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Vector irradiance, E, referred to a plane with a normal, n, is defined as follows:
| En | = n ∫4π L(ξ) dΩ | |
| = n ∫4π L(ξ) |n · ξ| dω | ||
| = n ∫4π L(θ, φ) cosθ sinθ dθ dφ | (1) |
where L is the radiance, ξ is a direction unit vector, Ω is the projected solid angle, i.e. the solid angle relative to the orientation of the reference plane:
| Ω = |n · ξ| ω | (2) |
with ω being the solid angle. Notation a · b specifies, as usually, the dot product of vectors a and b. Note that the magnitude of the vector irradiance corresponds to the irradiance as defined by Eq. 1 in Radiometry: Irradiance by accounting for the direction of the incident electromagnetic radiation relative to the reference plane.
In analogy with the scalar irradiance, by considering radiation incident on a horizontal area from the upper and lower hemispheres, one can further define the downwelling (downward) and upwelling (upward) irradiances, respectively (for example, Morel and Smith 1982):
| Ed = ∫Ξd L(ξ) |n · ξ| ω | (3) |
| Eu = ∫Ξu L(ξ) |n · ξ| ω | (4) |
where Ξd and Ξu are respectively the solid angles subtended by the lower and upper hemispheres. These irradiances are also referred as the downward and upward plane irradiances respectively (for example, Mobley CD 1989). Note that the difference of these irradiances is the magnitude of the vertical component of the vector irradiance (for example, Mobley 1994). Indeed, if n points, say, downwards (the reference plane is horizontal), we have:
| En | = n ∫4π L(ξ) dΩ | |
| = n [ ∫Ξd L(ξ) |n · ξ| - dω ∫Ξu L(ξ) |n · ξ| dω ] | ||
| = n ( Ed - Eu ) | (5) |
The magnitude of En, i.e. Ed - Eu, indicates in this case the magnitude and orientation of the flow of radiation energy along the vertical direction.
One can think of En, expressed by Eq. 1, as a component of a vector and generalize the concept of the vector irradiance to three dimensions (for example, Maffione RA et al 1993) by defining that vector as follows:
| E = Ex + Ey + Ez | (6) |
where the three terms on the right side of the above equation represent projections of vector E onto the x, y, and z axis of a Cartesian coordinate system, respectively. Each of this vectors represents the net energy flow along the respective axis of that coordinate system.
See also: Radiometry: Scalar irradiance.
| CITATION: Jonasz M. 2006. Radiometry: Vector irradiances (www.tpdsci.com/Tpc/RdmIrdVctER.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 22-Mar-2006 Modified: 02-Mar-2007 Peer-reviewed: 05-Mar-2007 |
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