Addressing challenges in uncertainty quantification. The case of geohazard assessments

Abstract

By describing critical tasks in quantifying uncertainty using geohazard models, we analyse some of the challenges involved. Under the often-seen condition of very limited data and despite the availability of recently developed sophistications to parameterise models, a major challenge that remains is the constraining of the many model parameters involved. However, challenges also lie in the credibility of predictions required in the assessments, the uncertainty of input quantities, and the conditional nature of the quantification on the choices and assumptions made by analysts. Addressing these challenges calls for more insightful approaches that are yet to be developed; however, clarifications and reinterpretations of some fundamental concepts together with practical simplifications may be required first, and these are discussed in this paper. The research aims at strengthening both the foundation of geohazard risk assessments and its practice.

Full text

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Addressing challenges in uncertainty quantification. The case
of geohazard assessments
Ibsen Chivata Cardenas1, Terje Aven1, Roger Flage1
1Department of Safety, Economics and Planning, University of Stavanger, Stavanger, 4021, Norway
Correspondence to: Ibsen Chivata Cardenas (ibsen.chivatacardenas@uis.no)
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Abstract. By describing critical tasks in quantifying uncertainty using geohazard models, we analyse some of the
challenges involved. Under the often-seen condition of very limited data and despite the availability of recently
developed sophistications to parameterise models, a major challenge that remains is the constraining of the many
model parameters involved. However, challenges also lie in the credibility of predictions required in the
assessments, the uncertainty of input quantities, and the conditional nature of the quantification on the choices and
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assumptions made by analysts. Addressing these challenges calls for more insightful approaches that are yet to be
developed; however, clarifications and reinterpretations of some fundamental concepts together with practical
simplifications may be required first, and these are discussed in this paper. The research aims at strengthening both
the foundation of geohazard risk assessments and its practice.
1 Introduction
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Uncertainty quantification, UQ, helps determine the uncertainty of a system’s responses when some quantities and
events in such system are unknown. Using models, system’s responses can be calculated analytically, numerically,
or by random sampling (Monte Carlo method, rejection sampling, Monte Carlo sampling using Markov chains,
importance sampling, subset simulation) (Metropolis and Ulam, 1949; Brown, 1953; Ulam, 1961; Hastings, 1970).
Given the high-dimensional and spatial nature of hazard events and associated quantities, sampling methods are
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frequently used because they result in a less expensive and more tractable uncertainty quantification in comparison
with analytical and numerical methods. In the sampling procedure, specified distributions of the input quantities
and parameters are sampled and respective outputs of the model are recorded, then the process is repeated as many
times as may be required to achieve the desired accuracy (Vanmarcke, 1984). Eventually, the distribution of the
outputs can be used to calculate probability-based metrics, such as expectations or probabilities of critical events.
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Model-based uncertainty quantification using sampling is now more often used in geohazard assessments, e.g.,
Uzielli and Lacasse (2007), Wellmann and Regenauer-Lieb (2012), Rodríguez-Ochoa et al. (2015), Pakyuz-
Charrier et al. (2018), Huang et al. (2021), Luo et al. (2021), Sun et al. (2021a).
In this paper, we consider recent advances in UQ and analyse some remaining challenges. For instance, we note
that, despite the availability of sophistications to parameterise models used for UQ, a major problem persists,
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namely the constraining of the many parameters involved. In practice, based solely on historical data, only some
parameters can be constrained (e.g., Albert, Callies, and von Toussaint, 2022). Another challenge is that model
outputs are not only conditional on the choice of model parameters, but also on input quantities, including initial
and boundary conditions. For example, a geological system model could be specified to include some structures
in the ground and geological boundary conditions (Juang et al., 2019). Such systems are usually time dependent
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and spatial in nature involving, e.g., possibly changing conditions (e.g., Chow, Li, and Koh, 2019). Incorporating
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uncertainties related to such conditions complicates the modelling and demands further acquisition of data. Next,
models could be accurate at reproducing data from past events but may be inadequate for unobserved outputs or
predictions, as might be the case when predicting, e.g., extreme velocities in marine turbidity currents, which are
driven by emerging and little understood soil and fluid interactions (Vanneste et al., 2019). Overlooking these
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challenges in a geohazard assessment implies that the quantification will only reflect some aspects of the
uncertainty involved. These challenges are, unfortunately, neither exhaustively nor clearly discussed in the
geohazard literature. Options and clarifications addressing these challenges are underreported in the field , yet
analysing these challenges can be useful in treating uncertainties consistently and providing meaningful results in
an assessment. This paper’s aim is thus to bridge the gap in the literature by providing an analysis and clarifications
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enabling a useful quantification of uncertainty.
It should be emphasised that, in this paper, we consider uncertainty quantification in terms of probabilities. Other
approaches to measure or represent uncertainty that have been studied by, for example, Zadeh (1968), Shafer
(1976), Ferson and Ginzburg (1996), Helton and Oberkampf (2004), Dubois (2006), Aven (2010), Flage et al.,
(2013), Shortridge, Aven, and Guikema (2017), Flage, Aven, and Berner (2018), and Gray et al. (2022a,b), will
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not be discussed here. The discussion about the complications in UQ related to computational issues generated by
numerical approximations including, for instance, sampling procedures are also beyond the scope of the current
work.
The remainder of the paper is as follows. In Section 2, based on recent advances, we describe how uncertainty
quantification using geohazard models can be conducted. Next, some remaining challenges in UQ are identified
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and illustrated. Options to address the challenges in UQ are discussed in Section 3. A simplified example, further
illustrating the discussion, is found in Section 4, while the final section provides some conclusions from this study.
2 Quantifying uncertainty using geohazard models
In this section, we make explicit critical steps in uncertainty quantification, UQ. We describe a general approach
to UQ that considers uncertainty as the analysts’ incomplete knowledge about quantities or events. The UQ
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approach described is restricted to probabilistic analysis. Emphasis is made on the choices and assumptions usually
made by analysts.
A geohazard model can be described as follows. We consider a system (e.g., debris flow) with a set of input
quantities X (e.g., sediment concentration, entrainment rate) whose relationships to the output quantities Y (e.g.,
runout volume, velocity, or height of flow) can be expressed by a set of models Ɱ. Analysts identify or specify X,
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Y, and Ɱ. A vector Θɱ (including, e.g., friction, viscosity, turbulence coefficients) parameterises a model ɱ in Ɱ.
The parameters Θɱ determine specific functions among a family of potential functions modelling the system.
