Covariance matrix and dense pilot-point set-up #470
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Hi, Considering the way PEST-IES operates, limitations to pilot-point usage (density) is no longer as big of a problem as it has been with past PEST-methods. However, when using a quite dense pilot point pattern. Say for instance a model of 3 km by 3 km and a square pattern of pilot points with 100 m distance between them. If I were to construct a covariance matrix using the following workflow. where the tpl-file is generated for the pilot points respective location. make a dataframe called df_pp using pyemu helper and the pilot point template file hkpp.dat.tpldf_pp = pyemu.pp_utils.pp_tpl_to_dataframe(os.path.join(tmp_d,"mod0000.Kx.KL1.1.pp.tpl")) v = pyemu.geostats.ExpVario(contribution=1.0, a=2500.0) prior_cov = gs.covariance_matrix(df_pp.x, df_pp.y, df_pp.parnme) This gives me the following covariance matrix.
I'm not entirely sure of how to interpret it. I'm guessing since the matrix is built for each pilot-point (?) it gives the following PEST-IES run the option to have quite large in-between differences, heterogenity. If i would like to provide pest instructions on smoothing out the parameter field for HK while using the initial values from a zones of constant hydraulic conductivity (based on known geological units) where pest is allowed to change the values of each zone within ranges reasonable for the given geology, how would one approach the problem and how would one go about constructing the covariance matrixes? Is there a better work flow, say prior ensambles? I'm a bit lost so advice would be helpful! Sincerely D |
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Replies: 2 comments 7 replies
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Can we see what If I have a model where there are "known" (in quotes bc they are never really known) zones of hydraulic property boundaries, then I usually add a set of zone-type parameters (estimating the value of HK in each zone) and then use a set of pilot point multipliers that can interpolate across zone boundaries (since we really never know exactly where those boundaries are). This results in having the expert knowledge about the zones represented in the prior ensemble, but without being so strict that there is no mechanism to disregard/modify those zone boundaries. The key to those multiplier pilot points is to set their bounds so that they yield an acceptable range of resulting property values, but this is very problem specific... |
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I'll paste the df_pp below (long). Yeah, known is pretty far stretched. But as expert knowledge goes over a fairly large domain, it is the best available information based on years of field surveying. So lets just say, the type of soil is "known" (in quotes) and is as close to the truth as one can get at this stage.. Add to this the value range for each soil type based on literature values and we have at least some idea of what to aim for. Goal would then be to use PEST to apply PEST calibration techniques to interpret the hydraulic conductivity parameter-field for each geologic zone. thus allow for heterogeneity within each zone within reasonable parameter bounds for each unit, while holding each unit separate in terms of hydraulic properties. Also, conceptually I would expect the k-array to be quite smooth and not make these large changes over short distances, thus I would like some form of regularization applied. I'm not sure I entirely understood the following part
I wouldn't want to add a constant value over the zone with a single calibration multiplier for the whole area. I would want to interpolate the values with pilot points still. However, the zones would have different initial parameters and different ranges of parameter allowance. Reasonably one would want to smooth the transition between zones as well, but that is maybe something that can be done with your approach to add a set of pilot points across the border, or some regularization scheme.
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If you set the initial value for each pilot point to be the expected value in the zone, that would get you started and would give you smooth zone boundaries. And if you use a longer correlation length with the variogram, this will give you smoother and smoother heterogeneity. Otherwise, you could blank/zero out the off-diagonal entries in that covariance matrix that correspond to pilot points that are in different zones. This would cause sharper transitions at the zone boundaries. Its gets tricky if you have different parameter upper/lower bounds and/or variograms for each zone. In this case, you would need to loop over each zone, get the corresponding pilot points for said zone into a dataframe, pass that to |
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I think i managed to get everything set up correctly. Still struggling a bit with the prior information and understanding how it should be constructed for good behavior of my pest runs. However, I find my self facing other issues. Run in regularisation mode my k-field change very little (perhaps the way it tried to stay close to initial values weigh in to much?). If instead run in estimation i get expected changes but still it seems like it doent try out the full range of parameters. I havent faced this issue with pest-hp for instance. Any clues on what might go wrong in a pestpp-glm run? |
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In general, pest_hp has a much better capabilities for regularized minimum-error-variance parameter estimation - the regularization solver in pestpp-glm is not nearly as advanced so im not surprised that pest_hp does better than pestpp-glm. And just to reiterate, pestpp-ies is not focused on finding a single min-err-var parameter set so prior info equations do not apply - instead pestpp-ies wants to explore that non-uniqueness that regularization removes... |
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Right! I suppose my "ducking" from running in a full Python environment will be unsustainable in the long run since I have limited options to do forward simulations/predictive modeling with many parameter sets from within a GUI. But IF one still would attempt to utilize par-fields generated from IES in the above described situation. What would be a reasonable workflow? I have read through the relevant parts of the documentation several times but the concepts for IES still confuse me in some respects. For instance, IES generate one (1) .rei and .par file for each run however many ensembles you run. What do you make of that? is it the "lowest phi"-ensemble that ends up in that file or why not all of them? |
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IES writes the so-called "base" realization to par and rei files each iteration (you can read up the base realization - its just the control file parameter values), so that if you need a single model (who needs just a single model?!), you have easier access to the base realization. |
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Thanks! You'd probably be surprised of how many (especially consultants) who "needs" just one single model. Quotations as I'm not sure if people in general knows of what they actually need or not... |
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If you set the initial value for each pilot point to be the expected value in the zone, that would get you started and would give you smooth zone boundaries. And if you use a longer correlation length with the variogram, this will give you smoother and smoother heterogeneity. Otherwise, you could blank/zero out the off-diagonal entries in that covariance matrix that correspond to pilot points that are in different zones. This would cause sharper transitions at the zone boundaries.
Its gets tricky if you have different parameter upper/lower bounds and/or variograms for each zone. In this case, you would need to loop over each zone, get the corresponding pilot points for said zone into a da…