Scenario description: Temperature has been measured at $k+2$ different depths in a borehole. Measurements of the temperature were taken once each hour over a period of about 3 months. So observed data has the form $$\{(Y_{\text{top}, t}, Y_{1,t}, \ldots, Y_{k,t}, Y_{\text{bottom},t})\}_{t=1}^n$$

Using the top and bottom temperature measurements $\{(Y_{\text{top}, t}, Y_{\text{bottom},t})\}_{t=1}^n$ from the borehole as boundary values, a (for our purposes) black box model was used to estimate/simulate the temperatures at the intermediate temperatures, for each timepoint. This gives us values $$\{(Z_{1,t}, \ldots, Z_{k,t})\}_{t=1}^n$$

I.e. $Z_{j,t} = \hat{Y}_{j,t}$ in some sense. This black-box model is not dependent on time other than it having used the observed temperatures for the intermediate depths as starting values on the first timepoint.

Problem description: A third variable for which we have data $\{X_t\}_{t=1}^n$ (measured with some uncertainty) is thought to influence to influence the observed values differently than the simulated values; the hypothesis is that the $X$-variable (the independent variable) has a stronger effect on the observed values compared to the simulated values. The form in which the $X$-variable influences the other two variables (at each depth) has not been specified, but a linear relationship may be a good place to start. An additional complication is that the effect of $X$ on the temperature might be lagged - but perhaps this does not change the methodology much.

Similar questions, differences: Similar problems have been given answers e.g. here: combining the observed and simulated temperatures (at a certain intermediate depth, say), and regressing these values on $X$ together with an interaction term between $X$ and a dummy variable for observed vs. simulated. What I suppose makes my question different is the time-series nature of the data: both observed and simulated data are heavily autocorrelated, and particularly for data from small depths there may be a day-night effect.

My question: How do I measure if the (independent) variable $X$ influences the variable $Y$ more than/less than $Z$, possibly taking into account the time-series nature of the data and the fact that $Z$ is an estimate of $Y$?

Plot of data at 25 cm depth: (y-axis values removed due to confidentiality) Plot of data at 25 cm depth

Note: measured and simulated temperatures are smoother at greater depths.


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