Including model uncertainty in non-linear least squares minimization The problem
I have experimental data $Y$ with heteroscedastic and normally distributed uncertainties characterized by covariance matrix $C_{exp}$. I want to fit the data using model $F(X, \beta)$ where $F$ is a  nonlinear function mapping independent variable $X$ into theoretical curve $\hat{Y}$ using parameter vector $\beta$.
The model function $F(X, \beta)$ has the following form:
$F(X, \beta) = \beta_1S_1(X) + \beta_2S_2(X) + \beta_3f(X,\beta_4,\beta_5)$
where $\{\beta_i\}$ is the set of parameters I am trying to determine from the fitting; $S_1$ and $S_2$ are curves that were measured independently and are characterized with covariance matrices $C_1$ and $C_2$, respectively; $f(X,\beta_4,\beta_5)$ is a nonlinear theoretical function.
$C_{exp}$, $C_1$, and $C_2$ are three different full-rank positive definite matrices.
Each of the parameters in the $\{\beta_i\}$ set is important, i.e. none of them are so-called nuisance parameters.
To fit the data one needs to construct and minimize a $\chi^2$ function that normally looks like this:
$\chi^2 = (F(X,\beta) - Y)^T C_{exp}^{-1} (F(X,\beta) - Y)$
The issue with this approach is that it does not include model uncertainties - recall the curves $S_1$ and $S_2$ which have non-negligible uncertainties characterized with covariances $C_1$ and $C_2$. Therefore, the minimization of this $\chi^2$ generally will not yield correct optimal values and associated confidence intervals.
The question
How do I construct my $\chi^2$ function (cost/objective) in the light that portions of the model have uncertainty?
My proposed solution
The idea is to construct a total covariance matrix and use it in the definition of $\chi^2$. Since $S_1$ and $S_2$ go into the model additively and $\beta_1$ and $\beta_2$ are scaling parameters for these curves, the total covariance can be defined as follows:
$C_{tot} = C_{exp} + \beta_1^2C_1 + \beta_2^2C_2$
The new $\chi^2$ will be defined as follows:
$\chi^2 = (F(X,\beta) - Y)^T C_{tot}^{-1} (F(X,\beta) - Y)$
Does this look like a sound approach? Could anybody recommend some good literature on the subject of combining model and experimental uncertainty?
Thanks.
 A: Consider the non-linear regression model
$$y_i = \beta^0_1 s(x_i,\lambda^0) + \beta_2^0 f(x_i,\beta^0_3) + \epsilon_i$$
where it is assumed that $s(x,\lambda)$ and $f(x,\beta_3)$ are known functions and the superscript on parameters indicates true values for the parameters. Standard identifying assumption is that 
$$\mathbb E[y_i\lvert x_i] = \beta^0_1 s(x_i,\lambda^0) + \beta_2^0 f(x_i,\beta^0_3).$$
Clearly depending on the assumptions of functional form for $s$ and$f$ indetification may or may not be present. With $s=f$ it is offcourse not present. But here is an example where it works
set.seed(1)
N <- 10000
x1 <- rnorm(N)
x2 <- rnorm(N)
e <- rnorm(N)
b1 <- 1
lambda <- 0.5
b2 <- 1
b3 <- 0.1


y <- b1 * sin(lambda*x1) + b2 * exp(b3*x2) + e

g <- function(p)
    {   
        lambda <- p[1]
        b3 <- p[2]
        a1 <- sin(lambda*x1)
        a2 <- exp(b3*x2)
        model <- lm(y~a1+a2-1)
        error <- sum(model$residual^2)
        return(error)
    }

g(p=c(0.5,0.1))
nlm(g,p=c(lambda=1,b3=1),stepmax=0.3)

If this approach is not an option consider then the model where 
$$\mathbb E[y_i\lvert x_i] = \beta^0_1 s(x_i) + \beta_2^0 f(x_i,\beta^0_3).$$
for the sake of example let us assume that the kind goddess of truth grant you knowledge of $\beta_3^0$ not all you have to do is calculate $f(x_i,\beta_3^0)$ and then run the linear regression
$$y_i =  \beta^0_1 s(x_i) + \beta_2^0 f(x_i,\beta^0_3) + \epsilon_i$$
however you cannot do this because you do not know $s(x_i)$ but instead only some measurement output
$$\hat s_i = s(x_i) + u_i$$
where we assume that $u_i$ is independent of the true value $s(x_i)$. Pluggin $s(x_i)=\hat s_i - u_i$ into the regression gives you
$$y_i =  \beta^0_1 (\hat s_i - u_i) + \beta_2^0 f(x_i,\beta^0_3) + \epsilon_i$$ implying that
$$y_i =  \beta^0_1 \hat s_i + \beta_2^0 f(x_i,\beta^0_3) + \underbrace{(-\beta_1^0 u_i +\epsilon_i)}_{:=v_i}$$
where $cov(\hat s_i,v_i) \not= 0$ hence you face an endogenous regressor problem which is well known not simply to imply a wrong estimate of the variance but to imply that the OLS estimator is inconsistent. In non-linear models of the kind you seem to be suggesting estimation is also performed under minimization of the sum of squared errors under the same identifying assumption that the model correctly captures the conditional mean. Hence endogenous regressors which may very well be present in the case of measurement error potentially leads to an inconsistent estimator. 
The point is that without any further assumptions (than what you have specified in your question) a formula such as
$$C_{tot} = something$$
to simply correct the variance seems wishful thinking in the sense that the problem might be worse in the form of inconsistent estimator.
Offcourse one assumption you could make is to assume $s()$ function unknown and assume that
$$\hat s_i = s(x_i) + u_i,$$
where $u_i$ is independent of $\hat s_i$ but in that case you do not need to correct the variance either.
I think the following paper provides a decent place to start Nonlinear Models of Measurement Errors by XIAOHONG CHEN and HAN HONG and DENIS NEKIPELOV.
