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

The four basic 3-layer (VES) curve types are classified by the trend of resistivity with depth for \(\rho_1,\rho_2,\rho_3\):

Type A: \(\rho_1<\rho_2<\rho_3\) (rising) |  Type K: \(\rho_1<\rho_2>\rho_3\) (rise then fall)  |  Type H: \(\rho_1>\rho_2<\rho_3\) (fall then rise)  |  Type Q: \(\rho_1>\rho_2>\rho_3\) (falling)

A 4-layer curve type is written as two letters: the first letter is the 3-layer type of \((\rho_1,\rho_2,\rho_3)\), and the second letter is the 3-layer type of \((\rho_2,\rho_3,\rho_4)\). The key constraint is continuity at the \(\rho_2\text{-}\rho_3\) interface: the trend from \(\rho_2\) to \(\rho_3\) is a single physical fact and must agree between the two 3-layer sub-curves used to build the first and second letters.

Types A and H both start with \(\rho_2<\rho_3\) (A: \(\rho_1<\rho_2<\rho_3\); H: \(\rho_1>\rho_2<\rho_3\)) — so if the first letter is A or H, the second letter's own \(\rho_2\) vs \(\rho_3\) relation must also start with \(\rho_2<\rho_3\), which restricts the second letter to A or K only (both begin with \(\rho_2<\rho_3\)). It cannot be H or Q, since those begin with \(\rho_2>\rho_3\) at the (now relabelled) first interface.

Similarly, types K and Q both start with \(\rho_2>\rho_3\) — so if the first letter is K or Q, the second letter is restricted to H or Q only.

This gives exactly 8 physically possible 4-layer type-combinations: AA, AK, HA, HK, KH, KQ, QH, QQ. The other 8 combinations are impossible: AH, AQ, HH, HQ, KA, KK, QA, QK.

Checking each option against this rule:

(A) HQ — H requires second letter ∈{A,K}; Q is not allowed → impossible. AQ — A requires second letter ∈{A,K}; Q is not allowed → impossible. Both members of option (A) are impossible, so (A) is a correct choice.

(B) KQ — K requires second letter ∈{H,Q}; Q is allowed → possible. QH — Q requires second letter ∈{H,Q}; H is allowed → possible. Both are valid curve types, so (B) is not a correct choice.

(C) KH — K requires second letter ∈{H,Q}; H is allowed → possible. AA — A requires second letter ∈{A,K}; A is allowed → possible. Both valid, so (C) is not a correct choice.

(D) HH — H requires second letter ∈{A,K}; H is not allowed → impossible. KA — K requires second letter ∈{H,Q}; A is not allowed → impossible. Both members impossible, so (D) is a correct choice.

\(\boxed{\text{Options (A) HQ, AQ and (D) HH, KA are NOT possible four-layer curve types}}\)