Development of a Modified SMA Based MSCS-CN Model for Runoff Estimation

Abstract

The Soil Conservation Service Curve Number (SCS-CN) method developed by the USDA-Soil Conservation Service (SCS, 1972) is widely used for the estimation of direct runoff for a given rainfall event from small agricultural watersheds. The initial soil moisture plays an important role in re-structuring of the SCS-CN method and enables us to prevent unreasonable sudden jump in runoff estimation and this has prompted the concept of soil moisture accounting (SMA) procedure to develop improved SCS-CN based models. Applying the concept of SMA procedure and changed parameterization, Michel et al. Water Resour Res 41(2):1–6 (2005) developed an improved SCS-CN model (MSCS-CN), which could be thought of an improvement over the existing SCS-CN method; however, their model still inherits several conceptual limitations and inconsistencies. Therefore, in this study an attempt is made to propose an improved SMA based SCS-CN-inspired model (MMSCS-CN) model incorporating a continuous function for initial soil moisture and test its suitability over the MSCS-CN and SCS-CN model using a large dataset from US watersheds. Using, Nash and Sutcliffe efficiency (NSE) and root mean square error (RMSE) of these models, the overall performance is further evaluated using rank grading system, and it is found that the MMSCS-CN scores highest mark (95; overall rank I) followed by MSCS-CN with 61 (overall rank II), and SCS-CN model with 51 mark (overall rank III) out of the maximum 105. This study shows that the proposed MMSCS-CN model has several advantages and performs better than the MSCS-CN and the existing SCS-CN model.

Appendices

Appendix A

Equations (14a) & (14b) are re-written here as:

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

(a1)

$$ \mathrm{q}=\mathrm{p}\left[\frac{\left(\mathrm{P}+{\mathrm{V}}_0+S\right)\left(2\mathrm{P}+{\mathrm{V}}_0-\mathrm{I}\mathrm{a}\right)-\left(\mathrm{P}+{\mathrm{V}}_0\right)\left(\mathrm{P}-\mathrm{I}\mathrm{a}\right)}{{\left(\mathrm{P}+\mathrm{S}+{\mathrm{V}}_0\right)}^2}\right] $$

(a2)

P can be obtained from Eq. (a1) as:

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

(a3)

Using Eq. (a3), the different terms of Eq. (a2) can be written as:

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

(a4)

$$ \left(2\mathrm{P}+{\mathrm{V}}_0-{\mathrm{I}}_{\mathrm{a}}\right)=\frac{\mathrm{V}\left(\mathrm{S}+\mathrm{S}\mathrm{a}\right)-\mathrm{S}\mathrm{a}\left(\mathrm{S}+\mathrm{S}\mathrm{a}\right)}{\mathrm{S}+\mathrm{S}\mathrm{a}-\mathrm{V}}, $$

(a5)

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

(a6)

$$ \left(\mathrm{P}-\mathrm{I}\mathrm{a}\right)=\frac{\left(\mathrm{S}+\mathrm{S}\mathrm{a}\right)\left(\mathrm{V}-\mathrm{S}\mathrm{a}\right)}{\mathrm{S}+\mathrm{S}\mathrm{a}-\mathrm{V}}, $$

(a7)

$$ \mathrm{and}\kern2em {\left(\mathrm{P}+\mathrm{S}+{\mathrm{V}}_0\right)}^2=\frac{{\mathrm{S}}^2{\left(\mathrm{S}+\mathrm{S}\mathrm{a}\right)}^2}{{\left(\mathrm{S}+\mathrm{S}\mathrm{a}-\mathrm{V}\right)}^2} $$

(a8)

Using Eqs. (a4)–(a8) into Eq. (a2) and then simplifying results into

$$ \mathrm{q}=\mathrm{p}\left[\frac{\mathrm{VS}+\left(\mathrm{V}-\mathrm{S}\mathrm{a}\right)\left({\mathrm{S}}_{\mathrm{b}}-\mathrm{V}\right)}{\mathrm{S}\mathrm{Sb}}\right] $$

(a9)

This is the desired expression of q and p, i.e., Eq. (15).

Appendix B

Coupling of Eqs. (13) and Eq. (15) results into

$$ \frac{\mathrm{dV}}{\mathrm{dt}}=\mathrm{p}\left[1-\frac{\mathrm{VS}+\left(\mathrm{V}-\mathrm{S}\mathrm{a}\right)\left({\mathrm{S}}_{\mathrm{b}}-\mathrm{V}\right)}{\mathrm{S}\mathrm{Sb}}\right] $$

(b1)

After re-arranging, Eq. (b1) is expressible as:

$$ \frac{\mathrm{dV}}{\mathrm{dt}}=\mathrm{p}\left[1-\frac{\mathrm{V}}{{\mathrm{S}}_{\mathrm{b}}}+\left(\frac{\mathrm{V}}{{\mathrm{S}}_{\mathrm{b}}}-\frac{{\mathrm{S}}_{\mathrm{a}}}{{\mathrm{S}}_{\mathrm{b}}}\right)\left(1-\frac{\mathrm{V}}{{\mathrm{S}}_{\mathrm{b}}}\right)/\left(1-\frac{{\mathrm{S}}_{\mathrm{a}}}{{\mathrm{S}}_{\mathrm{b}}}\right)\right] $$

(b2)

or

$$ \frac{\mathrm{dV}}{\mathrm{dt}}=\mathrm{p}\left[\frac{{\left({\mathrm{S}}_{\mathrm{b}}-\mathrm{V}\right)}^2}{\mathrm{S}\mathrm{Sb}}\right] $$

(b3)

Again, re-arranging Eq. (b3) and applying appropriate lower and upper limits of integration results into

$$ {\displaystyle \underset{\mathrm{V}=\mathrm{V}0}{\overset{\mathrm{V}}{\int }}\frac{\mathrm{dV}}{{\left({\mathrm{S}}_{\mathrm{b}}-\mathrm{V}\right)}^2}}={\displaystyle \underset{\mathrm{t}=0}{\overset{\mathrm{t}}{\int }}\frac{\mathrm{pdt}}{\mathrm{S}\mathrm{Sb}}} $$

(b4)

On integrating Eq. (b4), we get

$$ \frac{1}{\left({\mathrm{S}}_{\mathrm{b}}-\mathrm{V}\right)}-\frac{1}{\left({\mathrm{S}}_{\mathrm{b}}-{\mathrm{V}}_0\right)}=\frac{\mathrm{P}}{\mathrm{S}\mathrm{Sb}} $$

(b5)

Now, substituting V from Eq. (12) into Eq. (b5) and rearranging leads to

$$ \frac{\left(\mathrm{P}-\mathrm{Q}\right)}{\left[\left({\mathrm{S}}_{\mathrm{b}}-{\mathrm{V}}_0\right)-\left(\mathrm{P}-\mathrm{Q}\right)\right]\left({\mathrm{S}}_{\mathrm{b}}-{\mathrm{V}}_0\right)}=\frac{\mathrm{P}}{\mathrm{S}\mathrm{Sb}} $$

(b6)

Equation (b6) can be further simplified as:

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

(b7)

This is the desired expression of Q and P, i.e., Eq. (19).