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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Uncertainty analysis of a 2-D slice model
Possible paper titles
Step1:  Model set-up
Subsequently, we will use a model from the "Atlas of Structural Geophysics" as an example model.
Step2:  BUG!!!!
Note
Step3:  We now define the parameter uncertainties
Step4:  And, in a next step, perform the model sampling
Step5:  Save the model data for later re-use (e.g. to extend the data set)
Step7:  Calculation of cell information entropy
(Include note on
Step8:  The algorithm works on the simple idea that we do not explicitly require the single outputs at each location, but only the relative probability values. This may not matter too much for single entropy estimates (uni-variate), but it will matter a lot for multivariate cases, because we do not need to check all possible outcomes! Note that all outcomes with zero probability are simply not considered in the sorting algorithm (and they do not play any role in the calculation of the entropy, anyway), and that's exactly what we want to have!
Step9:  We now visualise the cell information entropy, shown in Fig. (). We can here clearly identify uncertain regions within this model section. It is interesting to note that we can mostly still identify the distinct layer boundaries in the fuzzy areas of uncertainty around their borders (note
Step10:  Here again an example of single models (adjust to visualise and probably include something like a "table plot" of multiple images for a paper!)
Step11:  And here the "mean" lithologies (note
Step12:  And here a bit more meaningful
Step13:  Idea
Step14:  Problem now
Step15:  Try own "entropy colormap"
Step16:  For comparison again
Step17:  And the difference, for clarity
Step18:  Clearly, the highset reduction is in the area around the borehole, but interestingly, the uncertianty in other areas is also reduced! Note specifically the reduction of uncertainties in the two neighbouring fold hinges.
Let's check some other positions (and drilling "depths")
Step19:  We also just include one timing step to estimate the approximate simualtion time
Step20:  Intersting! Only a local reduction around the drilling position, however
Step21:  Interesting! And now both combined
Step22:  We can see that now only a part on the left remains with significant uncertainties. So, let's "drill" into this, as well
Step23:  Additional idea to speed up computation (especially for higher multivariate examples)
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Python Code:
from IPython.core.display import HTML
css_file = 'pynoddy.css'
HTML(open(css_file, "r").read())
%matplotlib inline
# here the usual imports. If any of the imports fails, 
# make sure that pynoddy is installed
# properly, ideally with 'python setup.py develop' 
# or 'python setup.py install'
import sys, os
import matplotlib.pyplot as plt
import numpy as np
# adjust some settings for matplotlib
from matplotlib import rcParams
# print rcParams
rcParams['font.size'] = 15
# determine path of repository to set paths corretly below
repo_path = os.path.realpath('../..')
import pynoddy
reload(pynoddy)
import pynoddy.history
import pynoddy.experiment
reload(pynoddy.experiment)
rcParams.update({'font.size': 15})
Explanation: Uncertainty analysis of a 2-D slice model
Possible paper titles: 
Posterior analysis of geological models/ geophysical inversions? with information theory
...with measures from information theory
...with information theoretic measures
...with information measures
...with information
Ensemble analysis...
Posterior and ensemble analysis...
(Note: reference to posterior analysis, e.g. in Tarantola paper!)
Include:
Analysis of pynoddy models:
simple example of graben (because the uncertainty reduction is so counter-intuitive), 
and maybe more complex example of folding structure?
Analysis of object modelling results? (i.e. the typical "Stanford channels"? but: only two outcomes, so not so meaningful... better example?)
Analysis of posterior ensemble from geophysical inversion (Greenstone model? Other examples from Mark & Mark?)
Journal? Math. Geo? Tectonophysics (note: relevance to strucural geological models!)
Include theory: error bounds on information measures!
End of explanation
from pynoddy.experiment import monte_carlo
model_url = 'http://tectonique.net/asg/ch3/ch3_7/his/typeb.his'
ue = pynoddy.experiment.Experiment(url = model_url)
ue.change_cube_size(100)
sec = ue.get_section('y')
Explanation: Model set-up
Subsequently, we will use a model from the "Atlas of Structural Geophysics" as an example model.
