A conceptual look at knowledge gained in an investigation

 

The idea is to treat the curve as a knowledge-gain function, where:

• x = investigative input, effort, time, cost, or integrated investigation intensity

• y = usable spatial knowledge, or defensible confidence in site understanding

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Then: dy/dx  Is the marginal knowledge gain per unit investigative effort.

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Interpretation by project stage

Early stage: low dy/dx

• Desk study, initial assumptions, rough conceptual model.

• At this stage, effort may not immediately produce much reliable spatial knowledge. Some effort is spent identifying what is not known. There may even be confusion reduction rather than true knowledge expansion.

• This is an important point: early effort is still valuable, but the apparent rate of gain in defensible knowledge may be small.

Transitional stage: increasing dy/dx

• Knowledge gaps are identified, investigation becomes targeted, measurements begin to answer the right questions.

• This is where each added unit of effort may yield large reductions in uncertainty. This is often the most efficient part of the investigation.

Mature stage: peak dy/dx

• The investigation is well-designed, the main controls on variability are understood, and data are being transformed into robust spatial interpretation.

• This is the zone of maximum return on investigative effort.

Late stage: declining dy/dx

• Additional sampling, modelling refinement, and monitoring still help, but each new increment contributes less than before.

• This is the diminishing-returns zone

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​A useful alternative perspective is that dy/dx does not merely depend on effort quantity. It depends on effort quality.  So more rigorously dy/dx

is high when investigation is:

• Correctly targeted

• Spatially representative

• Well-attributed

• Properly analysed

• Linked to the governing uncertainties

and low when effort is:

• Ad hoc

• Biased

• Redundant

• Poorly located

• Not tied to the key conceptual uncertainties

This is probably the most important insight. The derivative is not just “how much work is being done,” but “how much useful understanding is being extracted from that work.”

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​The second Derivative

A further useful idea is the second derivative d2y/dx2

This indicates whether the rate of knowledge gain is accelerating or decelerating.

​Conceptually:

• d2y/dx2 > 0: investigation is becoming more effective, often because the conceptual model is improving and sampling is becoming better targeted

• d2y/dx2 = 0: point of maximum efficiency growth, often near the inflection region

• d2y/dx2 < 0: diminishing returns have begun

This can be very helpful if the aim is to argue for targeted investigation design rather than simply more investigation.

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Area under the curve

The Spatial Certainty Curve can be viewed as a cumulative expression of spatial knowledge developed through investigation. Its gradient, dy/dx, represents the rate at which useful knowledge is gained as investigative effort increases.

The area under this rate curve represents the cumulative increase in spatial knowledge over a given range of effort, rather than the total possible knowledge.

Where spatial knowledge is expressed qualitatively rather than as a measured index, this relationship should be understood as a conceptual guide rather than a strict mathematical quantity.

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​Important caution

A common assumption would be that knowledge always increases smoothly with effort. In reality that is not always true.

Sometimes there may be temporary negative effects in perceived confidence:

• Early investigation may reveal that prior assumptions were wrong

• Confidence may drop before reliable knowledge rises

• The curve may therefore include a “false confidence collapse” before the main rise

This is actually a very valuable idea for site investigation, because many projects begin with unjustified certainty. In those cases the first benefit of investigation is not increased confidence, but removal of false confidence.

So in a more realistic conceptual model:

• Apparent confidence may fall first

• Defensible knowledge then rises

• Later gains diminish

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