Accordingly, a model ɱ can be described as a multi-output function with, e.g., Y = {runout volume, velocity,
height of flow}, and we can write Eq. (1) based on Lu and Lermusiaux (2021):
ɱ: Xs,t˟ Θɱ→Ys,t (1)
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ɱ ≡ (Eɱ, SGɱ, BCɱ, ICɱ) (2)
where y as realisations of Y are the model responses when elements in X take the values x at a spatial location s 
S and a specific time t  T, and parameters θɱ  Θɱ are used. In Eq. (1), Xis the set of input quantities,
T is the time domain, S is the spatial domain, Θɱ󰡁 corresponds to a parameter vector, and
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Y is the set of output quantities, with d = 0, 1, 2, or 3. The system is fully described if ɱ is specified in terms
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of a set of equations Eɱ (e.g., conservation equations), the spatial domain geometry SGɱ (e.g., extension, soil
structure), the boundary conditions BCɱ (e.g., downstream flow), and the initial conditions ICɱ (e.g., flow at t =
t0), see Eq. (2).
In the sampling approach to uncertainty quantification, specified probability distributions reflecting analysts’
uncertainty about input quantities are sampled many times, and the distribution of the produced outputs can be
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calculated. The output probability distribution for a model ɱ can be denoted as f(y|x,θɱ,ɱ) for realisations y, x, θɱ,
ɱ of Y, X, Θɱ, and Ɱ, respectively.
Betz (2017) has suggested that the parameter set is fully described by a parameter vector Θ in Eq. (3):
Θ={Θɱ,ΘX,Θε,Θo} (3)
in which, Θɱ relates to parameters of the model ɱ, ΘX are parameters linked to the input X, Θε is the vector of the
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output-prediction error ε, and Θo is the vector associated with observation/measurement errors, when historical
records are used. More explicitly, to compute an overall joint probability distribution, we may have the following
distributions:
• f(y|x,θɱ,ɱ) is the distribution of Y when X takes the values x, and parameters θɱ  Θɱ and a model ɱ 
Ɱ are used to compute y;
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• f(x|θX,ɱ) is the conditional distribution of X given the parameters θX  ΘX and the model ɱ. Note that
each ɱ defines which elements in X are to be considered in the analysis;
• f(x|x
,θo) is a distribution of X given the observed quantities X
 = x
 and the observations/measurement error
parameters θo  Θo;
• additionally, one can consider f(y*|y,θε,ɱ), which is a distribution of Y*, the future system’s response,
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conditioned on the model output y and the output-prediction error vector θε Θε. The output-prediction
error ε is the mismatch between the model predictions and non-observed system’s responses y*. ε is used
to correct the imperfect model output y (Betz, 2017; Juang et al., 2019).
If, for example, the parameters Θɱ are poorly known, a prior distribution π(θɱ|ɱ) weighing each parameter value
θɱ for a model ɱ is usually specified. A prior is a subjective probability distribution quantified by expert judgement
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that represents uncertainty about the parameters prior to considering information in data (Raices-Cruz, Troffaes,
and Sahlin, 2022). When some data is available, e.g., in the form of measurements ɗ = {Ŷ = ŷ, X
 = x
}, as a part of
different sources of data, i.e., ɗ  Ɗ, such parameter values θɱ or their distributions π(θɱ|ɱ) can be constrained by
back-analysis methods. Back analysis methods include matching experimental measurements ŷ and calculated
model outputs y using different assumed values 󰡁
󰆒, i.e., more formally, values for θɱ can be calculated as follows
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(based on Liu et al., 2022):
θɱ = argmin[ŷ-y(x
,󰡁
󰆒)] (4)
The revision or updating of the prior π(θɱ|ɱ) with data ɗ to obtain a posterior distribution described as π(θɱ|ɗ,ɱ)
is also an option in back analysis. The updating can be calculated as follows (based on Juang et al., 2019; Liu et
al., 2022):
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󰇛󰔿󰡁󰇜 󰇛󰔿󰇜󰇛󰡁󰇜
󰕏󰇛󰔿󰇜󰇛󰡁󰇜
(5)
where ℒ(θm|ɗ) = f(ɗ|θm) is a likelihood function, a distribution which weighs ɗ given θɱ.
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Similarly, we can constrain any of the distributions above, e.g., f(y|x,θɱ,ɱ), or f(x|θX,ɱ) to obtain f(y|x,θɱ,ɗ,ɱ) and
f(x|θX,ɗ,ɱ), respectively.
For a geohazard problem, it is possible to specify several competing models, e.g., distinct geological models with
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diverse boundary conditions, see Eq. 2. If available knowledge is insufficient to determine the best model, different
models ɱ can be considered, and the respective overall output probability distribution for, e.g., uncorrelated
models is computed as (Betz, 2017; Juang et al., 2019):
f(y|x,Θ,Ɗ,Ɱ) = ∑ f(y|x,θ,ɗ,ɱ)ω(ɱ|Ɗ,Ɱ) (6)
f(y|x,θ,ɗ,ɱ) = ʃ f(y|x,θ,ɱ)π(θ|ɗ,ɱ)dθ (7)
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In Eq. (6), ω(ɱ|Ɗ,Ɱ) is another distribution weighing each model ɱ in Ɱ.
The various models Ɱ, their inputs X, parameters Θ, and outputs Y, experimental data ɗ can be coupled all together
through a Bayesian network, as has been suggested by Sankararaman and Mahadevan (2015) or Betz (2017). One
possible configuration of a network coupling elements in Ɱ, X, Θ, Y is illustrated in Figure 1.
Figure 1: A configuration of a network coupling some elements in Ɱ, X, Θ, Y, Y*
The previous description of a general approach to UQ considers uncertainty as that reflected in the analysts’
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incomplete knowledge about quantities or events. In UQ, to measure or describe uncertainty, subjective
probabilities can be used and constrained using historical observations ɗ. It is also explicitly shown that model
outputs are conditional on historical observations ɗ made available and models Ɱ chosen by analysts including
the selection of several parameters Θ and initial and boundary conditions, BCɱ and ICɱ. Based on the above
description, in the following, we analyse some of the challenges that arise when conducting UQ.
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As mentioned, back-analysis methods help constrain some elements in Θ; however, given the considerable number
of parameters (see Eq. 1-3) and data scarcity, constraining Θ is often only achieved in a limited fashion. Back-
analysis is further challenged by the potential dependency among Θ or Ɱ and between Θ and SGɱ, BCɱ, ICɱ. We
also note that, back analysis, or more specifically, inverse analysis, faces problems regarding non-identifiability,
non-uniqueness, and instability. Non-identifiability occurs when some parameters do not drive changes on the
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inferred quantities. Non-uniqueness arises because there may be more than one set of fitted or updated parameters
that adequately reproduce observations. Instability in the solution arises from errors in observations and the non-
linearity of models (Carrera and Neuman, 1986). Alternatively, in specifying, e.g., a joint distribution f(x,θ) to be
sampled, analysts may consider the use of, e.g., Bayesian networks (Albert, Callies, and von Toussaint 2022).
However, under the usual circumstance of a lack of information, establishing such a joint distribution is
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challenging and requires, in many instances, that analysts encode many additional assumptions (e.g., prior
distributions, likelihood functions, independence, linear relationships, normality, stationarity of the quantities and
X
θo
X
θX
θm
Y
θε
Y*
ɱ:x˟θɱ
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parameters considered), see e.g., Tang, Wang, and Li (2020); Sun et al. (2021b); Albert, Callies, and von Toussaint
(2022); Pheulpin, Bertrand, and Bacchi (2022). A more conventional choice is that x or θ are specified using the
maximum entropy principle in an attempt to specify the least biased distributions possible on the given information
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(Jaynes, 1957). Such distributions are subject to the system's physical constraints based on some available data.
