082,0.121], [ 1.6,0.246,0.097],[ 1.6,0.328,0.071],[ 1.6,0.41,0.],[ 1.5,0.084,0.046],
       [ 1.5,0.252,0.035],[ 1.5,0.336,0.073],[ 1.5,0.42,0.],[ 1.3, 0.084,0.005],[ 1.3,0.252,0.018],[ 1.3,0.336,0.021],
       [ 1.3,0.42,0.],[ 1.1,0.084,0.018],[ 1.1,0.252,-0.002],[ 1.1,0.336,-0.004],[ 1.1,0.42,0.],[ 0.8,0.064,0.012],
       [ 0.8,0.192,0.007], [ 0.8,0.256,0.002],[ 0.8,0.32,0. ],[ 0.5,0.05,0.004], [0.5,0.15,-0.002], [ 0.5,0.2,-0.001],
       [ 0.5,0.25,0.], [ 0.3,0.04,0.002],[ 0.3,0.16,-0.004], [ 0.3,0.2,0.], [ 0.15,0.,0.]])
x=xyz[:,0]
y=-xyz[:,1]
z=xyz[:,2]

# plot ugly heatmap
# ------------------
fig = make_subplots()
fig.add_trace(go.Heatmap(x=x,y=y, z=z, type='heatmap', colorscale='bluered', zsmooth=False))
fig.show()
fig.add_trace(go.Heatmap(x=x,y=y, z=z, type='heatmap', colorscale='bluered', zsmooth='best'))
fig.show()

# Try to get uglyness better by krigging interpolation
# ----------------------------------------------------
# set kriging grid
# gridx = cross section of river, in small steps of 0.01 m
gridx = np.arange(min(x), max(x)+0.01, 0.01)
# gridy = depth , in small steps of 0.01 meter
gridy = np.arange(0, max(y)+0.01, 0.01)

# create Universal Kriging-object,
UK = UniversalKriging(x,y,z,variogram_model="linear",drift_terms=["regional_linear"])
# calculate Z-values matrix and variance
krigz, ss = UK.execute("grid", gridx, gridy)

# plot interpolated heatmap
# ---the heatmap is right, the technique is wrong---------------
fig = make_subplots()
fig.add_trace(go.Heatmap(x=x,y=y, z=krigz, type='heatmap', colorscale='bluered', zsmooth=False))
fig.show()

[ChironeX1976]

I have been studying the tutorials on this website (and others).
I succeeded in producing a plot of the speed in a cross section of a river based on krigging.
It is still wrong in certain points.
How can i force the method to get a value in a certain point?
Should i take another krigging method?
Please, help. It's for the fish in there :-)
ugly heatmap, before krigging:

nicer heatmap, after krigging:

my new code:

