# Supplement 1  {#supplement-1 .unnumbered}

# Algorithm Theoretical Basis Document for {#algorithm-theoretical-basis-document-for .unnumbered}

####### **AMSR-E Ocean Algorithms** {#amsr-e-ocean-algorithms .unnumbered}

Frank J. Wentz and Thomas Meissner

# Introduction

This document is a supplement to the Algorithm Theoretical Basis
document (ATBD) for AMSR-E Ocean Algorithms \[*Wentz and Meissner*,
2000\]. This supplement focuses on the structure of the retrieval
algorithm, which is someone different than described in *Wentz and
Meissner* \[2000\].

The ocean algorithm is based on the physical Radiative Transfer Model
(RTM) described in *Wentz and Meissner* \[2000\]. The RTM consists of an
atmospheric absorption model for water vapor, oxygen and liquid cloud
water and a sea surface emissivity model that parameterizes the
emissivity as function of sea surface temperature, sea surface salinity,
sea surface wind speed and direction. The details of the physical RTM
have been described in *Wentz and Meissner* \[2000\]. Some components of
the model have been updated. These are the dielectric constant of sea
and cloud water \[*Meissner and Wentz, 2004*\], the isotropic wind
induced sea surface emissivity \[*Meissner and Wentz, 2006*\] and the
wind directional signal of the sea surface emissivity \[*Meissner and
Wentz 2002, Meissner and Wentz 2006*\].

The RTM is used to Monte Carlo simulate ocean brightness temperatures
(T~B~) at the top of the atmosphere (TOA) over a large training data set
of ocean-atmosphere scenes. The simulated T~B~ are then used to train
multi-stage regressions.

This Supplement 1 to the ATBD describes and explains in detail the
structure and training method of retrieval algorithms. The retrievals
are done by a set of two-stage regression algorithms. The first-first
stage is a set of regressions trained with global data. The first-stage
retrievals provide reasonable good estimates of SST, wind, vapor, and
cloud. However because of the non-linear relationships between
brightness temperature and the geophysical parameters, the retrievals
coming from the global regressions are not optimal. In the second stage,
we use a set of regressions that has been specially trained for a
localized range of environmental conditions in the neighborhood of the
first-stage retrievals. A similar methodology is used for SST, wind,
vapor, and cloud. Here we use the SST and wind speed algorithm as an
example.

# Basis of the SST and Wind Speed Algorithm

Between 4 and 11 GHz, the vertically polarized brightness temperature
(T~B~) of the sea-surface has an appreciable sensitivity to SST. For
example, at the AMSR-E frequency of 6.9 GHz, the derivative of the v-pol
T~B~ with respect to SST is about 0.5. It is this sensitivity that is
the basis for microwave SST retrievals.

In addition to SST, T~B~ depends on the sea-surface roughness and on the
atmospheric temperature and moisture profile. Note that sea-surface
roughness, which is tightly correlated with the local wind, is usually
parameterized in terms of the near-surface wind speed and direction.
Fortunately, the spectral and polarimetric signatures of the
surface-roughness and the atmosphere are quite distinct from the SST
signature, and the influence of these effects can be determined (and
retrieved) given simultaneous observations at multiple frequencies and
polarizations. For AMSR-E there are 10 channels corresponding to 5
frequencies (7, 11, 19, 24 and 37 GHz) and 2 polarizations (v-pol and
h-pol). This channel set is more than sufficient to separate the SST,
wind, and atmosphere signals (vapor, cloud, and rain).

# Formulation of the Basic SST and Wind Algorithm

The AMSR-E SST retrieval algorithm is a physically based regression that
expresses SST in terms of brightness temperature (T~B~). The algorithm
operates in two stages. In the first stage, initial estimates of SST and
wind speed are obtained from the following expressions:

![](media/image1.wmf) (1a)

![](media/image2.wmf) all channels except the 24 GHz channels (1b)

![](media/image3.wmf) for the two 24 GHz channels (1c)

where subscript *j* = 1 or 2 denotes either SST (*p~11~*) or wind speed
(*p~12~*). The leading subscript *1* denotes this is a first-stage
retrieval. The summation is over the 10 AMSR-E channels, and *T~Bi~*
refers to the brightness temperature for the i^th^ channel. The terms
*a~ij~* and *b~ij~* are regression coefficients that are determined from
a radiative transfer model (RTM) as is discussed below.

