A Coupled-Path Retrieval Algorithm for the Hurricane Imaging Radiometer (HIRad)

            The stepped frequency microwave radiometer (SFMR) is currently used for measuring surface wind speed and rain rates from hurricane reconnaissance aircraft. SFMR uses a single nadir-pointing antenna to measure the surface wind speed directly below the aircraft. The Hurricane Imaging Radiometer (HIRad) is an improved version of the SFMR that is currently under development by NASA, NOAA, the University of Central Florida and the University of Michigan. HIRad improves upon SFMR's spatial sampling by observing the surface over a wide cross-track swath as the aircraft flies over a hurricane.

HIRad is a passive C-band radiometer that operates at 4, 5, 6, and 6.6 GHz. From HIRad's observations, surface wind speed and rain rate are retrieved over a wide swath of about 70 km with 1 – 5 km resolution. HIRad is capable of measuring wind speeds of 10-85 m/s and rain rates of 5-100 mm/hr. HIRad has been a part of NASA's GRIP (Genesis and Rapid Intensification Processes) airborne campaign and is currently a part of the HS3 (Hurricane Severe Storm Sentinel) campaign. Flights over Hurricanes Earl and Karl during GRIP provide observations for developing a more rigorous wind speed and rain rate retrieval algorithm.

            A situation unique to HIRad creates a need for a more rigorous, geophysically-based retrieval algorithm. HIRad's observing geometry is such that the horizontal resolution is comparable to the freezing level height over portions of the cross-track field of view. This unusual condition introduces new complications to the retrieval algorithm that estimates surface wind speed and rain rate from the brightness temperature (TB) observations using an inversion of the forward radiative transfer model (FRTM). In particular, when HIRad observes a hurricane's eye wall at high incidence angles, the FRTM cannot assume that the rain columns which contribute to the upwelling and downwelling components of the net observed TB are the same. Figure (1) compares the associated geometry of the radiative transfer propagation paths for SFMR, typical spaceborne radiometers, and HIRad.

Figure 1a-c: The diagrams above show the relative geometry of the associated radiative transfer propagation paths for SFMR, typical spaceborne imagers with a conically scanning beam and a constant incidence angle, and HIRad. (Respectively from left to right.) This diagram is not to scale. Boxes represent the relative size of the rain column/ horizontal resolution for each situation.

FRTMs are embedded in retrieval algorithms. Generally, microwave FRTMs use the following radiative transfer model for the observed TB

                       TB = T_UP + e^-τ(ɛT_SFC + (1 - ɛ)( e^-τT_cos + T_DOWN))                    (1)

where T_UP is the upwelling atmospheric brightness emitted along the antenna line of sight, T_SFC is the physical sea surface temperature, T_DOWN is the downwelling atmosphere brightness, ɛ is the emissivity of the sea surface, and T_cos is the cosmic TB. e^-τ is the transmissivity of the atmospheric column.

For situations like those shown in Figure (1a) and (1b), upwelling and downwelling propagation paths can be reasonably approximated as passing through the same atmospheric column. However, in HIRad's high incidence angle situations, the upwelling and downwelling propagation paths can pass through significantly different horizontal portions of the atmosphere. In this case, the TB is modeled by a modified FRTM given by

                       TB = T_UP + e^-τUP(ɛT_SFC + (1 - ɛ)( e^-τDOWNT_cos + T_DOWN))                    (2)

The TB and transmissivity of the downwelling and upwelling paths are now different than those in eqn. (1). The transmissivity depends on the distribution of rain seen over the propagation path. In addition to the unmodified input variables of wind speed, sea surface temperature, EIA and frequency, the modified FRTM takes a weighted sum of the rain seen by HIRad as a new input. Each observed rain rate is weighted by a fractional area corresponding to the overlap of HIRad's propagation path through an observation bin. For a more robust TB, downwelling and upwelling rain rates are both accounted for separately and input into the modified FRTM. These weighted upwelling and downwelling rain rates are used to model the transmissivity for each propagation path separately.

With increasing EIA, the FRTM should consider a larger distribution of neighboring observations when modeling the TB observed at a given angle. Figure (2) shows the number of neighboring observations to be considered with increasing EIA. These calculations are made assuming 2 km horizontal resolution, 5 km freezing level height, and a 20 km aircraft height.

Figure 2: The number of neighboring observations that need to be considered when modeling the TB that HIRad observes with increasing EIA.

At HIRad's maximum EIA of approximately 70 degrees, a FRTM should consider up to seven neighboring rain column segments in a typical observation.  In a span of seven observations, HIRad could easily go from seeing a calmer scene to one with an intense rain band. Given the typical flight patterns flown during the GRIP field campaign, this observing situation is actually more typical than not. A new retrieval algorithm has been developed to handle this condition.

            Using the updated FRTM described above, an iterative least squares estimator is used to retrieve surface wind speed and rain rate from HIRad's TB observations. During each iteration, the Jacobian matrix associated with the FRTM is populated using a ‘Frechet derivative' for each retrieved variables at every EIA. A ‘Frechet derivative' is numerical derivative representing a change in simulated TB per small change in a retrieved variable. The populated Jacobian represents a weighting function that describes the rain column segments at different EIAs which contribute to the TB observed at a single EIA.

            To summarize, a new geophysically-based retrieval algorithm will be presented. Key to this algorithm is the ability to account for a distribution of rain seen along the path of HIRad's field of view.  Performance of this retrieval algorithm on data from GRIP and HS3 missions will be shown.