Weighting Functions and Atmospheric Soundings: Part I Ralf Bennartz Cooperative Institute for Meteorological Satellite Studies University of Wisconsin Madison
Outline What we want to know and why we need it? What does a satellite really observe (a.k.a. The radiative transfer equation) Weighting functions Recap: What is important?
What do we want to know? Weather forecasting? What data is needed for NWP model initialization (assimilation)? Source: Kelly and Thepaut, 207 ECMWF Newsletter 113
Why are satellite data important for forecasts? Forecast skill strongly dependent on satellite data E.g. Southern hemisphere: Day 3-4 forecasts now as good as Day 1 forecasts without satellite data. Source: Kelly and Thepaut, 207 ECMWF Newsletter 113
What does a satellite observe? Detects number of photons per exposure time at a given wavelength (or wvl range)traveling from viewing direction into detector No. of photons, direction, per time RADIANCE We need to physically and quantitatively understand the relation between observed radiance and state of the atmosphere Radiative Transfer Equation
What can happen to photons? 1. Photons can be ABSORBED 2. Photons can be EMITTED 3. Photons can be SCATTERED Let s now build a budget equation for a beam of photons I(θ, ) traveling in a certain direction (θ, ): Done.. di ds = Sources - Sinks Theoretical physics is so great
Unfortunately, we need a bit more detail Sink I: Absorption Proportional to how many photons are traveling in the first place (I) Proportional to how effectively the medium absorbs radiation, let s call this proportionality constant the volume absorption coefficient (β A ) di = β A I I(s) = I ds 0 e β A s A { = I τ 0 (s) Beer s Law (exponential decay) Transmission
Unfortunately, we need a bit more detail Source I: Emission Proportional to Planck function of medium (gas) at temperature T Proportional to how effectively the medium emits radiation If the medium is in local thermodynamic equilibrium this is also described by to the volume absorption coefficient (β A ) di ds E = β A B(T ) This cannot be integrated so easily as we do not know T along the way..and, if the medium emits (β A >0), it also absorbs. so, we have to put absorption and emission together
Schwarzschild s Equation Alright di ds = β A B I ( ) This is the differential form of Schwarzschild s Equation Applies to radiative transfer, IF scattering can be neglected Good news: For many applications in the infrared and microwave scattering actually can be neglected Bad news: If we want to understand atmospheric soundings and weighting functions, we will have to integrate Schwarzschild s Equation.
Non scattering θ S Temperature
Integrating Schwarzschild s Equation μ = cos(θ S ) : Cosine of Zenith angle δ (z) = β A dz : Optical Depth z τ (z,μ) = e δ (z)/μ : Transmission
Integrating Schwarzschild s Equation di = β A ( B I ) ds I = I 0 τ 0 + B(T ) β A(z) τ (z) dz μ 0 1424 3 W (z) W (z) = β A(z) τ (z) = dτ μ dz 0 I = I 0 τ 0 + B(T ) W (z) dz
Integrating Schwarzschild s Equation μ = cos(θ S ) : Cosine of Zenith angle δ (z) = β A dz : Optical Depth z τ (z,μ) = e δ (z)/μ : Transmission W (z,μ) = dτ dz : Weighting Function
Integrating Schwarzschild s Equation μ = cos(θ S ) : Cosine of Zenith angle δ (z) = β A dz : Optical Depth z τ (z,μ) = e δ (z)/μ : Transmission W (z,μ) = dτ dz : Weighting Function
Weighting Functions
Weighting Functions
Weighting Functions
Weighting Functions
Recap Non-scattering radiative transfer can be described using Schwarzschild s Equation Very useful in the infrared and microwave Absorption coefficient tells us how strongly a gas absorbs/emits. Transmission (between two points A and B) tells us what fraction of radiation will survive (i.e. not be absorbed Weighting function tells us where the radiation observed originated in the atmosphere. Allows us to relate observed radiance to layers/levels in the atmosphere Next: What gases absorb where and how strongly?