$$S^2_{1-x_2}\to \R[x_0,x_1]_{f}$$ defined by $x_i\mapsto 2x_i/f$ for $i=0,1$ and $x_2\mapsto (f-2)/f$. Using the exact sequence of localization (see [@BalSurvey Theorem 1.5.5]), we get an exact sequence $$\xymatrix@C=1.4em{0\ar[r] & H^0(S^2,W)\ar[r] & H^0(S^2_{1-x_2},W)\ar[r] & H^1_{1-x_2}(S^2,W)\ar[r] & H^1(S^2,W)\ar[r] & \ldots}$$ First we compute the Witt cohomology of $S^2_{1-x_2}$, which is the same as the Witt cohomology of $\R[x_0,x_1]_{f}$. Using the localization sequence again, we have an exact sequence: $$\xymatrix@C=1.4em{0\ar[r] & H^0(\A^2,W)\ar[r] & H^0(\R[x_0,x_1]_{f},W)\ar[r] & H^1_f(\A^2,W)\ar[r] & H^1(\A^2,W)\ar[r] & \ldots}$$ By homotopy invariance, $H^0(\A^2,W)=\Z$ and $H^i(\A^2,W)=0$ if $i\geq 1$. Because $f$ is a regular parameter and $\R[x_0,x_1]/{f}=C^1$, we have $H^i_f(\A^2,W)\simeq H^{i-1}(C^1,W)$ by Lemma \[reg\_param\].

The sequence and Lemma \[comput\_c1\] imply that $H^i(\R[x_0,x_1]_f,W)=0$ if $i\geq 1$. To compute $H^0(\R[x_0,x_1]_f)$, observe that the map $H^0(\R,W)\to H^0(\R[x_0,x_1]_f,W)$ admits a retraction given by the choice of any rational point. Therefore we get $H^0(\R[x_0,x_1]_f,W)\simeq \Z\oplus \Z/4$, the factor $\Z/4$ being generated by the class of the form $\langle f\rangle$.

It remains to compute the groups $H^i_{1-x_2}(S^2,W)$. Remark that $1-x_2$ is a regular parameter in $S^2_{x_1}$ and that $S^2_{x_1}/(1-x_2)\simeq \C[x_1,x_1^{-1}]$. Again, we have an exact sequence of localization $$\xymatrix@C=1.4em{0\ar[r] & H^1_{1-x_2}(S^2,W)\ar[r] & H^1_{1-x_2}(S^2_{x_1},W)\ar[r] & H^2_{1-x_2,x_1}(S^2,W)\ar[r] & \ldots.}$$ Now the only point of codimension $2$ with support on $(1-x_2,x_1)$ is the 