import plotly.graph_objs as go
from plotly.subplots import make_subplots
import numpy as np
import gstools as gs
from matplotlib import pyplot as plt
# data preparation: x y z values are a cross section of the speed of riverflow
riverbed = np.array([[0.15, 0.], [0.3, -0.2], [0.5, -0.25], [0.8, -0.32], [1.1, -0.42], [1.3, -0.42], [1.5, -0.42],
                     [1.7, -0.33], [1.8, -0.28], [1.9, -0.07], [2., 0.]])
xyz = np.array([[ 2.,0.,0.],[ 1.9, 0.07,0.002], [1.9,0.21,-0.002],[ 1.9,0.28,-0.006],[ 1.9,0.35,0.],[ 1.8,0.066,0.052],
       [ 1.8,0.198,0.02 ],[ 1.8,0.264,0.012],[ 1.8,0.33,0.],[ 1.7,0.07,0.138],[ 1.7,0.21,0.11 ],[ 1.7,0.28,0.097],
       [ 1.7,0.35,0.], [ 1.6,0.082,0.121], [ 1.6,0.246,0.097],[ 1.6,0.328,0.071],[ 1.6,0.41,0.],[ 1.5,0.084,0.046],
       [ 1.5,0.252,0.035],[ 1.5,0.336,0.073],[ 1.5,0.42,0.],[ 1.3, 0.084,0.005],[ 1.3,0.252,0.018],[ 1.3,0.336,0.021],
       [ 1.3,0.42,0.],[ 1.1,0.084,0.018],[ 1.1,0.252,-0.002],[ 1.1,0.336,-0.004],[ 1.1,0.42,0.],[ 0.8,0.064,0.012],
       [ 0.8,0.192,0.007], [ 0.8,0.256,0.002],[ 0.8,0.32,0. ],[ 0.5,0.05,0.004], [0.5,0.15,-0.002], [ 0.5,0.2,-0.001],
       [ 0.5,0.25,0.], [ 0.3,0.04,0.002],[ 0.3,0.16,-0.004], [ 0.3,0.2,0.], [ 0.15,0.,0.]])
x=xyz[:,0]
y=-xyz[:,1]
z=xyz[:,2]
#plot ugly heatmap
# ------------------
fig = make_subplots()
fig.add_trace(go.Heatmap(x=x,y=y, z=z, type='heatmap', colorscale='bluered', zsmooth=False))
fig.add_trace(go.Scatter(x=x,y=y, mode='markers+text',text=z, name='speed', showlegend=True,
                         marker=dict(size=20, color = z, colorscale = 'bluered', showscale = False )))
fig.add_trace(go.Scatter(x=riverbed[:,0],y=riverbed[:,1],mode='lines'))
fig.show()
#########################
# 2. Estimation of the variogram
#########################
# Choose i_afstand and i_binstap based
# on visual interpretation of data (plot)
# fast testplot: to estimate nominal distance from graph
plt.scatter(x, y)
plt.xlabel('m from side')
plt.ylabel('depth')
plt.show()
# based on estimation of the parameters i_afstand and i_binstap (by iteration)
# you should get a slope in the datapoints getting to a plateau:
# left: correlated distances, right: uncorrelated distances
# start by estimating, plot and change till the variogram is right
# estimation of distance of correlated values
i_afstand = 0.55
# make pairs of numbers going from zero to a distance with discrete steps
i_binsstap = 0.1
bins = np.arange(start=0, stop=i_afstand, step=i_binsstap)
bin_center, gamma = gs.vario_estimate(pos=(x, y),
                                      field=z,
                                      bin_edges=bins)
# fast testplot: the variogram
plt.scatter(bin_center, gamma)
plt.xlabel('distance between pairs')
plt.ylabel('gamma')
plt.show()
# variogram is estimated? next step.
# ##############################################################################################
# # 3. Fit variogram with model
#   model based on "gs.Stable" function
#   Wackernagel, H. “Multivariate geostatistics”, Springer, Berlin, Heidelberg (2003)
# ##############################################################################################
#  create model object with 2 dimensions, and standard parameters, based on Stable-function
model = gs.Stable(dim=2)
# fit the variogram with a stable model.
fit_parameters, pcov, r2_score = model.fit_variogram(bin_center, gamma,
                                                     nugget=False,
                                                     return_r2=True)
# fast testplot: fitted curve with data
plt.figure()
ax = model.plot(x_max=max(bin_center))
ax.scatter(bin_center, gamma)
plt.xlabel('afstand tussen punten')
plt.ylabel('gamma')
plt.show()
# variogram is fitted with model? Next step.
##################################################
# # 4. Kriging based on model
###################################################
# grid definition, based on size of river
gridx = np.arange(0.15, 2, 0.2) # start, einde, stap
gridy = np.arange(-0.0, -0.45, -0.2)
OK = gs.krige.Ordinary(model, [x, y], z, exact=True)
OK.structured([gridx, gridy])
# fast testplot for kriged data "river is upside down"
plt.figure()
ax = OK.plot()
ax2 = plt.scatter(x,y)
plt.show()
# # plot nicer heatmap
# # ------------------
fig = make_subplots()
fig.add_trace(go.Heatmap(x=gridx,y=gridy, z=OK.all_fields[0].transpose(), type='heatmap', colorscale='bluered', zsmooth='fast'))
# rivierbed - line
fig.add_trace(go.Scatter(x=x,y=y, mode='markers+text',text=z, name='speed', showlegend=True,
                         marker=dict(size=20, color = z, colorscale = 'bluered', showscale = False )))
fig.add_trace(go.Scatter(x=riverbed[:,0],y=riverbed[:,1],mode='lines'))
fig.show()

[ChironeX1976]

Answered by @LSchueler in analogue question: see https://github.com/GeoStat-Framework/GSTools/discussions/225#discussion-3739475