Derived the scs equation in hydrology

The total rainfall ($P$) is partitioned into initial abstraction ($I_a$), direct runoff ($Q$), and actual retention ($F$).

$$P = I_a + Q + F$$

For runoff to occur, $P > I_a$. If $P \le I_a$, then $Q = 0$.
The rainfall available for runoff and retention is $(P - I_a)$. Therefore, we can write:

$$P - I_a = Q + F$$

The SCS method assumes that the ratio of actual retention ($F$) to the potential maximum retention ($S$) is equal to the ratio of actual runoff ($Q$) to the total rainfall available for runoff and retention ($P - I_a$).

$$\frac{F}{S} = \frac{Q}{P - I_a}$$

Based on empirical studies, the SCS found that the initial abstraction ($I_a$) can be approximated as a fraction of the potential maximum retention ($S$).

$$I_a = 0.2S$$

This means that, on average, about 20% of the potential maximum retention occurs as initial abstraction before runoff begins.

From Step 2, we can express $F$ as:

$$F = S \left( \frac{Q}{P - I_a} \right)$$

Substitute this expression for $F$ and the empirical relationship for $I_a$ (from Step 3) into the water balance equation from Step 1 ($P - I_a = Q + F$):

$$P - 0.2S = Q + S \left( \frac{Q}{P - 0.2S} \right)$$

Now, we need to solve for $Q$:

$$P - 0.2S = Q \left( 1 + \frac{S}{P - 0.2S} \right)$$ $$P - 0.2S = Q \left( \frac{P - 0.2S + S}{P - 0.2S} \right)$$ $$P - 0.2S = Q \left( \frac{P + 0.8S}{P - 0.2S} \right)$$

Rearranging to solve for $Q$:

$$Q = \frac{(P - 0.2S)(P - 0.2S)}{P + 0.8S}$$ $$Q = \frac{(P - 0.2S)^2}{P + 0.8S}$$

This equation is valid only when $P > I_a$, i.e., $P > 0.2S$. If $P \le 0.2S$, then $Q = 0$.

To make the parameter $S$ more practical and dimensionless, the SCS introduced the Curve Number (CN). CN is an index that represents the runoff potential of a watershed and ranges from 0 to 100. The relationship between $S$ (in mm) and CN is:

$$S = \frac{25400}{\text{CN}} - 254$$

If $S$ is in inches, the relationship is:

$$S = \frac{1000}{\text{CN}} - 10$$

Substituting the expression for $S$ in terms of CN into the runoff equation (using $S$ in mm):

$$Q = \frac{(P - 0.2 \left( \frac{25400}{\text{CN}} - 254 \right))^2}{P + 0.8 \left( \frac{25400}{\text{CN}} - 254 \right)}$$

This is the full form, but typically, the equation is used with $S$ directly calculated from CN.

The final SCS runoff equation is:

$$\boxed{Q = \frac{(P - I_a)^2}{P - I_a + S}}$$

where $I_a = 0.2S$, so:

$$\boxed{Q = \frac{(P - 0.2S)^2}{P + 0.8S}}$$

This equation is applicable for $P > 0.2S$. If $P \le 0.2S$, then $Q = 0$.
The potential maximum retention $S$ is related to the Curve Number (CN) by:

$$S = \frac{25400}{\text{CN}} - 254 \quad (\text{for } S \text{ in mm})$$

or

$$S = \frac{1000}{\text{CN}} - 10 \quad (\text{for } S \text{ in inches})$$