Seasonal difference operator (${\Delta }_S=1-B^S$)

For an integer period $S$, ${\Delta }_S{X}_t=X_t-X_{t-S}$ uses the backshift operator as $1-B^S$. It annihilates a deterministic period-$S$ mean. If $X_t=m_t+{\epsilon }_t$ with $m_{t+S}=m_t$ and strong white noise, the result is the stationary moving average ${\epsilon }_t-{\epsilon }_{t-S}$.

If $X_t=c_t{\epsilon }_t$ with a deterministic periodic scale $c_t=c_{t-S}$ and strong white noise of variance ${\sigma }^2$, then ${\Delta }_S{X}_t=c_t({\epsilon }_t-{\epsilon }_{t-S})$. Its variance is $2{\sigma }^2{c}_t^2$, still seasonal when $c_t^2$ varies. A periodic scale model or variance standardization is more appropriate than blindly applying differencing.