# Inverted value is not consistent with expectation

We have a group of observations

$$y = f(x_1, x_2, x_3) \enspace .$$

We have also a forward model $y = f(x_1, x_2)$. The forward model does not include $x_3$ because $x_3$ might include dozens of parameters which we do not understand clearly.

First, we inverted parameter $x_1$ with the forward model $y = f(x_1, 0)$ embedded in our inversion strategy. The best model can fit 80% of the observed data sets. Then we inverted both parameters $x_1$ and $x_2$ with the forward model $y = f(x_1, x_2)$. However, the misfit did not reduce and the inverted $x_2$ is always quite close to zero.

Based on previous laboratory and numerical experiments, $x_2$ should have significant effects on the observation and its value is obvious larger than zero. So we fix $x_2$ as the experimental observed value and inverted for the optimized model. The misfit from the best inverted model is larger than that with $x_2$ fixed at zero. How can we understand these inversion results? Do they mean that $x_2$ is balanced by $x_3$ or that there was overfitting in inversions with $x_1$ only?

• Welcome to SciComp Exchange. Can you provide more information about yor data and models? So far, it is quite vague. Aug 15 '15 at 2:44
• Is $x_3$ controlled in your experiments? Aug 15 '15 at 16:06
• Thanks. Previous inversions only inverse for x1 while previous forward models demonstrate that x2 has significant effects on the observation. x3 might have significant effects on the observation. However, we still do not know its physics mechanism and can not put it into our model(for us Earth's mantle dynamics model). Aug 15 '15 at 20:36
• The model is a series of partial difference equations (PDEs). We solve these PDEs with prescribed parameters (x1, x2, some part of x3) to predict various observations, e.g. the gravitational signal. x1 is not a scalar, it is a vector actually. Aug 15 '15 at 20:40