Symbol

Description

h

Mountain elevation

$\eta$

Possible erosion height change in time interval Δt

r

Rate of uplift

$h_L$

Mountain base elevation

$\gamma \left(h-h_L\right)\Delta t$

Erosion probability in time interval Δt

$p\left(h\right)$

Probability density of mountain heights, $h^{b-1}{a}^b\exp \left(-ah\right)/\varphi \left(a\mathrm{,}b\right)$

a and b

Model parameters

$\gamma \left(b\mathrm{,}z\right)$

Lower incomplete gamma function, ${\displaystyle {\int }_0^z}{t}^{b-1}\exp \left(-t\right)\text{d}t$

$\varphi \left(a\mathrm{,}b\right)$

$\gamma \left(b,a{h}_{\max }\right)-\gamma \left(b,a{h}_{\min }\right)$

${\gamma }^{-1}\left(b\mathrm{,}z\right)$

Inverse lower incomplete gamma function, ${\gamma }^{-1}\left(b\mathrm{,}z\right)=y$ if $z=\gamma \left(b\mathrm{,}y\right)$

$h_i$

Value of ith height in data set of size N where $h_1\ge h_2\ge \cdots \ge h_N$

$h_{\max }$

Maximum height in data set, $h_1$

$h_{\min }$

Minimum height in data set, $h_N$

$\overline{h}$

${\displaystyle {\sum }_{i=1}^N}{h}_i/N$

$\overline{h^2}$

${\displaystyle {\sum }_{i=1}^N}{h}_i^2/N$

$\mathcal{L}\left(a\mathrm{,}b\right)$

Likelihood function, ${\displaystyle {\sum }_{i=1}^N}\left(\left(b-1\right)\log \left(h_i\right)-a{h}_i\right)+N\log \left(a^b/\varphi \left(a,b\right)\right)$

$G\left(i\right)$

Theoretical model height for mountain of height rank i,

${\gamma }^{-1}\left(b,\varphi \left(a,b\right)x_i+\gamma \left(b,a{h}_{\min }\right)\right)/a$ with $x_i=\left(N+1-i\right)/N$