Question

Which pairs of the following combinations of the H-, Q-, K- and A- type of resistivity sounding curves is/are NOT possible to generate four-layer models?

• A. HQ and AQ
• B. KQ and QH
• C. KH and AA
• D. HH and KA

Hint:

A 4-layer type 'XY' is valid only if the ρ2-ρ3 trend implied by the end of X matches the ρ2-ρ3 trend implied by the start of Y — A/H (rising ρ2→ρ3) can only be followed by A/K, and K/Q (falling ρ2→ρ3) can only be followed by H/Q.

Answer 1

The Correct Option is A, D

An equivalent, purely visual way to reach the same answer is to track the sign of the slope of the apparent-resistivity curve on log-log paper across each interface. Assign a '+' if resistivity trend rises across an interface and a '-' if it falls. Curve type A = (+,+), K = (+,-), H = (-,+), Q = (-,-), where the pair records the sign of the change across the \(\rho_1\text{-}\rho_2\) interface followed by the \(\rho_2\text{-}\rho_3\) interface.

For a valid 4-layer curve "XY", the *second* sign of X (the \(\rho_2\text{-}\rho_3\) trend) must equal the *first* sign of Y (also the \(\rho_2\text{-}\rho_3\) trend) — they describe the exact same physical interface, so they cannot disagree.

A → (+,+): second sign '+'. H → (-,+): second sign '+'. K → (+,-): second sign '-'. Q → (-,-): second sign '-'.

So A and H (second sign '+') can only be followed by a type whose first sign is '+', i.e. A(+,+) or K(+,-) — giving AA, AK, HA, HK. And K and Q (second sign '-') can only be followed by a type whose first sign is '-', i.e. H(-,+) or Q(-,-) — giving KH, KQ, QH, QQ.

Any combination violating this sign-matching rule is geometrically impossible to realize with a real layered earth model: HQ, AQ, HH, KA, AH, KK, QA, QK are all sign-mismatched and therefore excluded — reproducing exactly options (A) HQ/AQ and (D) HH/KA as the impossible pairs, i.e. answer A;D.