According to Horton, which expression correctly represents the infiltration capacity curve (with usual symbols fo, fc, k, t)?

Introduction:
The Horton infiltration capacity curve describes how infiltration capacity declines from an initial high value to an asymptotic lower value during a storm. It is fundamental in rainfall–runoff modeling and baseflow separation tasks in hydrology.

Given Data / Assumptions:

• Symbols: fo = initial infiltration capacity at t = 0; fc = ultimate (asymptotic) capacity; k = decay constant; t = time.
• We need the standard Horton form.

Concept / Approach:
Horton postulated an exponential decay of infiltration capacity with time under rainfall supply. The accepted form approaches fc as t increases and equals fo at t = 0.

Step-by-Step Solution:
Step 1: State the form that satisfies boundary conditions f(0) = fo and f(∞) = fc.Step 2: The function f = fc + (fo - fc) * e^-k t satisfies f(0) = fc + (fo - fc) = fo.Step 3: As t → ∞, e^-k t → 0, thus f → fc, matching physical behavior.

Verification / Alternative check:
Check dimensions (all terms have dimensions of infiltration rate) and sign of exponent (negative to ensure decay).

Why Other Options Are Wrong:

• f = ft + (fo - fc) * e^-kt: 'ft' is not standard; intended constant should be fc. Using ft causes ambiguity.
• f = ft - (fo - fc) * e^-kt: wrong sign relative to required boundary conditions.
• f = fc * (fo - fc) * e^kt: incorrect structure and positive exponent implies growth.

Common Pitfalls:

• Confusing fc (final) with fo (initial).
• Using a positive exponent which implies increasing, not decreasing, capacity.

Final Answer:
f = fc + (fo - fc) * e^-kt.