The information entropy of a probability distribution measures the amount of information contained in the
distribution. The larger the entropy, the less information is provided by the distribution. Thus, by maximising the
entropy over a suitable set of probability distributions, one finds the distribution that is least informative in the
sense that it contains the least amount of information consistent with the system's constraints. Note that a
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distribution is sought over all the candidate distributions subject to a set of constraints. This principle has been
questioned, since its validity and usefulness lie in the proper choice of physical constraints (Jaynes, 1957; Yano
2019). Doubts are also raised regarding the potential information loss when using the principle. Analysts usually
strive in making use of all knowledge available and avoiding unjustified information loss (Christakos, 1990; Flage,
Aven, and Berner, 2018).
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Options to address the parametrisation challenge also include surrogate models, parameters reduction, and model
learning (e.g., Lu and Lermusiaux, 2021; Sun et al., 2021b; Albert, Callies, and von Toussaint, 2022; Degen et al.
2022; Liu et al., 2022). Surrogate models are learnt in order to replace a complicated model with an inexpensive
and fast approximation. Parameters reduction is achieved based on either principal component analysis or global
sensitivity analysis to determine which parameters significantly impact model outputs and are essential to the
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analysis (Degen et al., 2022; Wagener, Reinecke, and Pianosi, 2022). Remarkably, versions of the model learning
option do not need any prior information about model equations Eɱ but require local verification of conservation
laws in the data ɗ (Lu and Lermusiaux, 2021). These approaches still require large past data sets, sourced
systematically, which is a frequent limitation in geohazard assessments. More importantly, however is that, like
many models, even those exhaustively validated, the credibility of unobserved surrogate model outputs can always
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be questioned, since, for instance, records may miss crucial events or models may fail to reproduce outputs caused
by recorded abrupt changes (e.g., extreme velocities of turbidity currents) (Alley, 2004; Woo, 2019). An additional
point is the issue of incomplete model response, which refers to a model not having a solution for some
combinations of the input variables (Cardenas, 2019; van den Eijnden, Schweckendiek, and Hicks, 2021).
In bypassing the described challenges when quantifying uncertainty, simplifications are usually enforced,
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sometimes unjustifiably, in the form of assumptions, denoted here by the set Ą, and this set can include one or
more of the assumptions listed in Table 1. Note that the set of assumptions can be increased with those assumptions
imposed by the use of specific models Ɱ (e.g., conservation of energy, momentum or mass, Mohr-Coulomb’s
failure criterion).
Table 1. Some enforced assumptions in UQ for geohazard assessments
Predictions (non-observed outputs) of Y are credible, despite models only reproducing responses based on
historical data ɗ={Ŷ=ŷ,X=x
}, ɗ  Ɗ.
A model has a solution for any combination of the input quantities X.
Elements in X are fully specified.
Elements in X are mutually independent.
The joint distribution f(x,θ) distributes according to the maximum entropy principle.
If measurements are available, some input quantities X are set to specific values x= x
.
Input quantities X are set to constant values x0, that is X=x0.
Some θ are set to specific point values and are mutually independent.
Some θ are independent of SGɱ, BCɱ, ICɱ.
SGɱ, BCɱ, ICɱ are set to be constant.
When some data ɗ is available in the form of measurements {ŷ, x
}, likelihood functions ℒ[ɱ(θ|ɗ)] are mutually
independent.
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3 Addressing the challenges in uncertainty quantification
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From the previous section, we saw that it is very difficult in geohazard assessments to meet data requirements for the
ideal parameterisation of models. Further, we have noted that, although fully parameterised models could potentially be
accurate at reproducing data from past events, these may turn out to be inadequate for unobserved outputs. We also made
explicit that predictions are not only conditional on Θ (including priors, likelihood functions and linked
hyperparameters) but possibly also on SGɱ, BCɱ, ICɱ, see Eq. (1-7). Ultimately, assumptions made also condition
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model outputs. More importantly, note that when only some model quantities or parameters can be constrained by data
ɗ, the modelling will only reflect some aspects of the uncertainty involved. If the above challenges remain unaddressed,
UQ lacks credibility. To address such challenges and provide increased credibility, clarifications and
reinterpretation of some fundamental concepts together with practical simplifications may be required, and these
are discussed in the following. Table 2 shows the major challenges found and how they are addressed in related
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literature, while in Table 3 some clarifications or considerations put forward by us are displayed. The discussion
in this section builds on previous analysis by Aven and Pörn (1998), Apeland, Aven, and Nilsen (2002), Aven and
Kvaløy (2002), Nilsen and Aven (2003), Aven and Zio (2013), Khorsandi and Aven (2017), and Aven (2019).
Table 2. Major challenges and options to address them in geohazard assessments
Challenges, CH
Options to address the challenges, O
Challenges related to the model outputs and system responses
CH1. Predictions of y lack credibility since these are
model outputs not recorded in the data ɗ,
CH2. A model does not have a solution for a feasible
combination of the input quantities X.
O1. Credibility of predictions is judged in terms of physical
consistency checks (Wagener, Reinecke, and Pianosi, 2022) and
by examining the ability of models to reproduce disruptive
changes recorded in the data (Alley, 2004).
O2. Predictions by Bayesian forecasting methods. Based on a prior
distribution for y, a posterior distribution of y is obtained by
including the information provided by the model prediction in the
form of model likelihood (Montanari and Koutsoyiannis, 2012).
Challenges related to input quantities
CH3. Data available ɗ may not include all the historic
crucial events or disruptive changes,
CH4. Some input quantities X remain unknown
(unidentified) to analysts during an assessment,
CH5. The distribution f(x) or the bounds of x are
unknown,
CH6. Some input quantities X may be mutually
dependent.
O3. Using the maximum entropy principle to specify the
distributions based on the choice of constraints regarding the
physics of the phenomena involved. Constraining the distributions
by data including data other than measurements to reduce
unjustified information loss (Jaynes, 1957; Christakos, 1990; Betz,
2017; Yano 2019).
O4. Counterfactual analysis in which alternative events to
observed facts including disruptive changes are ʻimaginedʼ,
assumed, and explored to obtain alternative system’s responses
using models (Pearl, 1993; Woo, 2019).
O5. Exhaustive investigation of input uncertainty using the
assumptions deviation approach to specify input distributions
(Aven, 2013).
Challenges related to the parameters and models
CH7. The distribution f(θ) or the bounds of θ are
unknown,
CH8. Some θ may be dependent on SGɱ, BCɱ, or ICɱ,
CH9. Likelihood functions ℒ[ɱ(θ|ɗ)] may be mutually
dependent,
CH10. Models ɱ in Ɱ may be mutually correlated.
O3. Using the maximum entropy principle as described above.
