Simplified SMA-inspired 1-parameter SCS-CN model for runoff estimation

Abstract

This study proposes a simplified 1-parameter SCS-CN model (M5) based on Mishra-Singh (2002) model and soil moisture accounting (SMA) procedure for surface runoff estimation and compares its performance with the existing SCS-CN method (SCS, 1956) (M1), Michel 1-P model (Water Resour Res 41:1-6, 2005) (M2), Sahu 1-P model (Hydrol Process 21:2872-2881, 2007) (M3), and Ajmal et al. model (J Hydrol 530:623-633, 2015) (M4) using large rainfall–runoff dataset of 48,763 events from123 USDA-ARS watersheds. The performance of models was evaluated using three statistical error indices such as Nash-Sutcliffe efficiency (NSE), root mean square error (RMSE), percentage bias (PBIAS), and rank and grading system (RGS). Based on the results obtained, the models can be ranked as follows: M5 > M4 > M3 > M1 > M2, i.e., model M5 outperformed all the remaining four models M1–M4 and hence is recommended for field applications.

Appendices

Appendix 1

Equation (17) can be rewritten using the value of Q from Eq. (16) as:

$$ V={V}_0+P-\left[\frac{\left(P-{I}_{\mathrm{a}}\right)\left(P-{I}_{\mathrm{a}}+{V}_0\right)}{\left(P-{I}_{\mathrm{a}}+{V}_0+S\right)}\right] $$

(A1)

The runoff rate (q) can be obtained by differentiating Eq. (16) with respect to time t which yields:

$$ q=\frac{\left(P-{I}_{\mathrm{a}}+{V}_0\right)\left(P-{I}_{\mathrm{a}}+{V}_0+S\right)+S\left(P-{I}_{\mathrm{a}}\right)}{{\left(P-{I}_{\mathrm{a}}+{V}_0+S\right)}^2}p,\kern0.5em \mathrm{if}\ P>{I}_{\mathrm{a}} $$

(A2)

Now, deriving the value of P from Eq. (A1) and substituting into Eq. (A2) yields:

$$ q=\frac{2S\left(V-{I}_{\mathrm{a}}-{V}_0\right)-{\left(V-{I}_{\mathrm{a}}-{V}_0\right)}^2+{SV}_0}{S\left(S+{V}_0\right)}p,\mathrm{if}\ V>{V}_0+{I}_{\mathrm{a}},=0,\mathrm{otherwise} $$

(A3)

Appendix 2

The complete model of soil moisture store can be obtained as:

$$ v=\frac{{\left(V-{S}_{\mathrm{a}}-S\right)}^2}{S\left(S+{V}_0\right)}p $$

(B1)

Eq. B1 can be rewritten as:

$$ \frac{dV}{{\left(V-{S}_{\mathrm{a}}-S\right)}^2}=\frac{pdt}{S\left(S+{V}_0\right)} $$

(B2)

After integration, we get

$$ \int \frac{dV}{{\left(V-{S}_{\mathrm{a}}-S\right)}^2}=\frac{1}{S\left(S+{V}_0\right)}\int pdt $$

(B3)

Since the soil moisture varies from V₀ to V for a storm of time t

$$ \underset{V_0}{\overset{V}{\int }}\frac{dV}{{\left(V-{S}_{\mathrm{a}}-S\right)}^2}=\frac{1}{S\left(S+{V}_0\right)}\underset{0}{\overset{t}{\int }} pdt $$

(B4)

$$ {\left[\frac{-1}{\left(V-{S}_{\mathrm{a}}-S\right)}\right]}_{V_0}^V=\frac{P}{S\left(S+{V}_0\right)} $$

$$ \frac{1}{S_{\mathrm{a}}+S-V}-\frac{1}{S_{\mathrm{a}}+S-{V}_0}=\frac{P}{S\left(S+{V}_0\right)} $$

(B5)

Now, putting V=V₀ + P-Q in the Eq. (B5), we obtain

$$ Q=P\left[1-\frac{{\left({S}_{\mathrm{a}}+S-{V}_0\right)}^2}{\left(P\left({S}_{\mathrm{a}}+S-{V}_0\right)+S\left(S+{V}_0\right)\right)}\right] $$

(B6)

Appendix 3

Table 4 NSE (%), RMSE and PBIAS resulted by applications of models in 123 watersheds