Johansson et al. (2007) derive different formulas to estimate the volume of a lake approximating it to that of a solid of revolution. Their equation (9) provides an expression of the radius of such a solid of revolution as a function of depth ($z$) and a shape parameter ($H_d$): $$ r(z)=r_{circle}\Bigg(\frac{H_d^{-z/z_{max}}-H_d^{-z_{max}/z_{max}}}{H_d^{-z_0/z_{max}}-H_d^{-z_{max}/z_{max}}}\Bigg) $$ where $r_{circle}=\sqrt{A_{max}/\pi}$, $A_{max}$ is the lake surface area, $z_{max}$ is the maximum depth and $z_0=0$. Then they give the formula for the area at depth $z$: $$ A(z)=A_{max}\Bigg(\frac{H_d^{-z/z_{max}}-H_d^{-z_{max}/z_{max}}}{H_d^{-z_0/z_{max}}-H_d^{-z_{max}/z_{max}}}\Bigg)^2 $$ And we can obtain the volume by integrating $A(z)$.

However, the authors of the paper don't explicitly derive the formula in the equation (9). They say that

it is important that we first normalise the features of the geometric body (i.e., the idealised lake), both vertically and horizontally.

But I cannot see how they arrived to this expression. I wonder also if this is some known result.

Update: addition of further information in response to a comment

The shape parameter $H_d$ is defined in the paper as hypsographic development parameter. The authors do not provide an explicit definition of $H_d$, but it is conceptually related to the volume development parameter $V_d=3V_{max}/A_{max}z_{max}$, where $V_{max}$ is the maximum volume. The lake basin shape is linear for $V_d=1$, convex for $V_d<1$ and concave for $V_d>1$.

The parameter $H_d$ seems to be related also with the convexity of the hypsographic curve (area-depth curve) and the presence or not of an inflexion point (see the figure, showing A(z) as defined above for different values of $H_d$). Calculation of $A(z)$ for different values of $H_d$

  • $\begingroup$ Since the formula mostly involves "a shape parameter" but there is no definition of what this shape parameter represents (I have no desire to spend 40 bucks to find out, thank you), there is no way to answer your question. $\endgroup$ – Paul Sinclair Nov 24 '17 at 20:24
  • $\begingroup$ @PaulSinclair I have added some more information. Does that help? Also, if I have understood you well, I need to concentrate on what the shape parameter represents in order to solve my question. Is that it? $\endgroup$ – lodebari Nov 27 '17 at 11:03

Asking this question and thinking on the comment by @Paul Sinclair aided me to find the answer. It is easier to understand how Johansson et al. derived their formula by simplifying an rearranging the $r(z)$ equation as $$\frac{r(z)}{r_{circle}}=\frac{H_d^{\frac{z_{max}-z}{z_{max}}}-1}{H_d-1}$$

Apparently, the authors assumed the relation between $r$ and $H_d$ can be estimated through a power function, e.g. $$r(z)=aH_d^y+b$$ (What $H_d$ represents and why a power function are other questions, maybe not mathematically relevant.) They normalise vertically by taking $y=\frac{z_{max}-z}{z_{max}}$, the relative elevation above the bottom, which makes $y$ vary between 0 and 1. They normalised horizontally by dividing by the surface radius, i.e., $r(z=0)=r_{circle}={aH_d+b}$, so that $$\frac{r(z)}{r_{circle}}=\frac{H_d^{\frac{z_{max}-z}{z_{max}}}+b/a}{H_d+b/a}$$ Since the radius is zero at the maximum depth ($r(z_{max})=0$), we have $b/a=-1$.


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