O6. A joint distribution of Θ, SGɱ, BCɱ, ICɱ, X for each ɱ, can be
specified by encoding other assumptions (e.g., prior distributions,
likelihood functions, independence, linear relationships,
normality, stationarity) in Bayesian networks (Albert, Callies, and
von Toussaint, 2022).
O7. Using surrogate models, parameters reduction, and model
learning (Lu and Lermusiaux, 2021; Albert, Callies, and von
Toussaint, 2022).
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Table 3. Some clarifications and considerations to address the challenges in UQ
C1. Uncertainty reflects that analysts’ knowledge about quantities or events is incomplete.
C2. Models are simplifications, mainly used for understanding the performance of the system and approximating its
responses; they are part of the knowledge of the system, and they do not introduce uncertainty.
C3. The focus is on the quantification of the uncertainty of the system responses rather than on the accuracy of a model
reproducing recorded data.
C4. Predictions are conditional on the model(s) chosen together with a number of assumptions made by analysts.
C5. The specification of the joint distribution f(x,θ) cannot solely rely on the use of maximum entropy principle, but on the
full scrutiny of background knowledge Қ, and such a distribution is better specified using knowledge-based probabilities.
C6. Some elements in the parameter set Θ are not quantities or properties of the system as such, and there could not be
uncertainty about them.
C7. Analysts may choose a model or a set of them, which are believed or judged to be the best credible models.
Among the clarifications, we consider a major conceptualisation suggested by the literature, which is the definition
of uncertainty. Uncertainty refers to incomplete information or knowledge about a hypothesis, quantity, or the
occurrence of an event (Society for Risk Analysis, 2018). In Table 3, we denote this clarification as C1. Embracing
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this definition has some implications for uncertainty quantification using geohazard models. We make use of these
implications to address the major complications and challenges found. For instance, if uncertainty is measured in
terms of probability, one such implication is that analysts are discouraged to use the so-called frequentist
probabilities due to the fact these probabilities do not measure uncertainty or lack of knowledge. Rather such
probabilities reflect frequency ratios representing fluctuations or variation in the outcomes of quantities.
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Frequentist probabilities are of limited use because these assume that quantities vary in large populations of
identical settings, a condition which can hardly be justified for rather few geohazard quantities due to both the
often one-off nature of many geohazard features and the impossibility to verify or validate data by, e.g., a large
number of repeated tests. Thus, considering the usual constraints in data, as well as the nature of geohazard events,
a more meaningful and practical approach can be suggested to actually measure uncertainty, namely the use of
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knowledge-based (also referred to as judgemental or subjective) probabilities (Aven 2019). A knowledge-based
probability is understood as an expression of the degree of belief in the occurrence of an event or quantity by a
person assigning the probability and conditional on the available knowledge Қ. Such knowledge Қ includes not
only data in the form of measurements ɗ made available to the analysts at the time of the assessment, but also other
data sources in Ɗ together with the models Ɱ chosen by the analysts for the prediction, as well as the modelling
210
assumptions Ą made by those analysts. Accordingly, we have in total that, to describe uncertainty about, e.g., x or
y, probabilities are assigned based on Қ and, therefore, we acknowledge that those probabilities are conditional on
Қ. In the previous section, we have made explicit the conditional nature of uncertainty on measured data ɗ Ɗ
and models Ɱ and wrote, explicitly, for the overall output probability distribution, the expression f(y|x,Θ,Ɗ,Ɱ),
(see Eq. 6). If assumptions Ą are also acknowledged as conditioning uncertainty, we write more explicitly
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f(y|x,Θ,Ɗ,Ɱ,Ą) or equivalently f(y|x,Θ,Қ). We can therefore write:
f(y|x,Θ,Қ) = f(y|x,Θ,Ɗ,Ɱ,Ą) (8)
The meaning of this expression can be elucidated next. If, in a specific case, we would write f(y|x,θ,Ɗ), it means
that Ɗ summarises all the knowledge that analysts have to calculate y given (realised or known) x and θ. The full
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expression in Eq. 8 implies, accordingly, that to calculate y, and when knowing x and θ, the background knowledge
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includes Ɗ, Ɱ, Ą. Note that Қ can also be formed by observations, justifications, rationales, and arguments, thus
Eq. 8 can be further detailed to include these aspects of Қ. Structured methods exist to assign knowledge-based
probabilities (see, e.g., Apeland, Aven, and Nilsen, 2002; Aven 2019). Here we should note, however, that since
models form part of the available background knowledge Қ, models can also inform these knowledge-based
probability assignments. It follows that, based on knowledge-based input probabilities, an overall output
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probability distribution calculated using models is also of subjective character or knowledge based, although
uncertainty quantification using sampling could provide some frequency resemblance or interpretation (Jaynes,
1957). Some of the implications of using knowledge-based probabilities are described throughout this section.
According to the left column in Table 2, the focus of the challenges relates to the model outputs, more specifically
predictions (CH1 and CH2), input quantities (CH3-CH6), parameters (CH7-CH9), and models (CH10). We recall
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that as defined, uncertainty quantification helps determine the system’s response uncertainty based on specified
input quantities using models. Accordingly, the focus of an assessment is on the potential system’s responses.
When it comes to quantifying uncertainty, the focus is often on uncertainty about future non-observed responses
Y*, which are approximated by the model output Y considering some specified input quantities X. We recall that
then Y* and X* are quantities that are unknown at the time of the analysis, but will, if the system being analysed
235
is implemented, take some value in the future, and possibly become known. Thus, during an assessment Y* and
X* are the uncertain quantities of the system since we have incomplete knowledge about Y* and X*. Based on
these grounds, the output-prediction error ε (described in the previous section) being the mismatch between the
model prediction values y and non-observed system’s responses values y*, usually suggested to correct the
imperfect model output y, can only be specified on the basis on the scrutiny of Қ.
240
Another consequence of considering the definition of uncertainty put forward, which solely links uncertainty to
quantities or events, and taking into account that models are only mathematical artefacts, is that models, as such,
are not to be linked to uncertainty. Models, per se, do not introduce uncertainty, but they are likely to be inaccurate.
Accordingly, another major distinction is to be set in place. We recall that models, by definition, are
simplifications, approximations of the system being analysed; they express or are part of the knowledge of the
245
system and should therefore be solely used for understanding the performance of the system rather than for illusory
perfect predictions. In Table 3, we denote the latter clarification as C2.