End of explanation
sec.block.shape
ue.plot_section('y')
plt.imshow(sec.block[:,50,:].transpose(), origin = 'lower left', interpolation = 'none')
tmp = sec.block[:,50,:]
tmp.shape
ue.set_random_seed(12345)
ue.info(events_only = True)
Explanation: BUG!!!!
Note: there is either a bug in pynoddy or in Noddy itself: but the slice plotting method fails: actually, not a slice is computed but the entire model (and the extent is also not correct!). Check with Mark in course of paper prep!
End of explanation
param_stats = [{'event' : 2, 
              'parameter': 'Amplitude',
              'stdev': 100.0,
              'type': 'normal'},
              {'event' : 2, 
              'parameter': 'Wavelength',
              'stdev': 500.0,
              'type': 'normal'},
              {'event' : 2, 
              'parameter': 'X',
              'stdev': 500.0,
              'type': 'normal'}]
ue.set_parameter_statistics(param_stats)
Explanation: We now define the parameter uncertainties:
End of explanation
ue.set_random_seed(112358)
# perfrom random sampling
resolution = 100
sec = ue.get_section('y')
tmp = sec.block[:,50,:]
n_draws = 5000
model_sections = np.empty((n_draws, tmp.shape[0], tmp.shape[1]))
for i in range(n_draws):
    ue.random_draw()
    tmp_sec = ue.get_section('y', resolution = resolution, 
                             remove_tmp_files = True)
    model_sections[i,:,:] = tmp_sec.block[:,50,:]
Explanation: And, in a next step, perform the model sampling:
End of explanation
import pickle
f_out = open("model_sections_5k.pkl", 'w')
pickle.dump(model_sections, f_out)
Explanation: Save the model data for later re-use (e.g. to extend the data set):
End of explanation
def entropy(diff_array, n_samples):
    Determine entropy from diffarray using switchpoints
    switchpts = np.where(diff_array > 0 )[0] + 1
    switchpts = np.append(0, switchpts)
    switchpts = np.append(switchpts, n_samples)
    pdiff = np.diff(switchpts)
    j_prob = pdiff / float(n_samples)
    # calculate entropy
    h = 0.
    for jp in j_prob:
        h -= jp * np.log2(jp)
    return h
Explanation: Calculation of cell information entropy
(Include note on: theory of entropy calculation)
(Include in this paper: estimates on error bounds?)
Here now the function to calculate entropy from a data array in general. What we will need to do later is to pass all results at a single position as a "data array" and we can then estimate the information entropy at this position.
This function already expects a sorted array as an input and then uses the (ultra-fast) switchpoint method to calculate entropy:
End of explanation
# sort data (axis = 0: sort along model results!)
mssort = np.sort(model_sections, axis = 0)
# create difference matrix
mssort_diff = mssort[1:,:,:] - mssort[:-1,:,:]
n_samples = model_sections.shape[0]
# and now: for all!
h = np.empty_like(mssub[0,:,:])
for i in range(100):
    for j in range(40):
        h[i,j] = entropy(mssort_diff[:,i,j], n_samples)
h[50,30]
Explanation: The algorithm works on the simple idea that we do not explicitly require the single outputs at each location, but only the relative probability values. This may not matter too much for single entropy estimates (uni-variate), but it will matter a lot for multivariate cases, because we do not need to check all possible outcomes! Note that all outcomes with zero probability are simply not considered in the sorting algorithm (and they do not play any role in the calculation of the entropy, anyway), and that's exactly what we want to have!
End of explanation
plt.imshow(h.transpose(), origin = 'lower left',
          cmap = 'gray', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
Explanation: We now visualise the cell information entropy, shown in Fig. (). We can here clearly identify uncertain regions within this model section. It is interesting to note that we can mostly still identify the distinct layer boundaries in the fuzzy areas of uncertainty around their borders (note: highlight in Figure!). However, additional aspects of uncertainty are now introduced: (a) the uncertainty about the x-position of the folds (see parameters: event 2, parameter x) is now clearly visible, and (b) uncertianties now seem to concentrate on the fold hinges. However, this is not so clear in the left part of the model, where the fold hing seems to be the least uncertain part. (check why: is this where the fold is actually fixed (even though still uncertain). My current interpretation: the fold location is fixed somewhere near this point, and so the wavelength uncertainty does not play a significant role. Furthermore, the fold is quite "open" at this position (i.e.: low angle between hinges) and therefore lateral shifts do not play a significant role.
