  Tikhonov regularization is studied in the case of linear pseudodifferential
operator as the forward map and additive white Gaussian noise as the
measurement error. The measurement model for an unknown function $u(x)$ is
\begin{eqnarray*} m(x) = Au(x) + \delta\hspace{.2mm}\varepsilon(x),
\end{eqnarray*} where $\delta>0$ is the noise magnitude. If $\varepsilon$ was
an $L^2$-function, Tikhonov regularization gives an estimate \begin{eqnarray*}
T_\alpha(m) = \text{argmin}_{u\in H^r}\big\{\|A u-m\|_{L^2}^2+
\alpha\|u\|_{H^r}^2 \big\}\end{eqnarray*} for $u$ where $\alpha=\alpha(\delta)$
is the regularization parameter. Here penalization of the Sobolev norm $
\|u\|_{H^r}$ covers the cases of standard Tikhonov regularization ($r=0$) and
first derivative penalty ($r=1$).
  Realizations of white Gaussian noise are almost never in $L^2$, but do belong
to $H^s$ with probability one if $s<0$ is small enough. A modification of
Tikhonov regularization theory is presented, covering the case of white
Gaussian measurement noise. Furthermore, the convergence of regularized
reconstructions to the correct solution as $\delta\rightarrow 0$ is proven in
appropriate function spaces using microlocal analysis. The convergence of the
related finite-dimensional problems to the infinite-dimensional problem is also
analysed.