Regarding the challenges CH1 and CH2, we should note that geohazard analysts are often more interested in
predictions rather than known system outputs. For instance, predictions are usually required to be calculated for
input values that are not contained in the validation data. We consider that predictions are those model outputs not
250
observed or recorded in the data, i.e., extrapolations out of the range of values covered by observations (e.g.,
mudflow extreme velocities). Thus, the focus is on the quantification of the uncertainty of extreme system’s
responses rather than on the accuracy of a model reproducing recorded data. This is the clarification C3 in Table
3. Considering this, models are yet to provide accuracy in reproducing observed outputs but, more importantly,
afford credibility in predictions. Such credibility is to be assessed, mainly, in terms of judgements since
255
conventional validation cannot be conducted using non-observed outputs. Recall that model accuracy usually
relates to the comparison of model outputs with experimental measurements (Roy and Oberkampf, 2011; Aven
and Zio, 2013) and is the basis for validating models. Regarding credibility of predictions, Wagener, Reinecke,
and Pianosi (2022) have reported that such credibility can be mainly judged in terms of the physical consistency
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of the predictions, by checks rejecting physically impossible representations of the system. The credibility of
260
predictions may also include the verification of the ability of models to accurately reproduce disruptive changes
recorded in the data (Alley, 2004). However, as we have made explicit in the previous section, models’ predictions
are conditional on a considerable number of critical assumptions and choices made by analysts (see Table 1 and
clarification C4 in Table 3). Therefore, predictions can only be as good as the quality of the assumptions made.
The assumptions could be wrong, and the examination of the impact of these deviations on the predictions must
265
be assessed. To provide credibility of predictions, such assumptions and choices should be justified and scrutinised,
ref. option O5 in Table 2. The option O5 addresses the challenge CH1; however, when conducting UQ using
models, O5 has a major role when investigating input uncertainty, which is discussed next.
A critical task in uncertainty quantification is the quantification of input uncertainty. As shown in Table 2, such
uncertainty is not only to be associated with the lack of knowledge of the distribution f(x) or bounds of x, CH5,
270
and some other sources are to be considered. For example, input uncertainty may originate from the situation that
historic crucial events or disruptive changes are missing in the records, CH3, or from the condition in which critical
quantities in X may remain unidentified to analysts during an assessment, CH4. We recall that analysts can
unintendedly fail to identify relevant elements in X due to insufficiencies in data or limitations of existing models.
For example, during many assessments, trigger factors that could bring a soil mass to failure could remain unknown
275
to analysts (e.g., Hunt et al. 2013; Clare et al. 2016; Leynaud et al. 2017; Casalbore et al. 2020). Uncertainty
quantification based on models requires simulating sampled values from X, and elements in X can possibly be
mutually dependent; however, the joint distribution of X, namely f(x), is often also unknown. This is the challenge
CH6. Considering the potential challenges CH3 to CH6, to specify f(x) we cannot solely rely on the use of the
maximum entropy principle, as described in the previous section, since it may fail to advance an exhausted
280
uncertainty quantification in the input, e.g., by missing relevant values not recorded in the measured data. This
would undermine the quality of predictions and therefore uncertainty quantification. Recall that the principle
suggests the use of the least informative distribution among candidate distributions constrained solely on
measurements. Using counterfactual analysis, as described in Table 2, is an option, but this will also fail in
providing quality in predictions, since this analysis focuses on the analysis of counterfactuals (a part of the data
285
Ɗ) and little on the overall knowledge available Қ. Note that system’s knowledge Қ includes, e.g., among others,
the assumptions made in the quantification of uncertainty, such as those shown in Table 1. Further note that such
assumptions not only relate to data, but also to input quantities, modelling, and predictions. Thus, it appears that
the examination of these assumptions should, desirably, be at the core in quantifying uncertainty in geohazard
assessments, as suggested in Table 2, option O5. The assessment of deviations of assumptions has been suggested
290
originally by Aven (2013) and Khorsandi and Aven (2017). An assumption deviation risk assessment evaluates
different deviations, their associated probabilities of occurrence, and the effect of the deviations. The major
distinctive features of the assumption deviation risk assessment approach are the evaluation of the credibility of
the knowledge Қ supporting the assumptions made, together with the questioning of the justifications supporting
the potential for deviations. The examination of Қ can be achieved by assessing the justifications for the
295
assumptions made, the amount and relevance of data or information, the degree of agreement among experts, and
the extent to which the phenomena involved are understood and can be modelled accurately. Note that justifications
might be in the form of direct evidence becoming available, indirect evidence from other observable quantities,
supported by modelling results, or possibly inferred by assessments of deviations of assumptions. This approach
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is succinctly demonstrated in the following section. Accordingly, we suggest specifying f(x) in terms of
300
knowledge-based probabilities in conjunction with the investigation of input uncertainty using the assumptions
deviation approach. This is identified as consideration C5 in Table 3.
Another point to consider is that, when uncertainty reflects analysts’ (lack of) knowledge about quantities or events
and is measured in terms knowledge-based probabilities, analysts should be aware that conditionality among
elements in X only implies that increased knowledge about, e.g., a quantity X1 will change the uncertainty about
305
another quantity X2, if X2 is conditional on X1. The expression that denotes this is conventionally written as X2|X1.
This interpretation may be exploited by analysts when specifying, e.g., the joint distribution f(x,θ). Analysts, for
example, may simplify the analysis when, according to the scrutiny of Қ, increased knowledge about, e.g., X1 will
not result in increased knowledge about another quantity, e.g., X2, and, for example, if a distribution f(y|x1,x2) is to
be specified, we may have then that f(y|x1,x2) reduces to f(y|x1)f(y|x2), according to probability theory. Apeland,
310
Aven, and Nilsen (2002) have illustrated how conditionality in the setting of knowledge-based probabilities can
inform the specification of a joint distribution.
The parameterisation problem, which involves the challenges CH7 to CH9 in Table 2, warrants exhaustive
consideration. Addressing these challenges also requires some reinterpretation. To start, we recall that, parameters,
by definition, are coefficients determining specific functions among a family of potential functions modelling the
315
system. Those parameters, eventually, constrain a model’s responses. Recall that, y as realisations of Y are the
model responses when X takes the values x and some parameters θ  Θ and models ɱ Ɱ are used. Thus, as
shown in the previous section, any output y is conditional on θ and so is the uncertainty attached to y. Further note
that, taking into consideration the definition of uncertainty previously stated, which attributes uncertainty to events,
quantities, or hypothesis solely, we may say that, unless a parameter is associated with a quantity or property of
320
the system, non-quantity parameters, being merely artefacts in the models, are not to be linked to any uncertainty.
This is identified as clarification C6 in Table 3. Using the uncertainty definition previously put forward and looking
at Eq. 3, analysts may consider that, for instance, parameters θε, which are linked to the output-prediction error ε,
some model parameters in Θɱ, the vector associated with observation/measurement errors Θo and the overall
attached hyperparameters linked to probability distributions (including priors, likelihood functions), are not
325
quantities or properties of the system as such; they are modelling artefacts, and therefore, it is questionable to
consider uncertainty about them. Thus, focused on the uncertainty of the system responses rather than model
inaccuracies, the analysts may consider strictly assigning uncertainty to those parameters that represent physical
quantities and fix with single values those that are not quantities as such. To help identify those parameters to
which some uncertainty can be linked, we can scrutinise, e.g., the physical nature of these. Note, however, that in
330
fixing parameters to a single value, we still can make use of back analysis procedures, as mentioned previously.
