The filter kernel G {\displaystyle G} uses cutoff length and time scales, denoted Δ {\displaystyle \Delta } and τ c , {\displaystyle \tau _{c},} respectively. Scales smaller than these are eliminated from ϕ ¯ . {\displaystyle {\overline {\phi }}.} Using this definition, any field ϕ {\displaystyle \phi } may be split up into a filtered and sub-filtered (denoted with a prime) portion, as