In a drainage-basin water balance, runoff is obtained from precipitation after accounting for initial recharge (initial abstractions) and groundwater accretion. Which relationship is correct?

Introduction:
Basin runoff represents the part of precipitation that appears as streamflow after deducting initial abstractions (interception, depression storage, and early infiltration) and considering baseflow contributions from groundwater. Understanding the sign convention for each term is essential for consistent water-balance equations.

Given Data / Assumptions:

• Precipitation (P) supplies water to the basin.
• Initial recharge (initial abstractions, Ia) represents early losses before runoff forms.
• Groundwater accretion (G) denotes net baseflow that augments streamflow during/after events.
• Other long-term terms like evapotranspiration and change in storage are omitted here to focus on the three listed quantities.

Concept / Approach:
A simplified event water-balance for direct runoff and baseflow can be expressed as: Runoff R = P - Ia + G where P increases streamflow potential, Ia reduces it as an initial loss, and G (if positive as accretion) adds to streamflow through groundwater contribution.

Step-by-Step Solution:
Step 1: Start from inputs and losses: P is input; Ia is an early-time loss. Step 2: Include groundwater accretion as a gain to channel flow. Step 3: Combine with signs: R = P - Ia + G.

Verification / Alternative check:
Check limiting cases: If G = 0 (no baseflow), then R = P - Ia, which matches classic initial-loss concepts. If Ia = 0 and G > 0 (e.g., wet antecedent conditions), runoff increases by the baseflow amount, consistent with observed hydrographs where groundwater sustains flow.

Why Other Options Are Wrong:

• Precipitation + initial recharge - groundwater accretion: Adds a loss and subtracts a gain; sign convention reversed.
• Precipitation - groundwater accretion - initial recharge: Subtracts the baseflow gain; incorrect.
• Initial recharge + groundwater accretion + precipitation: Treats all as gains; ignores that Ia is a loss.
• Precipitation + groundwater accretion + initial recharge: Same issue—adds the loss term.

Common Pitfalls:

• Interchanging terms: 'initial recharge' (initial abstractions/losses) versus later groundwater contributions.
• Ignoring the time scale: event-based balances differ from long-term averages where evapotranspiration and storage changes play major roles.

Final Answer:
Precipitation - initial recharge + groundwater accretion.