Analysts may have some additional basis to specify parameters values when the background knowledge available
Қ is scrutinised to verify that Қ, including not only data measurements but other sources of data, models,
assumptions made, and arguments, strongly supports a specific parameter value. Based on this interpretation,
setting the values of these non-quantity-based parameters to a single value reduces, considerably, the complications
335
in quantifying uncertainty. It also follows that, analysts are encouraged to make explicit that model outputs are
conditional on these fixed parameters, as well as on the model or models chosen, as we have shown in the previous
section. The latter interpretation also leads us to argue that the focus of the quantification is on the uncertainty of
the system response rather than the inaccuracies of the models. This, of course, implies that, in a practical sense
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and in the context of geohazard assessments, once a clear differentiation between parameters and input quantities
340
is made, and a model or a set of them believed or judged to be the best models providing the most credible
predictions are chosen, uncertainty quantification can then proceed. This parsimonious modelling approach is
identified as consideration C7 in Table 3. This latter consideration addresses, to an extent, the challenge CH10.
In the following section, we further illustrate the above discussion by analysing a documented case in which UQ
in a geohazard assessment was informed by modelling using sampling procedures.
345
4 Case analysis
To further describe the proposed considerations, we analyse a case reported in the specialised literature. The case
deals with the quantification of uncertainty of geological structures, namely uncertainty about the subsurface
stratigraphic configuration. Conditions in the subsurface are highly variable, whereas site investigations only
provide sparse measurements. Consequently, subsurface models are usually inaccurate. At a given location,
350
subsurface conditions are unknown until accurately measured. Soil investigation at all locations is usually
impractical and uneconomical, and point-to-point condition variation cannot therefore be known (Vanmarcke,
1984). Such uncertainty means significant engineering and environmental risk to, e.g., infrastructure built on the
surface. One way to quantify this uncertainty is calculating the probability of every possible configuration of the
geological structures (Tacher et al. 2006; Thiele et al. 2016; Pakyuz-Charrier et al. 2019). Sampling procedures
355
for UQ are helpful in this undertaking. We use an analysis and information by Zhao et al. (2021), which refer to a
site located in the Central Business District, Perth, Western Australia, where 6 boreholes were executed. The case
has been selected taking into account its simplicity to illustrate the points of this paper, but at the same time, it
provides details to allow some discussion. Figure 2 displays the system being analysed.
Figure 2: Borehole logs in colours and longitudinal section reported by Zhao et al. (2021) located in the Central Business
District, Perth, Western Australia. The records correspond to information of the six boreholes. Three types of materials
are revealed by the boreholes including sand (yellow), clay (magenta), and gravel (blue).
In the system under consideration, a particular material type to be found in a non-bored point, a portion of terrain
360
that is not penetrated during soil investigation, is unknown and thus uncertain. The goal is, therefore, the
computation of the probability of encountering a given type of soil in these points. In Zhao et al. (2021) the focus
is on calculating the probabilities of encountering clay in the subsurface, and the approach advocated was a
sampling procedure to generate many plausible configurations of the geological structures and evaluating their
4
5
6
Vertical direction (metres)
Horizontal direction (metres)
40
60
80
0
-10
-20
-30
0
20
1
2
3
y=?
y=?
y=?
y=?
y=?
One of many potential
configurations of the clay
SGm
BCm=?
X
θh=?
θv=?
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probabilities based on their frequencies. To calculate the probability of encountering a given type of soil c, p(y=c),
365
in a non-penetrated point in the ground, Zhao et al., (2021) used a function that depends on two correlation
parameters, namely the horizontal and vertical scale of fluctuation θh and θv. Note that spatial processes and their
properties are conventionally assumed as spatially correlated. Such spatial variation may presumably be
characterised by correlation functions, which depend on a scale of fluctuation parameter. The scale of fluctuation
measures the distance within which points are significantly correlated (Vanmarcke, 1984). Eq. 9 describes the
370
basic components of the model chosen by Zhao et al., (2021) (specific details are given in the Appendix to this
paper):
ɱ: Xs˟ Θɱ→Ys →p(y=c) (9)
where X is the collection of all quantities at borehole-points sx which can take values x from the set {sand, clay,
gravel}, according to the setting in Figure 2. Y is the collection of all quantities with values y at non-borehole
375
points sy. The values y together with the values x are sampled and probabilities computed on the basis of a chosen
model using the parameters θh=11,1 and θv=4,1 metres, θh, θv  Θɱ. Using the maximum likelihood method, the
parameters were determined based on the borehole data revealed at the site. In the determination of parameters,
the sampling of uniform and mutually independent distributions of θh and θv was the procedure advocated. The
system is further described by a set of equations Eɱ (a correlation function and a probability function), the spatial
380
domain geometry SGɱ (a terrain block of 30 x 80 metres), and the boundary conditions BCɱ (the conditions at the
borders). More details are given in the Appendix to this paper. Since this system is not considered time dependent,
the initial conditions ICɱ were not specified.
The summary results reported by Zhao et al. (2021) are shown in Figure 3. In Figure 3, the most probable
stratigraphic configuration along with spatial distribution of the probability of the existence of clay is displayed.
385
The authors focused their attention on this sensitive material, which likely represents a risk to infrastructure built
on the surface.
(a)
(b)
Figure 3: Zhao et al. (2021) findings shown in their Figure 9. (a) Most probable stratigraphic configuration. (b)
Spatial distribution of the probability of the existence of clay. Authorisation of reproduction: License number
5351891245835. Distances in metres.
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Zhao et al., (2021) stated that “characterisation results of the stratigraphic configuration and its uncertainty are
consistent with the intuition and the state of knowledge on site characterisation”. Next, throughout Zhao et al.’s
390
(2021) analysis the following assumptions were enforced (Table 4), although these were not explicitly disclosed
by the authors.
Table 4. Assumptions enforced by Zhao et al., (2021)
Predictions (non-observed outputs) are credible.
Likelihood functions ℒ[ɱ(θ|ɗ)] were set to be mutually independent.
For the determination of parameters and model, f(θh,θv) distributes according to the maximum entropy principle
and θh and θv are mutually independent.
Specified elements X are complete.
Input quantities X were set to the measured values, i.e., x = x
 (no inaccuracies in data).
θ are independent of SGɱ and BCɱ.
SGɱ, BCɱ were set to be constant.
Unfortunately, the authors did not report enough details on how the majority of these assumptions are justified.
We, however, should note that providing these justifications was not the objective of their research. Yet, we analyse
395
here how these can be justified by both scrutinising the supportive knowledge Қ and using some elements of the
assumption deviation approach, described in the previous section. Table 5 summarises the analysis conducted and
only reflects the most relevant observations and reservations identified by us. Accordingly, the information in
Table 5 may not be exhaustive but still useful for the desired illustration. Table 5 displays some observations by
us related to the credibility of the knowledge Қ. The examination of Қ is achieved by assessing the amount and
400
relevance of data or information, the extent to which the phenomena involved are understood and can be modelled
accurately, the degree of agreement among experts, and the justifications for the assumptions made. Observations
regarding the justifications for potential deviations of assumptions also form the analysis.
