First angular moment of a radiance field ($\int \mathbf{\omega }L(\mathbf{\omega })d\Omega$)

The first angular moment of a radiance field is the vectorwith units $\mathrm{W}{\mathrm{m}}^{-2}$ for frequency-integrated radiance. It records a directional imbalance. Here $\mathbf{\omega }$ is the viewing direction toward the arriving light, as in dipole anisotropy of a radiance field; using propagation directions instead reverses the vector. With that convention change, this is the bolometric vector form of the radiative flux, an existing radiation-field moment.

Opposite ideal planar receivers measure this moment without assuming a pure dipole. Since $\max (0,x)-\max (0,-x)=x$, the cosine-response irradiance formula givesThree orthogonal receiver pairs therefore determine the three components. For original differences $(0.60,-0.20,0.30)\mathrm{W}{\mathrm{m}}^{-2}$, the moment has those components and magnitude $0.70\mathrm{W}{\mathrm{m}}^{-2}$. The derivation requires the same spatial field, stable gains and known angular response. The primary optical guide describes cosine-receptor characterization rather than assuming it perfect.^[ref]

A zero first moment does not establish an isotropic radiance field. For example, two equal oppositely directed collimated beams have cancelling vector moments while remaining strongly directional. Conversely, any nonzero first moment rules out isotropy. The scalar all-direction integral $J=\int Ld\Omega$ bounds the moment by $|\mathbf{M}|\le J$ for nonnegative radiance, by the triangle inequality and $|\mathbf{\omega }|=1$.

Within the restricted dipole anisotropy of a radiance field model, $\mathbf{M}=(4\pi /3)L_0\mathbf{b}$ determines its dipole vector. More general angular structure is not reconstructed by these three numbers. Opposed receivers reveal the first moment, not the entire directional distribution.