Equation (1) provides a reasonable good initial estimate of SST and wind
speed. However, the relationship of T~B~ versus SST and wind is
non-linear, and the simple form given by (1) is not capable of fully
representing these non-linearities. Hence, we add a second stage to the
retrieval algorithm. For the second stage, a large set of localized
retrieval algorithms are used. By 'localized', we mean the algorithm is
trained to perform well over a relatively narrow range of SST and wind
speeds. Localized algorithms are derived for 38 SST reference values
ranging from --3 to 34 C and for 38 wind speed reference values ranging
from 0 to 37 m/s. All SST/wind combinations are considered, thereby
giving a total of 1444 localized algorithms. Each algorithm is trained
to perform well over a SST/wind range of ±1.5°C and ± 2 m/s centered on
the reference SST and wind value. The reason for using such a large
number of algorithms is to ensure good continuity in the retrievals when
moving from set of reference SST/wind to the next. These localized
algorithms take the form:

![](media/image4.wmf) (2)

where the subscript *k* ranges from --3 to 34 and subscript *l* ranges
from 0 to 37 and denote the reference SST and the reference wind speed,
respectively. The second-stage retrieval is found from a linear
interpolation of the localized algorithms that are in the neighborhood
of the first-stage retrievals.

![](media/image5.wmf) (3)

![](media/image6.wmf) (4a)

![](media/image7.wmf) (4b)

![](media/image8.wmf) (4c)

![](media/image9.wmf) (4d)

![](media/image10.wmf) (5a)

![](media/image11.wmf) (5b)

![](media/image12.wmf) (6a)

![](media/image13.wmf) (6b)

# Training of the SST and Wind Algorithm

The algorithm regression coefficients are found by generating a large
ensemble of simulated brightness temperatures computed from a microwave
RTM for the ocean and intervening atmosphere \[*Wentz and* *Meissner*,
2000\]. The simulation is based on a set of 42,195 radiosonde soundings
launched from weather ships and small islands around the globe
\[*Wentz*, 1997\]. These radiosondes are used to specify the atmospheric
part of the RTM. A cloud layer of various columnar liquid cloud water
densities ranging from 0 mm to 0.3 mm and various cloud bottom and top
heights between 500 m and 2500 m is superimposed over the radiosondes.
For each radiosonde, a large range of the sea-surface conditions is
considered. The sea-surface wind speed is varied from 0 to 40 m/s. The
sea-surface temperature is varied ± 5.5°C about the Reynolds SST for the
island site. For each of these Earth scene, the RTM computes the set of
10 AMSR-E T~B~, and then noise is added to the T~B~ to replicated sensor
noise.

The simulated T~B~ are substituted in equation (1), and the regression
coefficients *a~ij~* and *b~ij~* are found via the standard least
squares method, which minimizes the variance between the SST and wind
value given by (1) and the true SST and wind value. Once the stage-1
algorithms are specified, training of the stage-2 algorithms can be
performed. In this case, regression coefficients *c~ijkl~* for 1444
separated algorithms are found, again using standard least squares. Each
algorithm is trained using just a small subset of the Earth scenes that
have a SST and wind value near the algorithm's reference SST/wind.

# Correction for Wind Direction Signal 

The surface emissivity and therefore also the T~B~ in each channel also
depend on wind direction relative to the looking azimuth. The wind
direction signal grows with wind speed and reaches up to 3 K peak to
peak amplitude above 10 m/s. \[*Meissner and Wentz 2002*, *Meissner and
Wentz 2006*\]. This leads to a significant error in the retrieved SST
and wind and a crosstalk between the accuracy error of the SST and wind
retrievals and relative wind direction. In order to correct for the wind
direction signal, the algorithm training is performed separately for a
discrete set of 19 values for relative wind directions between 0^o^ and
360^o^ in 20^o^ steps. When doing the retrievals, the wind direction
needs to be obtained form an auxiliary source. For operational
processing, we use the wind direction from the NCEP GFS forecast model.
For the final processing the forecast is substituted by the NCEP GDAS
analysis. The regression coefficients are linearly interpolated from the
19 reference values of the discrete wind directions to the actual NCEP
wind direction.

