this yields equation (\[eq:quadrel\]).

Similarly, it may be shown that: $$Q_{ijk}=Q_{ijk}^{(0)}+P_{ijk}\ .$$ However, $$Q_{1111}=Q_{1111}^{(0)}+P_{1111}+6Q_{11}^{(0)}P_{11}\ ,$$ with a similar expression for $Q_{2222}$, and $$Q_{1122}=Q_{1122}^{(0)}+P_{1122}+Q_{11}^{(0)}P_{22}+Q_{22}^{(0)}P_{11}$$ provided we assume the PSF is nearly circular.

If we further look only at nearly circular sources, then we may estimate the flexion using the forms in equations (\[eq:skewness\]) and (\[eq:arciness\]). Again, assuming unweighted moments, and zero PSF and intrinsic flexion we find: $$\tilde{{\cal F}}_i={\cal F}_i
\frac{9\xi^{(0)}-6(Q^{(0)2}_{11}+Q^{(0)2}_{22})}{9\xi-6(Q_{11}^2+Q_{22}^2)}
\label{eq:fcorrect}$$ Where ${\cal F}_i$ is an unbiased estimate of the flexion, and $\tilde{{\cal F}}_i$ is the estimated flexion if one does not include the correction for the PSF. However, the normalization constant may be estimated directly from combinations of the PSF 2nd and 4th moments, and the unweighted moments of the image. Since this term represents something like the overall radial profile of the source, the unweighted moments can be estimated even under noisy conditions.

Similarly, the second flexion may be estimated as: $$\tilde{{\cal G}}_i={\cal G}_i
\frac{\xi^{(0)}}{\xi}$$

Though we have derived these relations for a nearly circular source, we have found they provide a good correction even when the PSF and intrinsic image size are comparable, and when ellipticities for the source image are $\varepsilon\simeq 0.2$.

Simulated Lensing {#sec:simulate}
=================

