NASA Technical Memorandum 104566, Vol. 32

SeaWiFS Technical Report Series

Stanford B. Hooker, Elaine R. Firestone, and James G. Acker, Editors

Volume 32, Level-3 SeaWiFS Data Products:

Spatial and Temporal Binning Algorithms

Janet W. Campbell, John M. Blaisdell, and Michael Darzi

August 1995
NASA Technical Memorandum 104566, Vol. 32

SeaWiFS Technical Report Series

Stanford B. Hooker, Editor

NASA Goddard Space Flight Center

Greenbelt, Maryland

Elaine R. Firestone, Technical Editor

General Sciences Corporation

Laurel, Maryland

James G. Acker, Technical Editor

Hughes STX

Lanham, Maryland

Volume 32, Level-3 SeaWiFS Data Products:

Spatial and Temporal Binning Algorithms

Janet W. Campbell

University of New Hampshire

Durham, New Hampshire

John M. Blaisdell and Michael Darzi

General Sciences Corporation

Laurel, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

1995
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Abstract

The level-3 data products from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) are statistical data

sets derived from level-2 data. Each data set will be based on a fixed global grid of equal-area bins that

are approximately 9 × 9 km2. Statistics available for each bin include the sum and sum of squares of the

natural logarithm of derived level-2 geophysical variables where sums are accumulated over a binning period.

Operationally, products with binning periods of 1 day, 8 days, 1 month, and 1 year will be produced and archived.

From these accumulated values and for each bin, estimates of the mean, standard deviation, median, and mode

may be derived for each geophysical variable. This report contains two major parts: the first (Section 2) is

intended as a users’ guide for level-3 SeaWiFS data products. It contains an overview of level-0 to level-3 data

processing, a discussion of important statistical considerations when using level-3 data, and details of how to

use the level-3 data. The second part (Section 3) presents a comparative statistical study of several binning

algorithms based on CZCS and moored fluorometer data. The operational binning algorithms were selected

based on the results of this study.

1. INTRODUCTION

The level-3 data processing stage is the first stage in

which data from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) are spatially and temporally averaged. Prior

to this stage, a standard set of geophysical variables will

be derived for individual pixels. These level-2 variables include chlorophyll concentration, a diffuse attenuation coefficient, and water-leaving radiances in the visible bands

of SeaWiFS.

In generating level-3 data products, pixels containing

valid level-2 data will be mapped to a fixed spatial grid

whose resolution elements are 9 × 9 km2. These square

grid elements or bins are arranged in rows beginning at

the South Pole. Each row begins at 180◦ longitude and

circumscribes the Earth at a given latitude. There are

5,940,422 bins for each level-3 data set. Within each bin,

statistics will be accumulated for time periods of 1 day, 8

days (often referred to as the weekly product), 1 month,

and 1 year. There will be a global level-3 data product

archived for each day, 8-day period, calendar month, and

calendar year of the SeaWiFS mission.

The level-3 data products may be used to derive the

mean, standard deviation, and other statistical measures

for the standard level-2 variables, and for certain other

variables, such as primary productivity, which are functions of level-2 variables. The Coastal Zone Color Scanner (CZCS) North Atlantic monthly composite chlorophyll

images (Esaias et al. 1986 and Feldman et al. 1989) are examples of monthly means derived from level-3 CZCS data.

The purpose of binning data is to create reduced-volume

data sets appropriate for use in climate and basin-scale biogeochemical models. By averaging data over time periods

of several days or longer, problems of missing data can

be overcome. Although temporal and spatial resolutions

are reduced, compared with the level-2 data, the resulting

smoothed level-3 means are effective in depicting seasonal

patterns on regional and basin scales.

There are important statistical considerations that involve the use of level-3 data. Users should be aware of

these considerations, especially in situations where level-3

data are used in models to derive other variables. For example, to use a mean chlorophyll concentration (level-3

variable) in an algorithm to derive mean primary productivity might result in significantly biased results. Recommended procedures for using level-3 variables in models

are presented in this report.

The remainder of this report is divided into two parts.

The first part (Section 2) is intended to serve as a guide for

users of level-3 data products. Section 2.1 is an overview of

the processing from level-0 to level-3. Section 2.2 contains

a discussion of the important statistical considerations involved in using level-3 data, and Section 2.3 provides the

equations to be used to compute the mean, standard deviation, median, and mode of each level-3 variable. Equations

for computing statistics of level-4 variables, derived from

level-3 variables, are given in Section 2.4.

The second part (Section 3) documents a statistical

study based on CZCS data and moored fluorometer data

which compared alternative binning algorithms. Results

of this study were the basis for the selection of the binning

algorithm used. Three color plates compare the results of

alternative binning algorithms applied to seven representative CZCS scenes.

