Say I have a time series, with actual measurements of a variable $a$ in different locations. If I want to know the average of $p$ from time $t_1$ to $t_n$, I could say that $\overline{a_{p}}=\frac{\sum\limits_{t=1}^n a_{{p}_t}}{n}$, where $p$ is the coordinate of the point of measurement. So far, so good. Let's generalize and call this averaging function $m_p=f(a_p,t)$, where $f(a,t)$ is some deterministic function and $m$ is its result.
Now, say I want to calculate $m$ for a location $o$, where $o$ is a points for which I don't have actual measurements and $a_o$ was estimated through a an interpolating function $g(a,o)$. My intuition tells me that in this case, we should first estimate every $a_{o_t}$ before applying $f(a,t)$. In other words, considering that $a_o = g(a,o)$, we get $m_o$ by doing:
$m_o=f[g(a,o),t]=f(a_o,t)$
But I have seen so many papers that do the other way around. They first calculate $f(a,t)$ for every known $a$ and then interpolate the results using the same g() even though there is no guarantee that $m$ behaves like $a$, or:
$m_o=g[f(a,t),o]=g(m,o)$
In some cases, they use the same logic when instead of $t$, there is some other variable $b$, i.e., it's not a time series, but $f(a,b)$ is a deterministic function.
Are the models in those papers conceptually wrong? If they are wrong, is there a techcnical term to call this kind of mistake?