End of explanation
plt.imshow(mssub[70,:,:].transpose(), origin = 'lower left', interpolation = 'none')
Explanation: Here again an example of single models (adjust to visualise and probably include something like a "table plot" of multiple images for a paper!):
End of explanation
plt.imshow(np.mean(mssub, axis = 0).transpose(), origin = 'lower left', interpolation = 'none')
Explanation: And here the "mean" lithologies (note: not really a distinct meaning, simply showing the average litho ids - could be somehow interpreted as characteristic functions, though...).
End of explanation
# step 1: estimate probabilities (note: unfortunate workaround with ones multiplication,
# there may be a better way, but this is somehow a recurring problem of implicit
# array flattening in numpy)
litho_id = 4
prob = np.sum(np.ones_like(mssub) * (mssub == litho_id), axis = 0) / float(n_samples)
plt.imshow(prob.transpose(), 
           origin = 'lower left', 
           interpolation = 'none', 
           cmap = 'gray_r')
plt.colorbar(orientation = 'horizontal')
Explanation: And here a bit more meaningful: the analysis of single layer probabilities:
End of explanation
sys.path.append("/Users/flow/git/mutual_information/")
import hspace
reload(hspace)
Explanation: Idea: also include a "new" consideration: where to collect information to reduce uncertainty of a single layer? Could be identified by reducing layer fuzziness, for example!
Or: what are most likely positions/ locations of a specific unit, given collected information?
Analysis of multivariate condtional entropy
Later also:
"opposite" question, i.e.: if we would like to resolve uncertainty in a specific region: where to look best?
"complete" uncertainty (i.e.: joint entropy!)
greedy search for best spots for uncertainty reduction, simple (cell-wise), complex (related to potential drilling positions)
further ideas for "greedy search" to reduce uncertainties in a specific "search region" (i.e.: expected location of a deposit, etc.):
start with cell with highest (multivariate) mutual information
rank cells with highest I due to their own mutual information with other cells, which are not part of a defined "search region"
for simple, cell-wise: describe similarity to mapping! Maybe even a field example with data from Belgium? But this is still one or two MSc theses away...)
For the joint entropy analysis, we now use the new lexicographic (correct term?) sorting algorithm, implemented in the module hspace:
End of explanation
locs = np.array([1,2,3])
locs.shape
locs = np.array([[1,1],[1,2],[1,3]])
locs.shape
models_sub = model_sections[:10,:,:]
joint_units = []
for entry in models_sub:
    joint_val = ""
    for i, loc in enumerate(locs):
        joint_val += "%d" % entry[loc[0], loc[1]]
    joint_units.append(joint_val)
print joint_units
hspace.joint_entropy(model_sections, locs, n_samples)
# now: define position of "drill":
n = 10
drill_i = [60] * n
drill_j = range(39,39-n,-1)
drill_locs = zip(drill_i, drill_j)
# determine joint entropy of drill_locs:
h_joint_drill = hspace.joint_entropy(model_sections, drill_locs, n_samples)
# check with arbitrary additional position:
locs = drill_locs + [[50, 20]]
print locs
h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
print h_joint_drill
print h_joint_loc
print h_joint_loc - h_joint_drill
print h[50,30]
# Determine max value of initital entropies for colorbar scaling
h_max = np.max(h)
print h_max
n_max = int(np.ceil(2 ** h_max))
print n_max
print np.log2(n_max)
pwd
# Import Viridis colorscale
import colormaps as cmaps
plt.register_cmap(name='viridis', cmap=cmaps.viridis)
plt.register_cmap(name='magma', cmap=cmaps.magma)
plt.set_cmap(cmaps.viridis)
Explanation: Problem now: in hspace module, a single location information is required. Try to adapt?