Not surprisingly, the observations in our analysis are concentrated on the predictions’ credibility. Recall that the
focus of UQ is on the system’s response, which is approximated by model predictions (considerations C2 and C3
405
in Table 3). We note, for example, that, although the use of correlations is an accepted practice and a practical
simplification, correlation functions appear to be counterintuitive to model geological structures or domains and
do not help much in understanding the system (consideration C2 in Table 3). Recall that such structures are mainly
disjoint domains linked to a finite set of possible categorical (masses of soil or rock) rather than continuous
quantities. Next, the variation of such structures can occur by abrupt changes of materials, thus the use of, for
410
example, smoothed functions, as correlation functions may be, to represent them requires additional consideration.
Further, the physical basis of the correlation functions is not clear and physical models based on deposition
processes may be suggested (e.g., Catuneanu et al., 2019). We should further note that a potential justification for
the deviation of the assumption regarding the credibility of predictions is that knowledge from additional sources
such as surface geology, sedimentology, local geomorphic setting, and structural geology was not explicitly taken
415
into account in the quantification of uncertainty. The revision of this knowledge can contribute to reduce the
probability of the deviation in predictions. Based on the observations in Table 5, we can conclude that there is
potential to improve the credibility of predictions.
420
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Table 5. Examination of supporting knowledge Қ and justifications for the potential deviation of assumptions (part a)
The amount and relevance of
data or information
The extent to which the phenomena
involved are understood, and
accurate models exist
The degree of agreement
among experts
Justifications for the
assumptions made
Justifications supporting the
potential deviations
Assumption: Predictions of Y are credible
The analysis is only based
on borehole information;
however, such investigation is
exceptionally exhaustive, 6
boreholes
The physical basis for using
correlations is dubious and models based
on the deposition process can be
considered
The use of correlations is
an accepted practice in the
field
Exhaustive borehole
information
The knowledge of surface
geology, sedimentology, local
geomorphic setting, and
structural geology was not
explicitly incorporated into
UQ
Variation of geological structures can
occur by abrupt changes, thus the use of
smoothed functions to represent them
requires additional consideration
Global rather than local correlation
between spatial quantities has been used,
possibly misrepresenting geological
structures variation
Assumption: Likelihood functions ℒ[ɱ(θ|ɗ)] were set to be mutually independent
Assumption: f(θh,θv) distributes according to the maximum entropy principle and θh and θv are mutually independent
Data of the six boreholes has
been used to calibrate the
model chosen; however,
knowledge of surface geology,
sedimentology, local
geologic/geomorphic setting,
and structural geology was not
explicitly incorporated into the
analysis
Based on the maximum
likelihood method, a model
judged to be the best model
was chosen
Dependency between θh
and θv, cannot be supported
by general knowledge and
such dependency hardly can
be enforced
An increased revision of Қ,
could have been useful to
specify f(x) and f(θ) providing
a richer information than that
suggested by the maximum
entropy principle, regarding
how X or Θ take values
A joint distribution
f(x,θ,sgɱ,bcɱ) could have
been investigated when
determining the models and
parameters, but establishing
such a joint distribution is
challenging and requires, in
many instances, that
analysts encode many
additional assumptions,
which hardly can be justified
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Table 5. Examination of supporting knowledge Қ and justifications for the potential deviation of assumptions (part b)
The amount and relevance of
data or information
The extent to which the phenomena
involved are understood, and
accurate models exist
The degree of agreement
among experts
Justifications for the
assumptions made
Justifications supporting the
potential deviations
Assumption: Specified elements X are complete
Input quantities were
considered fully specified
Another type of material
was revealed by other
soundings in the area, i.e., silt
(see sources in Zhao et al.
2021), and depending on the
revision of Қ
, this fourth
material could be considered.
Assumption: Input quantities X were set to measured values x=x

Errors during surveys may
have resulted in horizontal
positioning inaccuracies
Data was judged to be
accurate
There is usually not data
basis to calculate these
errors
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The choices made by Zhao et al., (2021) regarding the use of parameters with fixed values together with the choice
425
for a single best model can be highlighted and illustrate the points raised in the considerations C6 and C7 (Table
3), respectively. The maximum likelihood method supported these choices, a back analysis method focused on the
matching of measurements and calculated model outputs using different assumed values for θh and θv. We highlight
that, a model judged to be the best model was chosen. This includes the specification of a particular spatial domain
geometry SGɱ. Investigating the impact of variation of SGɱ was considered unnecessary. There was no need to
430
specify several competing models, which is line with our consideration labelled as C7 in this paper.
Although Zhao et al. (2021) investigated the joint distribution f(x), which was sampled to calculate probabilities,
someone can suggest that the joint distribution f(x,θ,sgɱ,bcɱ) could have been produced when determining the
models and parameters for calculating probabilities. Nevertheless, we can argue that establishing such a joint
distribution is challenging and requires, in many instances, that analysts encode many additional assumptions (e.g.,
435
prior distributions, likelihood functions, independence, linear relationships, normality, stationarity of the quantities
and parameters considered).
A more crucial reservation derived from the analysis of potential deviations of assumptions, which might impact
considerably the credibility of predictions, comes from revisiting the knowledge sources of Zhao et al.’s (2021)
analysis, available from https://australiangeomechanics.org/downloads/. Another type of sensitive material was
440
revealed by other soundings in the area, more specifically, silt. Depending on the revision of Қ, this fourth
suspected material could be analysed in an extended uncertainty quantification of the system. Note that, originally,
the input quantities X were assumed to take values x from the set {sand, clay, gravel}. Such an assumption was
based on the records of six boreholes which were believed accurate. The latter illustrates the relevance of the
consideration C5 in Table 3.
445
Another interesting choice made by Zhao et al. (2021) is that they disregarded the possibility of incorporating into
the modelling measurement errors in the borehole data, probably because this data was judged to be accurate. We
recall in this respect that these errors reflect the inaccuracy of the ground model rather than the uncertainty about
the system. We further note that, as stated for consideration C6 (Table 3), we can hardly justify attaching
uncertainty to measurement error parameters, since measurement errors are not a property of the system. The same
450
can be said for the parameters θh and θv, which are not quantities of the system. Note that their physical basis is
questioned. We should note, however, that assuming coefficients for the parameters θh and θv is an established
practice (Vanmarcke, 1984, Lloret-Cabot et al., 2014, Juang et al., 2019). We also point out that uncertainty
quantification in this kind of systems is to an extent sensitive to the choice of scale of fluctuation values
(Vanmarcke, 1984), and that the use of a global rather than local correlation between spatial quantities can
455
potentially misrepresent geological structures variation. Accordingly, a further examination of the existing Қ can
justify undertaking an assessment of the impact of a deviation of assuming a local rather than global scale of
fluctuation.
Overall, the Zhao et al. (2021) analysis is to an extent based on the previously suggested definition of uncertainty,
ref. the consideration C1 in Table 3.
