SFMR Algorithm

The SFMR infers surface wind speed by using the blackbody radiance emitted by the ocean surface, as governed by Planck’s law (Uhlhorn and Black 2003). Using the Rayleigh-Jeans approximation to Planck’s law (applicable at microwave frequencies), a linear relationship between TB and physical temperature, T , is implied. The portion of energy emitted by the atmosphere and the sea surface is expressed as emissivity (ε)

ε = TB

T . (3)

Utilizing Kirchoff’s energy conservation law, the absorption and emission of a material in local thermodynamic equilibrium must be equal. Thus, radiation not emitted by the material (where scattering is neglected) must be transmitted,

ε = 1− τ, (4)

where τ is the transmissivity.

Figure 5 Radiative contributions to total TB as measured from a nadir-looking ra-diometer. The values in parentheses are percentages of contributions to the total TB. All values are computed using the assumed atmospheric profile found in the SFMR algorithm with zero wind and rain rate at a frequency of 5 GHz (Diagram from Appendix A of Uhlhorn and Black 2003).

As shown by Fig. 5, the apparent TB observed by the nadir-looking SFMR is the sum of the following radiative sources: (i) cosmic radiation that has not been attenuated by the atmosphere and is reflected by the sea surface (TCOS); (ii) downward atmospheric emissions that are reflected by the sea surface (TDOW N); (iii) emissions from the sea surface that are not attenuated by the intervening atmosphere (TOCEAN);

and (iv) upward emissions from the intervening atmosphere (TU P). The absorption, emission, and transmission of the atmosphere is predominantly due to absorption by oxygen and water vapor molecules, and absorption and scattering by liquid water constituents.

The atmospheric contribution TDOW N to the apparent TB can be considered to be the sum of the contributions from gaseous and hydrometeor constituents

TDOW N = (1− τr,∞)hTr,∞i + (1 − τa,∞)hTa,∞i , (5) where T is measured in Kelvin; the subscripts a and r indicate transmission by the atmosphere and rain, respectively; ∞ represents the contribution by the entire

at-mospheric column; and angle brackets denote a mass-weighted layer average. The total sky TB (TSKY) thus becomes the sum of TDOW N and the extraterrestrial source (TCOS)

TSKY = TDOW N + (1− τr,∞τ_a,∞) TCOS. (6) The ocean TBis directly related to SST and is given by TOCEAN = ε (SST ), where ε is the variable to be solved in the SFMR algorithm. The ocean and reflected sky contributions that are not attenuated by the intervening atmosphere can be represented as (1 − ) T^SKY. The upward emission of radiation from the atmospheric below the aircraft flight level is represented by

TU P =¡

1− τ^r,A/Cτa,A/C

¢ ­Ta,A/C

®, (7)

where A/C denotes the emission from atmospheric layers below the aircraft. The total TB observed by the nadir-looking SFMR aboard the aircraft is the sum of the upwelling components

TB =¡ τr,A/C

¢ ¡τa,A/C

¢[TOCEAN + (1− ε) T^SKY] + TU P. (8) Under calm winds and the typical rain-free tropical atmosphere with a nominal flight level, TOCEAN represents approximately 95 percent of the total apparent TB from the ocean surface.

Since the ocean absorbs only a portion of the incident radiation, the remainder is reflected and is represented as ε = 1 − Γ. The reflectivity of a smooth (specular) ocean surface can be expressed in functional form as

Γp = Γp(θ, f, SST, S), (9)

where p is the polarized state (horizontal or vertical), θ is the incident angle, f is the electromagnetic frequency, and S represents the ocean salinity. Since SFMR measurements are taken at nadir, the incident angle is zero and reflection is independent of polarization. Assuming the SST and salinity are known, the reflectivity can be calculated at each SFMR frequency.