End of explanation
# generate conditional entropies for entire section:
h_cond_drill = np.zeros_like(h)
for i in range(100):
    for j in range(40):
        # add position to locations
        locs = drill_locs + [[i,j]]
        # determine joint entropy
        h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
        # subtract joint entropy of drill locs to obtain conditional entropy
        h_cond_drill[i,j] = h_joint_loc - h_joint_drill
import simple_2D_example
from simple_2D_example import entropy_colormap
reload(simple_2D_example)
ecmap = simple_2D_example.entropy_colormap(h_max);
plt.imshow(h_cond_drill.transpose(), origin = 'lower left',
          cmap = 'viridis', interpolation = 'none', vmax=np.log2(n_max+0.02))
plt.colorbar(orientation = 'horizontal')
# half-step contour lines
contour_levels = np.log2(np.arange(1., n_max + 0.001, .5))
plt.contour(h_cond_drill.transpose(), contour_levels, colors = 'gray')
# superpose 1-step contour lines
contour_levels = np.log2(np.arange(1., n_max + 0.001, 1.))
plt.contour(h_cond_drill.transpose(), contour_levels, colors = 'white')
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,99])
plt.ylim([0,39])
Explanation: Try own "entropy colormap":
End of explanation
plt.imshow(h.transpose(), origin = 'lower left',
          cmap = 'viridis', interpolation = 'none', vmax=np.log2(n_max+0.02))
plt.colorbar(orientation = 'horizontal')
# half-step contour lines
contour_levels = np.log2(np.arange(1., n_max + 0.001, .5))
plt.contour(h.transpose(), contour_levels, colors = 'gray')
# superpose 1-step contour lines
contour_levels = np.log2(np.arange(1., n_max + 0.001, 1.))
plt.contour(h.transpose(), contour_levels, colors = 'white')
Explanation: For comparison again: the entropy of the initial model:
End of explanation
plt.imshow((h - h_cond_drill).transpose(), origin = 'lower left',
          cmap = 'viridis', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,99])
plt.ylim([0,39])
Explanation: And the difference, for clarity:
End of explanation
# define position of "drill":
n = 10
drill_i = [20] * n
drill_j = range(39,39-n,-1)
drill_locs = zip(drill_i, drill_j)
# determine joint entropy of drill_locs:
h_joint_drill = hspace.joint_entropy(model_sections, drill_locs, n_samples)
Explanation: Clearly, the highset reduction is in the area around the borehole, but interestingly, the uncertianty in other areas is also reduced! Note specifically the reduction of uncertainties in the two neighbouring fold hinges.
Let's check some other positions (and drilling "depths"):
End of explanation
%%timeit
locs = drill_locs + [[50,20]]
h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
# esimated total time:
ttime = 100 * 40 * 0.0476
print("Estimated total time: %.3f seconds or %.3f minutes" % (ttime, ttime/60.))
# generate conditional entropies for entire section:
h_cond_drill = np.zeros_like(h)
for i in range(100):
    for j in range(40):
        # add position to locations
        locs = drill_locs + [[i,j]]
        # determine joint entropy
        h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
        # subtract joint entropy of drill locs to obtain conditional entropy
        h_cond_drill[i,j] = h_joint_loc - h_joint_drill
# store results
f_out = open("h_cond_drill_i20_10.pkl", 'w')
pickle.dump(h_cond_drill, f_out)
f_out.close()
plt.imshow(h_cond_drill.transpose(), origin = 'lower left',
          cmap = 'gray', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
plt.imshow((h - h_cond_drill).transpose(), origin = 'lower left',
          cmap = 'RdBu', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
Explanation: We also just include one timing step to estimate the approximate simualtion time:
End of explanation
# define position of "drill":
n = 30
drill_i = [20] * n
drill_j = range(39,39-n,-1)
drill_locs = zip(drill_i, drill_j)
# determine joint entropy of drill_locs:
h_joint_drill = hspace.joint_entropy(model_sections, drill_locs, n_samples)
%%timeit
locs = drill_locs + [[50,20]]
h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
# esimated total time:
ttime = 100 * 40 * 0.130
print("Estimated total time: %.3f seconds or %.3f minutes" % (ttime, ttime/60.))
