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I am trying to perform a regression of GPR measurement response (X) vs concrete core density (Y). There is a known, constant error associated both these measurements. How would I incorporate this error into a simple linear regression?

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If it is a known error in both $X$ and $Y$, you can subtract errors from both $X$ and $Y$ to get $X'$ and $Y'$, and then fit the regression on $X'$ and $Y'$.

So, lets say there is a known constant error of $c_x$ in your $X$ and a known constant error of $c_y$ in your $Y$. (These are signed errors). Then you define $X' = X - c_x$ and $Y' = Y - c_y$, and fit the regression on $X'$ and $Y'$. The model would then be $$Y' = a + bX' + \epsilon, $$

where $a$ is the intercept, $b$ is the slope and $\epsilon$ is the random error with zero mean. Just to see how this regression related back to the old one, $$Y - c_y = a + b(X - c_x) + \epsilon \Rightarrow Y = c_y + a - bc_x + bX + \epsilon. $$

Thus we get that if you fit the regression on the original $X$ and $Y$, then the slope is unaffected by the constant errors and the $y$-intercept changes according to the constants.

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