As energy is transferred to the ocean surface by the wind, the scattering and emission properties become much more complicated. The ocean surface becomes roughened by capillary and short gravity waves as the wind stress increases. When a critical steepness is reached, the waves break and produce foam patches and streaks on the ocean surface. Foam emits microwave energy more readily than a specular ocean surface, and thus a fractional foam model must be incorporated. The SFMR algorithm assumes that wind speed-dependent and specular components make up the total emissivity of the ocean surface.

Black and Swift (1984) established the relationship between emissivity and hurricane-force winds through dual aircraft missions in which one aircraft was flown 0.5- to 1.5-km altitude and a second aircraft was flown at approximately 3 km in altitude to take in situ wind speed measurements. These measurements were reduced to near-surface wind speed values using the Powell (1980) boundary layer model. The emissivity of the wind-driven sea is determined by removing the emissivity of a specular sea surface from the total apparent emissivity. The specular Fresnel power reflection coefficient (Γ) is calculated using an algorithm developed by Klein and Swift (1977). The spec-ular Fresnel power reflection coefficient is added to the wind-driven excess to obtain the total emissivity.

The emissivity of the rain column is left to be determined. As in Appendix A of Uhlhorn and Black (2003), solving Eq. (8) for emissivity ( ) gives

= aτ_r,∞− b

and c = 0. Thus the approximate expression for emissivity from the ocean surface in the presence of rain becomes

Given the small ratio of rain droplet size to SFMR electromagnetic wavelength, scat-tering can also be neglected at even the high rain rates present in tropical cyclones.

Thus, rain rate can be estimated solely as a result of absorption processes. Trans-missivity of a rain column is a function of hydrometeor content that is proportional to electromagnetic frequency and rain rate. The relationship between transmissivity and absorption, κr, is given by

τr= exp (−κ^rh), (12)

where h is rain column depth. Rainfall absorption coefficient is derived using

κr = aR^b, (13)

where R denotes rain rate in mm h⁻¹, and a and b have been empirically determined.

Olsen et al. (1978) have shown a to be a function or frequency and rain rate with the following relationship:

a = gf^n(R)‚ (14)

where n ≈ 2.6R^0.0736 according to Atlas and Ulbrich (1977) and g = 1.87x10⁻⁶ Np km⁻¹ based on a calibration by Black and Swift (1984). The exponent b = 1.35 in Eq. (13) was determined by the C-band radar reflectivity measurements in hurricanes made by Jorgensen and Willis (1982). According to Uhlhorn and Black (2003), the SFMR is capable of measuring rain rates >5 mm h⁻¹. As shown in Fig. 12b of Uhlhorn and Black (2003), the spread of apparent TB increases between microwave frequencies with increased rain rate, such as in the eyewall of a tropical cyclone.

The retrieval of surface wind speed and rain rate from a set of SFMR TB measure-ments constitutes an inverse problem that requires the number of measuremeasure-ments to be greater than or equal to the number of parameters to be solved. Since the SFMR uses six frequencies, the solution is over-determined and a least squares inversion method is applied (Uhlhorn and Black 2003). Using a physical model designed by Pedersen (1990), an n-length vector of TB measurements (H) to an m-length vector of retrieved parameters (y) is

H_n= Wnm· y^m, (15)

where the matrix W consists of the partial derivatives of H with respect to y

W_nm= ∂Hn

∂ym

. (16)

A set of radiometer observations is given by

Hˆ_n = Wnm· y^m+ ²n = Hn+ ²n, (17) where ²n is an error vector and the top hat denotes an estimate of the true vector.

The solution vector for the parameter estimates is given by

ˆ y=¡

W^TW¢−1

W^TH,ˆ (18)

and is obtained from the condition that the sum of squared differences between the observed and model-predicted TB is minimized as follows:

Xn

Solutions are possible when the derivative matrix elements Wij are significantly different so that errors ²i are not spuriously amplified by the elements of¡

W^TW¢−1

. When wind speeds are <10 m s⁻¹ or rain rate <5 mm h⁻¹, the sensitivity of changes in TB observed by SFMR frequencies at nadir incident angle is typically too weak for the SFMR algorithm to converge to a solution.