# Earth Incidence Variations

The surface emissivity also depends on the Earth Incidence Angle (EIA).
Due to the oblateness of the Earth the EIA can deviate by about ±0.5°
from the nominal EIA of 55^o^ over the course of an AMSR-E orbit. The
method to compensate for the variation in EIA is similar to the method
to compensate for the wind direction effect described in the last
section. The SST and wind algorithms are trained for 5 discrete values
of EIA that are taken in 0.5^o^ steps around the nominal EIA of 55^o^.
In the retrievals, the regression coefficients are then linearly
interpolated from the discrete set to the actual value of the EIA.

# Salinity Correction 

All regressions are trained using a reference ocean surface salinity
![](media/image14.wmf) of 35 ppt when computing the ocean surface
emissivity. In order to accommodate the variability of the ocean
salinity we could follow a similar procedure than we have done for the
variations in relative wind direction and EIA. Because ocean salinity is
a slow varying function of time and location, it is somewhat simpler to
correct the measured AMSR-E T~B~ to the reference salinity, before doing
the retrieval. We precompute a T~B~ - correction climatology atlas,
which consists of the ratio ![](media/image15.wmf) for each channel at a
discrete set of locations (latitude and longitude) and times (month of
the year). For the calculation of this ratio we have used salinity
values from the NODC World Ocean Atlas, SST values from the Reynolds
climatology and typical values for the other geophysical parameters
(wind speed, water vapor and liquid cloud water). In the retrievals, the
T~B~ correction values are linearly interpolated in location and time to
the actual values of the AMSR-E measurement.

# Algorithm Flow Diagram

Figure 1 shows a schematic diagram of the SST algorithm training and
retrievals. The schematic diagram for the wind algorithm is essentially
the same.

![](media/image1.wmf)

######## Figure 1. Schematic diagram of the SST algorithm training and retrievals {#figure-1.-schematic-diagram-of-the-sst-algorithm-training-and-retrievals .unnumbered}

# Algorithms for Other AMSR-E Ocean Products

The SST and wind speed algorithms, which are described above, are
essentially the same. The only real difference is that the SST algorithm
uses all 10 AMSR-E lower channels, while the wind algorithm does not use
the 7 GHz channels. The reason is that 7 GHz is not significantly
improving the wind retrievals and hence by not using it, we can obtain
wind retrievals are a higher spatial resolution.

The columnar water vapor and columnar liquid cloud water algorithms also
have a similar structure. These two algorithms only use the 19, 23, and
37 GHz channels.

# References

Wentz, F.J., and T. Meissner, AMSR Ocean Algorithm Theoretical Basis
Document, Version 2. Remote Sensing Systems, Santa Rosa, CA, 2000.

Wentz, F.J., A well-calibrated ocean algorithm for SSM/I. J*. Geophy.
Res. (102), pp. 8703-8718, 1997*.

Meissner, T., and F.J. Wentz, The Complex Dielectric Constant of Pure
and Sea Water from Microwave Satellite Observations. *IEEE TGARS, vol.
42(9), pp. 1836 -- 1849, 2004.*

Meissner, T. and F. Wentz, Ocean Retrievals for WindSat: Radiative
Transfer Model, Algorithm, Validation. *CD with Proceedings of the 9^th^
Specialist Meeting on Microwave Radiometry and Remote Sensing
Applications, San Juan, Puerto Rico, 28 February -- 03 March 2006, IEEE
Catalog no. 06EX1174C, 2006.*

Meissner, T., and F.J. Wentz, An Updated Analysis of the Ocean Surface
Wind Direction Signal in Passive Microwave Brightness Temperatures.
*IEEE TGARS, vol. 40(6), pp. 1230 -- 1240, 2002.*