In addition, there are three appendices providing details for statisticians and programmers who may wish to

write codes to bin data. Appendix A explains the procedure used for mapping pixels to bins based on the center

latitude and longitude of the pixel, and for determining the

latitude and longitude coordinates of a bin. Appendix B

contains details of the weighting scheme used for weighting data from different orbits (times). Appendix C contains three pseudocodes that reveal how data are accumulated spatially (Space Binner Code), temporally (Time

Binner Code) and how means, standard deviations, and

1
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

other statistics are calculated from the binned data (Bin

Data Interpreter Code).

2. USERS’ GUIDE

2.1 Overview of Data Processing

As the name would suggest, the level of a data product

refers to the amount of processing that has been applied

to the data. Certain conventions have been adopted to

describe the major levels of processing.

2.1.1 Level-0 Data

Data recorded on board the satellite and subsequently

broadcast to ground receiving stations are called level-0

data. Data broadcast directly (without being recorded) are

also considered level-0 data. The recorded data provide

either local area coverage (LAC) or global area coverage

(GAC). This classification refers to the spatial resolution of

the data. In SeaWiFS LAC data, the spatial resolution is

1.1 km at nadir (directly beneath the satellite), and pixels

are contiguous.

The GAC data are comprised of individual pixels having the same spatial resolution as LAC data (1.1 km), but

the pixels are spaced at 4.4 km intervals. The GAC data

are created on board the satellite by selecting every fourth

pixel on every fourth scan line. This subsampling reduces

the volume of data required to provide global coverage.

A comparative study of alternative GAC sampling algorithms was reported by McClain et al. (1992).

Only a limited amount of LAC data will be recorded

on board SeaWiFS. However, LAC data will be continuously broadcast as high-resolution picture transmission

(HRPT) data to sites around the world which operate licensed ground-receiving stations. All HRPT data will be

LAC data.

2.1.2 Level-1a Data

The level-1a products include the raw image data and

all instrument and spacecraft telemetry, as in the level-0

data, together with appended instrument calibration and

navigation data. In addition, instrument telemetry and

selected spacecraft telemetry are reformatted and also appended.

Approximately 40 minutes of contiguous level-1 data

are produced on the daylight portion of each orbit. Operationally, this 40-minute swath may be subdivided into

two or more level-1 scenes. The division may occur when

the sensor tilt is changed, i.e., so each scene would nominally have a constant sensor tilt, or other criteria, e.g.,

maximum scan lines per scene, may dictate further subdivisions of the swath.

The level-1a data can be used to calculate calibrated

radiances in units of W m−2 μm−1 sr−1 in the 8 spectral

bands of SeaWiFS. This radiance received at the satellite

altitude is solar radiation backscattered from the Earth’s

atmosphere, ocean, clouds and land. Water-leaving radiance (the signal of interest) usually comprises less than

10% of the total signal.

2.1.3 Level-2 Data

Geophysical properties of the ocean and atmosphere

derived from level-1a data are considered level-2 data.

Level-2 data correspond to the original pixel positions;

there is no remapping. Each level-2 scene corresponds to

a level-1 scene and vice versa; there is no change in the

geographical coverage of each scene for operational products.

Before computing level-2 data, pixels are eliminated if

they contain clouds, sun glint, or other abnormalities. For

pixels that pass these screens, an atmospheric correction

algorithm (Gordon et al. 1983 and Gordon and Casta ̃no

1987) is applied to subtract the atmospheric scattering

components from the total radiance, and thus derive the

water-leaving radiances in bands 1–5. Then, bio-optical

algorithms (Clark 1981 and Gordon and Morel 1983) are

applied to the water-leaving radiances to derive in-water

properties.

Standard variables currently planned for computation

are:

LWN (λi) normalized water-leaving radiances in the

bands i = 1–5,

La(λi) atmospheric aerosol radiances in the bands

i = 6–8,

τa(865) aerosol optical thickness at 865 nm (band

8),

PIG CZCS-like pigment concentration (mg m−3),

CHL chlorophyll a concentration (mg m−3), and

K490 diffuse attenuation coefficient at 490 nm

(m−1).

2.1.4 Level-3 Data

The level-3 data are statistical data products derived

by binning level-2 GAC data. This is the first stage at

which data are both spatially and temporally averaged. A

level-3 product will be produced for each day, 8-day period

(week), calendar month, and calendar year of the SeaWiFS

mission. The 8-day periods are started from the first day

of each calendar year. Thus, there will be 46 weeks per

calendar year, with the last week having only 5 or 6 days

instead of 8.

