# altitude bounds derivations

1. altitude ranges from midpoints

symbol

description

unit

variable name

$$z(i)$$

altitude

$$m$$

altitude {:,vertical}

$$z^{B}(i,l)$$

altitude boundaries ($$l \in \{1,2\}$$)

$$m$$

altitude_bounds {:,vertical,2}

The pattern : for the dimensions can represent {time}, or no dimension at all.

\begin{eqnarray} z^{B}(1,1) & = & \frac{3z(1) - z(2)}{2} \\ z^{B}(i,1) & = & \frac{z(i-1) + z(i)}{2}, 1 < i \leq N \\ z^{B}(i,2) & = & z^{B}(i+1,1), 1 \leq i < N \\ z^{B}(N,2) & = & \frac{3z(N) - z(N-1)}{2} \end{eqnarray}

This formula applies if the harp option regrid_out_of_bounds is set to nan or to extrapolate. If the option is set to edge then the first and last boundary value are set to the midpoints ($$z^{B}(1,1) = z(1)$$, $$z^{B}(N,2) = z(N)$$).