  We present the second-order expression for the observed redshift, accounting
for all the relativistic effects from the light propagation and from the frame
change at the observer and the source positions. We derive the generic
gauge-transformation law that any observable quantities should satisfy, and we
verify our second-order expression for the observed redshift by explicitly
checking its gauge transformation property. This is the first time an explicit
verification is made for the second-order calculations of observable
quantities. We present our results in popular gauge choices for easy use and
discuss the origin of disagreements in previous calculations.
