Random variables, uncertainty and decisions... by Murray Lark

Here's Murray Lark, our environmental statistician, to talk about the uncertainty in three dimensions. How do different people interpret the same information and how can we make that information about uncertainty useful to you, the valuable user...
Rocks are three-dimensional objects, and computer technology has revolutionized the way in which geologists can present their understanding of the subsurface in 3D.  Where once geological information was presented on a flat map, with a few judiciously-chosen cross sections, a computer model can now represent the rocks beneath our feet in full 3D.  This is useful to planners, engineers, regulators and others who want to build a tunnel or explore the complexity of groundwater flows. 
However, all geological information has an uncertainty attached, and 3-D models are no exception to this.  In a recent paper, published open-access in the journal Solid Earth, we examined the uncertainty in interpretations of boreholes along a cross-section by a group of more than 20 geologists.  This is a key stage in the development of the 3-D model.  But in this blog I want to focus on the question, how can this information on uncertainty be made comprehensible and relevant to the end user?
It is relatively straightforward to make a statistical description of the uncertainty in some information.  Well, in fact it can be quite a complex task, but the real challenge is always to ensure that the resulting account of uncertainty is useable.  The figure (right), from the Solid Earth paper, shows a 95% confidence interval for the elevation of the base of the London Clay formation along a 4-km section.  At any location the probability is 95% that the base lies between the two blue lines.  The lines narrow near boreholes, and widen away from them.
This is interesting, but how should the user apply it?  In my view this can be done only by analysing the decision that the user wants to make.  We consider the hypothetical case of an engineer who wants to dig a tunnel along the route of the section, and to minimize the risk that the tunnel goes under the London Clay into the less-suitable Lambeth Group below.  The engineer might specify that the route of the tunnel should impinge on the Lambeth Group for no more than 1% of its length.  If the model is perfect, with no uncertainty, then the tunnel can be planned to follow the modelled base of the London Clay.  Because of the uncertainty, however, a safety margin should be applied.  We can use the uncertainty model to answer the question, "how close to the modelled base can the route be planned such that the risk of failing to meet the 1% criterion is small?"


Now this question cannot be answered from the confidence limits shown above, but it can be answered from the statistical model of uncertainty from which the limits are generated.  The graph (left) shows the probability of achieving the 1% criterion as a function of how far above the modelled base of the London Clay the route is planned.  This shows that the safety margin must be 8m or so if we are to have a high level of confidence.
This graph is tailor-made for the particular user question.  Other information users need to analyse their questions and then come up with risk assessments built on the uncertainty model.  In short, there are no identikit solutions, but many powerful ways to explore the implications of uncertainty for the users of information in earth sciences.

We will be talking about how to communicate uncertainty effectively to information users at the 2015 EGU general assembly in Vienna (12th-17th April).  Details of the session are below, and anyone interested is invited to submit an abstract by 7th January via the congress website.
SESSION: Communication of uncertain information in the earth sciences
(SSS11.5/ESSI3.6/HS12.7/NH9.17) Uncertainty impinges on most information that earth scientists generate, whether this is by observation, measurement, interpretation, process modelling or some combination of these.... read more on the EGU website.