Table of Contents
Alternative simple models
Simple models, motivation
Hydrogeologists solving practical groundwater problems face significant uncertainty, which is given by the lack of relevant data and their interpretation ambiguity. Mathematical models are used to understand the ongoing processes and to support decision-making. Because of the uncertainty of the data and its interpretation, it is appropriate to create multiple simple models – a “multiple model ensemble” (Uusitalo et al. 2015) – that can be later developed into more complex models. Even simple models can help to select a strategy of further exploration and data collection and to support preliminary decision-making.
In this context, simple models can be closed-form solutions and spatial analyses, which often make use of aggregated data and use simplified assumptions (e.g. geometry of the modelling domain is simplified to a single rectangle). Such models usually compute for example: balance (e.g. of water or chemicals), water flux (Darcy’s law and continuity equation), flux of solutes, or residence time. Slightly different are spatial computations performed mostly by geographical information systems (GIS) – e.g. the pumping rate based on the distance from the fringe of the contaminant plume. Available screening modelling tools for contaminant transport (e.g. length of steady-state contaminant plume) can be considered as simple ones too. To support the formulation of conceptual models it is necessary to visualize and analyze data and to perform data aggregation and common calculation and estimations (e.g. average hydraulic gradient, geochemical background, or redox conditions from chemical composition).
The issues mentioned above require an information system that facilitates the collection of data from various semi-structured sources, visualization, and analysis of data in order to create alternative conceptual models and implement the corresponding simple procedural models. Interoperability with third-party modelling software is also a requirement.
Dealing with Uncertainty — Multiple Simple Groundwater Models
Heterogeneous subsurface is usually characterized by scarce data – therefore, it is reasonable to fully exploit available data and develop multiple simple models. The strategy of multiple simple models and especially alternative (equifinal) simple models is demonstrated in three case studies. Those models have diverse structure, are based on diverse assumptions, or they test diverse hypotheses.
Case study 1
In the first case study, four alternative models represent four alternative hypotheses explaining loss of water in the Smědá river in the Czech Republic. One hypothesis (No 4) quantifies a natural process – the effluence to the subglacial channel. The water loss can be explained this way; therefore the hypothesis was not rejected. Other hypotheses explain the loss by seepage to the surface coal mine through diverse geological layers.
A closed-form solution with simplified assumptions (e.g. geometry of the modelling domain is simplified to a single rectangle) was used. The case study represents the equifinality principle.
- Glacifluvial gravel (not rejected) – green arrow
- Sandy layer (max. 10 l/s → rejected) – red arrow
- Crystalline rock (could not be quantified → not rejected) – yellow arrow
Surface water seeps the subglacial channel (a trench formed by the quaternary continental glacier filled with sediment). It is the only natural explanation – not caused by mining. Hydraulic conductivity 3∙10–3 m/s of the gravel in the subglacial channel would explain the water loss. Hydraulic conductivity was not measured. The hypothesis was not rejected.
A comprehensive explanation of the water loss is the combination of all hypotheses.
Case study 2
The second case study combines different simple models for different parts of a site contaminated by petroleum hydrocarbons. The maximal extent of the contaminant plume (downgradient from the contamination source) is estimated by computing steady-state plume length by three methods implemented in the CoronaScreen (Wilson et al. 2005) code. The maximal permitted pumping rate upgradient is calculated by a simple spatial computation based on the distance from the plume's fringe. The pumping rate is limited by recharge from precipitation and lateral influx of the capture zone of the exploited well.
Downgradient: Steady-state plume
Steady-state plume length = distance from the source of the pollution where the pollution is depleted by natural attenuation (biodegradation etc.)
CoronaScreen code (Wilson et al. 2005)
- Electron balance model
- Analytic model
- Moving 1D
CoronaScreen does not overestimate biodegradation because it does not overestimate mixing of contaminated water (donors of electrons) with pristine water (electron acceptors).
Upgradient: Limited pumping
Comparison of Case study 1 and Case study 2
Table: Comparison of the approaches used in the case studies 1 and 2.
|Case study 1 – Hypotheses testing||Case study 2 – Prediction for water resources management|
|Steady-state plume||Limited pumping|
|Processes||Darcy flux||Natural attenuation||Water balance|
|Domain||Different depth horizons||Downgradient||Upgradient|
|Apparatus||Closed-form solution||Closed-form solution, numerical model PHREEQC||Closed-form solution|
|Resulting quantity||Water flux||Area that could be contaminated||Maximal permitted pumping|
Tenability of the models in case study 1 and 2 is semi-quantitatively evaluated by the pedigree matrix of Refsgaard et al. (2006).