460
We should stress that the Zhao et al. (2021) uncertainty quantification refers specifically to the ground model
described at the beginning of this section. In other words, the probabilities displayed in Figure 3b are conditional
on the parameters chosen (θh=11,1 and θv=4,1 metres); the model selected (described by the Eq. 9, A-1 and A-2 in
the Appendix to this paper); the specified spatial domain geometry SGɱ (a terrain block of 30 x 80 metres); and
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ultimately the assumptions made (listed in Table 4). This information is to be reported explicitly to the users of
465
the results. This reflects the clarification C4 in Table 3.
Regarding the consideration of subjective probabilities, there has been agreement, to an extent, on their use in this
kind of UQ since Vanmarcke (1984). However, the use of knowledge-based probabilities in the extension
described here is recommended given the illustrated implications to advance UQ (as discussed in the previous
section and stated in consideration C5). For example, increased examination of Қ, might have resulted in using a
470
more informative distribution f(θh,,θv) other than the uniform distribution, and in turn, different values for θh and
θv, as well as a different model. Recall that the selection of the model and determination of parameters were based
on the maximum likelihood method, which only makes use of measured data ɗ. Note, however, that when Zhao et
al. (2021) calculated the probabilities, they sampled an improved joint distribution f(x) using the parameters θh and
θv and a chosen model.
475
In our analysis of Zhaoʼs et al. (2021) assessment, the examination of supporting knowledge Қ resulted essentially
in:
(i) judging the credibility of predictions;
(ii) providing justifications for undertaking an assessment of assumption deviations considering, e.g., further
analysis involving the modelling of a fourth material;
480
(iii) considering additional data other than the borehole records, such as surface geology, sedimentology, local
geomorphic setting, and structural geology;
(iv) analysing the possibility of distinct geological models with diverse spatial domain geometry and local
correlations; and
(v) ultimately, further examining the existing Қ.
485
5 Conclusions
In this paper, we have discussed challenges in uncertainty quantification, UQ, based on models, for geohazard
assessments. Beyond the parameterisation problem, the challenges include how to assess the quality of predictions
required in the assessments, the quantification of uncertainty in the input quantities, the consideration of the impact
of choices and assumptions made by analysts. Such challenges arise from the common-place situation of limited
490
data and the one-off nature of geohazard features. If these challenges are kept unaddressed, UQ lacks credibility.
Here, we have formulated seven considerations that may contribute to providing increased credibility in the
quantifications. For example, we proposed understanding uncertainty as lack of knowledge, a condition that can
only be attributed to quantities or events in the system under consideration. Another consideration is that the focus
of the quantification should be more on the uncertainty of the system response rather than the accuracy of the
495
models used in the quantification. We drew attention to the clarification that models, in geohazard assessments,
are simplifications used for predictions approximating the system’s responses. We have also considered that since
uncertainty is only to be linked to properties of the system, models, as such, do not introduce uncertainty.
Inaccurate models can, however, produce poor predictions, and under these circumstances, the uncertainty about
system response is to be judged as large. Such models should be rejected, and increased examination of background
500
knowledge will be required to credibly quantify uncertainty. We also put forward that there could not be
uncertainty about those elements in the parameter set that are not quantities or properties of the system. The latter
also has pragmatic implications concerning, e.g., how the many parameters in a geohazard system could be
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constrained in a geohazard assessment.
We went into details to show that predictions, and in turn UQ, are conditional on the model(s) chosen together
505
with the assumptions made by analysts. Based on the identified limitations of measured data to support the
assessment of the quality of predictions, we have proposed that the quality of UQ is to be judged based on crucial
tasks such as the exhaustive scrutiny of the knowledge coupled to the assessment of deviations of those
assumptions made in the analysis.
Key to enacting the proposed clarifications and simplifications is the full consideration of knowledge-based
510
probabilities. Based on the proposed examination of strength of knowledge, knowledge-based probabilities can be
assigned. Considering this type of probabilities will also help overcome the identified limitations of the maximum
entropy principle or counterfactual analysis to quantify uncertainty in input quantities. We have exposed that the
latter approaches are prone to produce unexhausted uncertainty quantification due to their reliance on measured
data, which can miss crucial events or overlook relevant input quantities.
515
Appendix
In this Appendix, the necessary details of the original analysis made by Zhao et al. (2021) are given. The following
are the basic equations Eɱ used by these authors.
󰇛󰇜󰥟
  󰥟


(A-1)
 󰇧







󰇨
(A-2)
where X is the collection of all quantities at borehole-points, which take values x, Y is the collection of all quantities
520
at non-borehole-points with values y,  is the value of correlation between a quantity value x at a penetrated
point sx Sx and the value y at a non-penetrated point sy Sy. 






is the horizontal distance between points sx and
sy, while  is the vertical one. θh and θv are the horizontal and vertical scale of fluctuation, respectively. Each
material class considered is associated exclusively with an element in the set of integers {1,2,…,C}. p(y=c) is the
probability of encountering a type of material c in a point sy. Such probability is initially approximated using Eq.
525
A-1. More accurate probabilities are computed on the basis of the repeated sampling of the joint distribution f(x,y),
which was approximated using Eq. A-1. Eq. A-1, described in short, approximates probabilities as the ratio of the
sum of correlation values calculated for a penetrated point in the set Sx and the set of non-penetrated points Sy for
a given material c to the sum of correlation values for all points and all materials.
Based on the stratigraphy data collected at borehole locations, the selection of the type of correlation function and
530
the scales of fluctuation took place using the maximum likelihood method. The authors considered three types of
correlation functions, namely squared exponential, single exponential, and second-order Markov. In this case, the
likelihood function ℒ(θm|ɗ) = f(ɗ|θm) represents the likelihood of observing ɗ at borehole locations, given the
spatial correlation structure θm. The squared exponential function yielded the maximum likelihood when the
horizontal scale of fluctuation and the vertical scale of fluctuation are set to 11.1 and 4.1 metres, respectively.
535
Hence, the squared exponential function correlation whose expression is Eq. A-2 in this Appendix was selected.
Eq. A-3 and A-4 correspond to the single exponential and second-order Markov functions, respectively.
https://doi.org/10.5194/gmd-2022-210
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19
 󰇧







󰇨
(A-3)
 






󰇧
󰇨󰇧







󰇨
(A-4)
Data availability
540
The research reported in this paper did not generated data.
Code availability
The research reported in this paper did not generate any code.
Authors' contributions
Ibsen Chivata Cardenas: Conceptualization, Methodology, Writing- Original draft preparation, Investigation,
545
Validation; Terje Aven: Investigation, Supervision, Writing- Reviewing and Editing; Roger Flage: Investigation,
Supervision, Writing- Reviewing and Editing.
Conflict of interest/Competing interests
The authors declare that they have no conflict of interest.
Funding
550
This research is funded by ARCEx partners and the Research Council of Norway (grant number 228107).
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