# generate conditional entropies for entire section:
h_cond_drill = np.zeros_like(h)
for i in range(100):
    for j in range(40):
        # add position to locations
        locs = drill_locs + [[i,j]]
        # determine joint entropy
        h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
        # subtract joint entropy of drill locs to obtain conditional entropy
        h_cond_drill[i,j] = h_joint_loc - h_joint_drill
plt.imshow(h_cond_drill.transpose(), origin = 'lower left',
          cmap = 'gray', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
plt.imshow(h.transpose(), origin = 'lower left',
          cmap = 'gray', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
plt.imshow((h - h_cond_drill).transpose(), origin = 'lower left',
          cmap = 'RdBu', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
# define position of "drill":
n = 30
drill_i = [60] * n
drill_j = range(39,39-n,-1)
drill_locs = zip(drill_i, drill_j)
# determine joint entropy of drill_locs:
h_joint_drill = hspace.joint_entropy(model_sections, drill_locs, n_samples)
%%timeit
locs = drill_locs + [[50,20]]
h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
# generate conditional entropies for entire section:
h_cond_drill = np.zeros_like(h)
for i in range(100):
    for j in range(40):
        # add position to locations
        locs = drill_locs + [[i,j]]
        # determine joint entropy
        h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
        # subtract joint entropy of drill locs to obtain conditional entropy
        h_cond_drill[i,j] = h_joint_loc - h_joint_drill
plt.imshow(h_cond_drill.transpose(), origin = 'lower left',
          cmap = 'gray', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
plt.imshow((h - h_cond_drill).transpose(), origin = 'lower left',
          cmap = 'RdBu', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
Explanation: Intersting! Only a local reduction around the drilling position, however: extending to the deeper layers, as well! Why?
Drill deeper:
End of explanation
# define position of "drill":
n = 30
drill_i = [60] * n + [20] * n
drill_j = range(39,39-n,-1) + range(39,39-n,-1)
drill_locs = zip(drill_i, drill_j)
# determine joint entropy of drill_locs:
h_joint_drill = hspace.joint_entropy(model_sections, drill_locs, n_samples)
%%timeit
locs = drill_locs + [[50,20]]
h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
# generate conditional entropies for entire section:
h_cond_drill = np.zeros_like(h)
for i in range(100):
    for j in range(40):
        # add position to locations
        locs = drill_locs + [[i,j]]
        # determine joint entropy
        h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
        # subtract joint entropy of drill locs to obtain conditional entropy
        h_cond_drill[i,j] = h_joint_loc - h_joint_drill
plt.imshow(h_cond_drill.transpose(), origin = 'lower left',
          cmap = 'gray', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
plt.imshow((h - h_cond_drill).transpose(), origin = 'lower left',
          cmap = 'RdBu', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
Explanation: Interesting! And now both combined:
End of explanation
# define position of "drill":
n = 30
drill_i = [60] * n + [20] * n + [5] * n
drill_j = range(39,39-n,-1) + range(39,39-n,-1) + range(39,39-n,-1)
drill_locs = zip(drill_i, drill_j)
# determine joint entropy of drill_locs:
h_joint_drill = hspace.joint_entropy(model_sections, drill_locs, n_samples)
%%timeit
locs = drill_locs + [[50,20]]
h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
# generate conditional entropies for entire section:
h_cond_drill = np.zeros_like(h)
for i in range(100):
    for j in range(40):
        # add position to locations
        locs = drill_locs + [[i,j]]
        # determine joint entropy
        h_joint_loc = hspace.joint_entropy(model_sections, locs, n_samples)
        # subtract joint entropy of drill locs to obtain conditional entropy
        h_cond_drill[i,j] = h_joint_loc - h_joint_drill
# store results
f_out = open("h_cond_drill_i62_20_5_30.pkl", 'w')
pickle.dump(h_cond_drill, f_out)
f_out.close()
Explanation: We can see that now only a part on the left remains with significant uncertainties. So, let's "drill" into this, as well:
End of explanation
plt.imshow(h_cond_drill.transpose(), origin = 'lower left',
          cmap = 'gray', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
plt.imshow((h - h_cond_drill).transpose(), origin = 'lower left',
          cmap = 'RdBu', interpolation = 'none')
plt.colorbar(orientation = 'horizontal')
# plot drilling positions above it:
dp = np.array(drill_locs).transpose()
plt.plot(dp[0], dp[1], 'ws')
plt.xlim([0,100])
plt.ylim([0,40])
Explanation: Additional idea to speed up computation (especially for higher multivariate examples): do not estimate value at locations where conditional entropy of subset (i.e.: often previously calculated)! (Check: theoretical reason/ justification!)
End of explanation
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