|Score||Supporting empirical evidence||Theoretical understanding||Representation of understood underlying mechanisms||Plausibility||Colleague consensus|
|Proxy||Quality and quantity|
|15||Not correlated and not clearly related (0)||Historical/field data uncontrolled experiments small sample direct measurements (3)||Well-established theory (4)||Aggregated parameterized meta model (2)||Highly plausible (4)||Competing schools (2)|
Keywords: ground water; multiple model ensemble; simplicity; equifinality
REFSGAARD, Jens Christian, Jeroen P. VAN DER SLUIJS, James BROWN and Peter VAN DER KEUR, 2006. A framework for dealing with uncertainty due to model structure error. Advances in Water Resources. 29(11), 1586–1597. ISSN 0309-1708. DOI: 10.1016/j.advwatres.2005.11.013
WILSON, R.D., S.F. THORNTON, A. HUETTMANN, M. GUTIERREZ-NERI and H. SLENDERS, 2005. CoronaScreen : Process-based models for natural attenuation assessment: guidance for the application of NA assessment screening models. 2005. University of Sheffield, UK; TNO Institute of Environmental Sciences, The Netherlands. https://www.sheffield.ac.uk/media/19256/download?attachment
Simplic: Spherical cow:
Quotations about simplicity and complexity of groundwater models
“Jamieson (2000) pointed out that scientists live in a highly competitive environment where funding for research is limited. Involvement in policy-modeling projects helps scientists present themselves as real-world problem solvers, which helps secure funding for their scientific pursuits.” (Clement 2011, p. 626)
“The simulation results can be used to construct reasonable qualitative arguments as to why certain processes or events can or cannot occur. However, it is important that we understand the limits of these tools and recognize that they are better viewed as computer-aided thinking tools rather than computer-aided prediction tools.” (Clement 2011, p. 627)
“In summary, ‘hydrogeologic science’ is not well suited to quantitative prediction and is best suited to providing hydrogeologists with theoretical and science-based intuition that they can apply when suggesting solutions to complex practical problems. Hydrogeologists are faced with being primarily descriptive scientists rather than quantitative scientists, and can employ quantification only to the extent they ‘believe’ is meaningful and useful. Some hydrogeologic problems cannot be solved – they are too complex. This must be admitted and efforts should rather be applied to solvable problems. It should be a major role of hydrogeologists to help managers define the practical questions that are possible to answer.” (Voss 2005, p. 6)
“There is a direct correlation between the power of personal computers (PCs) and the complexity of models. The question is: Has the added complexity increased the level of groundwater understanding?” (Bredehoeft 2010, p. 328)
“Oreskes (2003) noted that we tend to have more intuitive faith in complex models because they allow us to simulate more processes. However, as we add more processes (and parameters) to a model, the overall certainty of its predictions might decrease. Ironically, the “truer’ the model, the more difficult it is to show that it is “true’.” (Clement 2011, p. 625)
“Freyberg (1988) noted that in a modeling class he taught, predicted system response was better simulated with more parsimonious but less well-calibrated models than with models calibrated using a large number of parameters to obtain a good fit (a phenomenon often referred to as ‘point calibration’).” (Hunt et al. 2007, p. 254)
“Victor Baker, the former President of the Geological Society of America, said ‘allowing the public to believe that a problem can be resolved … through elegantly formulated … models is the moral equivalent of a lie’ (Pilkey and Jarvis 2007, p. 188).“ (Clement 2011, p. 627)
“Models lie and liars model.”
“During the 1960s, when the first numerical models appeared, there was great anticipation about their ability to solve a large number of practical problems in hydrogeology. This excitement has since undergone moderation. It is now recognized that numerical models can only be valuable in providing insights into the potential behavior of complex hydrogeological systems, and to test alternate hypotheses to better understand observed phenomena. Considering the inaccessibility of the Earth’s subsurface, the pervasive heterogeneity on many spatial scales, the strong interactions among fluid flow, deformation, heat flow, and chemical interactions, and lack of knowledge of future forcing functions, it will not be prudent to assume that numerical models will predict the future with confidence, even with the availability of the most powerful computing machines.” (Narasimhan 2005, p. 18)
“Bredehoeft’s statement, ‘For me the model is not an end in itself, but rather a powerful tool that organizes my thinking and my engineering judgment,’ is insightful. When we generate model results to very complex systems, the numbers are not as important as the patterns of behavior the numbers suggest. Often, the challenge is to assure ourselves that the results are credible. We meet this challenge by invoking our experience and intuition.” (Narasimhan 2010; Bredehoeft 2010, p. 328)
“[Groundwater models] … are tools that are of great value in helping us test alternate hypotheses and behavioral possibilities. If the possibilities are well constrained with data, we treat them as predictions […] The real value of the models is that they give a quantitative form to what we know qualitatively. In some cases, model output may reveal certain patterns we did not expect a priori. These “anomalies” occasionally help us comprehend the existence of unusual phenomena.” (Narasimhan 2010)
“Leavesley et al. (2002) proposed a new modelling paradigm: ‘this concept requires that we change the question of ‘which model is most appropriate for a specific set of criteria?’ to ‘what combination of process conceptualisations is most appropriate?’” (Branger et al. 2010, p. 1673)
