S3U

Science / Geology / Graduate / sge01

Nonunique Geological Mixtures and Independent Tracers

Use a simplified inverse problem to distinguish precision from identifiability.

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5 questions0m 0s

Prerequisites

Linear systems, mass balance, uncertainty, and inverse problems.

Learn the skill

An exact fit can still leave infinitely many geological interpretations. Count independent constraints, examine the nullspace, and consider measurement uncertainty and endmember assumptions before reporting a unique mixture.

Worked example

If two endmembers share one measured tracer value, that tracer cannot distinguish their fractions. Measuring it more precisely does not remove this structural ambiguity.

Model and Assumptions

An invented three-endmember mixture has fractions f1,f2,f3 >= 0 summing to 1. One conservative tracer has endmember values (0,1,2) and mixture value 1. A proposed independent tracer has endmember values (1,0,1). All balances are linear and endmembers fixed.

1. Which family fits the first tracer and total balance?
2. Would zero error on the first tracer alone ensure a unique mixture?
3. If the second tracer is exactly 0.6, what mixture results?
4. What if the second tracer lies between 0.4 and 0.6?
5. What could invalidate the linear interpretation?

Further inquiry

Derive the feasible line segment and the rank of the two-tracer system including total balance. Explain how uncertain endmember values alter the feasible region, and propose a test for post-mixing alteration.

Review criteria

  • Use nonnegativity to bound t.
  • Show which observation removes the null direction.
  • Distinguish structural assumptions from measurement uncertainty.