“Wolfram feels that science is far too ad hoc, in part because the models used are too complicated and/or unnecessarily organized around the limited primitives of traditional mathematics. Wolfram advocates using models whose variations are enumerable and whose consequences are straightforward to compute and analyze.” A New Kind of Science
“Booch et al. defined a model: ‘simplification of reality created to better understand the system being created’.” “Modeling should constitute a scientific expression of our ignorance rather than a claim to knowledge that we do not possess.” (Doherty 2011, p. 455)
“So why do we not embrace models as tools for encapsulating our knowledge and quantifying our ignorance? One reason is that human beings have always wanted to ‘see’ the future as if the veil of time were lifted. In pursuing his time-honored endeavor, complex models are the current prophetic tool of choice.” (Doherty 2011, p. 455)
“Simple models, on the other hand, are fast and stable—and they can dance.” (Doherty 2011, p. 455)
“However, all too often complexity wins the day—often for no other reason than to preemptively circumvent criticism that the model does not ‘look like’ what we imagine reality to look like.” (Doherty 2011, p. 455)
“Fools ignore complexity. Pragmatists suffer it. Some can avoid it. Geniuses remove it. …Simplicity does not precede complexity, but follows it. … In seeking the unattainable, simplicity only gets in the way. … You can't communicate complexity, only an awareness of it.” (Perlis 1982) – částečně citováno v hydrogeologickém kontextu: (Voss 2011a, p. 1281)
“Il semble que la perfection soit atteinte non quand il n’y a plus rien a ajouter, mais quand il n’ y a plus rien a retrancher. (It seems that perfection is reached not when there is nothing more to add, but when nothing more can be removed.) (Terre des Hommes [Land of People] by Antoine de Saint Exupéry, a writer, poet and aviator; Saint Exupéry 1939)” (Voss 2011a, p. 1458)
“Simplicity is the final achievement. After one has played a vast quantity of notes and more notes, it is simplicity that emerges as the crowning reward of art. (Frédéric Chopin, a musician and composer, quoted in If Not God, Then What? by Fost 2007)” (Voss 2011b, p. 1455)
“Managers need to be educated regarding what model analysis can and cannot provide. The model should generally not be what is contracted as a product, as is most often the case today; rather, an improvement of understanding of the system in question should be contracted, and particular advice sought from the analyst, who may or may not choose to employ groundwater modeling toward achieving this goal. Managers should buy advice from a competent hydrogeologist; they should not buy a groundwater model.” (Voss 2011b, p. 1457)
“The methods of science depend on our attempts to describe the world with simple theories. Theories that are complex become unstable, even if they happen to be true. Science may be described as the art of over-simplification: the art of discerning what we may with advantage omit.” (Popper 1982) Citováno podle (Hill a Tiedeman 2007, p. 268)
“If you can't reduce a difficult engineering problem to just one 81×11-inch sheet of paper, you will probably never understand it. (Ralph Brazelton Peck, a soil mechanics engineer, quoted in DiBiagio and Flaate2000).” (Voss 2011a, p. 1284)
“Models are to be used, not to be believed in! Dooge (1972)” Citováno podle Ebel a Loague (2006, p. 2887)
“Our models should be designed expressly to maximize the possibility of discovering that of which we are ignorant. Beck (2002)” Citováno podle: (Ebel a Loague 2006, p. 2887)
“Kirkby (1996): Models are thought experiments which help refine our understanding of the dominant processes acting … While most simulation models may be used in a forecasting mode, the most important role of models is as a qualitative thought experiment, testing whether we have a sufficient and consistent theoretical explanation of physical processes. The best model can only provide a possible explanation which is more consistent with known data than its current rivals. Every field observation, and especially the more qualitative or anecdotal ones, provides an opportunity to refute, or in some cases overturn, existing models and the theories which lie behind them.” (Ebel a Loague 2006, p. 2895)
“… application of distributed hydrological models is more an exercise in prophecy than prediction.” (Beven 1993, p. 41)
“… looking for more hydrological understanding through detailed distributed modelling is a dead-end track …” (Savenije 2001, p. 2835)
“The fact that [the model] is an approximation does not necessarily detract from its usefulness because models are approximations. All models are wrong, but some are useful.“ (Box a Draper 1987, s. 424) – as cited in Bakker (2013, p. 313).
Engineering models must compromise between simplicity and realism. “The best solution emphasizes the former without undue violence to the latter”. (DiToro 2001)
“…simple models are a good place to start because their transparent features provide clarity. A simple model is something to build on. In its sleek lines and limited assumptions, it can provide a base for elaboration while capturing the essence of a variety of more detailed possible explanations.” Levin (2007), as cited in Fatichi et al. (2016).
A wise man has great power; and a knowledgeable man increases strength; for by wise guidance you wage your war; and victory is in many advisors. Bible: Proverbs 24:5–6
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Good practice in groundwater modelling form the PEST guy
Czech version of this document is much more elaborate because it references part of my updated dissertation that contains both theoretical part and several case studies.