Each data product will contain statistics derived by

mapping level-2 data to a fixed global grid whose resolution elements (called bins) are approximately 9 × 9 km2.

The bins are arranged in rows beginning at 180◦ longitude

and circumscribing the Earth eastward at a given latitude.

There are 5,940,422 bins for each level-3 data product. Appendix A contains details related to the gridding scheme,

2
J.W. Campbell, J.M. Blaisdell, and M. Darzi

and the precise areal coverage and geographic location of

each bin.

Statistical data provided with the level-3 data products

will allow users to calculate the mean, standard deviation,

median, and mode for each level-2 variable listed above.

The procedures are described in Section 2.3, and pseudocodes for programming implementation are detailed in

Appendix C.

In addition to level-2 variables, statistical data will also

be provided for the ratio:

ICK ≡ CHL

K490

(1)

calculated at each pixel in the level-2 data set (but not

saved as a level-2 variable). This ratio, which appears in

several primary productivity algorithms (Balch et al. 1992,

Platt and Sathyendranath 1988, Eppley et al. 1985, Smith

and Baker 1978, and Bannister 1974), may be regarded as

the integral chlorophyll (units of mg m−2) integrated over

the upper optical depth. The rationale for including this

as a level-3 variable will be presented in Section 2.2.

In addition to the level-3 data products, a number of

standard level-3 image products will be produced. These

will include standard mapped images, which are equirectangular projections of means derived from the level-3 statistical data, and reduced resolution images intended for

browsing purposes.

2.1.5 Level-4 Data

In this report, variables derived from level-3 data will

be called level-4 variables. It is anticipated that level-3

data will be used as input to biogeochemical models where

the goal of the modeling is to estimate global fluxes of key

elements such as carbon and nitrogen. In such applications, it is important that the level-4 variable represent a

spatial-temporal mean, e.g., the average daily, weekly, or

monthly carbon flux. The practice of substituting means

into models to produce spatial-temporal means can result

in significantly biased results. This will be discussed further in Section 2.2.

The methods used to produce the level-3 SeaWiFS data

have been designed to overcome this problem for a large

class of level-4 variables. Procedures for computing unbiased estimates of the mean of level-4 variables will be

discussed in detail in the following sections.

2.2 Statistical Considerations

The question of how to bin SeaWiFS data revolved

around certain statistical issues. Many of the issues or

questions raised had come to light through the experience

of binning CZCS data into daily, monthly, and yearly composites. There were several proposed ways to average data,

and results would be significantly different depending on

the method chosen. It was further recognized that the

choice of method should depend on how level-3 SeaWiFS

data are to be used. The practice of using level-3 means in

equations to derive level-4 means was inappropriate, and,

therefore, this issue had to be addressed as well.

Following is a discussion of four major issues and the

summary of the decisions related to each. In many instances, decisions were based on a statistical analysis of

CZCS data and moored fluorometer time-series data. The

results of the statistical study are presented in Section 3.

The four issues were:

1. Should statistics be computed for CHL or for

log(CHL)? What about other level-2 variables?

2. What is the best method for estimating level-4

variables?

3. What statistics should be saved for each sampling domain?

4. Should the temporal statistics give equal weight

to all data falling within the sampling domain?

Or, should some accommodation be made to

compensate for the uneven temporal distribution of data?

2.2.1 CHL vs. log(CHL) Statistics

Chlorophyll measurements tend to be lognormally distributed, i.e., log(CHL) is normally distributed, in large

data sets of satellite or ship data (Fig. 1). Lognormal distributions occur commonly in biological processes where

the rate of change of a variable is proportional to its size

(Aitchison and Brown 1957 and Crow and Shimizu 1988).

One of the first issues addressed, therefore, was whether or

not statistics should be computed for CHL or for log(CHL).

The same question was also addressed for other variables.

It is fairly common practice to log-transform CHL measurements before using them in other derivations. For example, Chelton and Schlax (1991) used log-transformed

data in comparing time averages of chlorophyll data. The

CZCS pigment algorithm was derived by a linear regression

of log(CHL) versus log-transformed radiance ratios, and

CZCS pigment images are usually scaled according to the

logarithm of pigment. The mean derived by first averaging

log-transformed data and then inverting the transform is

the geometric mean. Is the geometric mean preferable to

the arithmetic mean?

It was agreed at the outset that the arithmetic mean

is the appropriate mean for most biogeochemical applications. The mean chlorophyll concentration, for example,

represents the mean biomass per unit volume which will

subsequently be multiplied by total volume (depth × area)

to estimate regional or global biomass. However, the sample mean derived from small samples might be a poor estimator of the true population mean.

Let X be a lognormally distributed variable (Fig. 2),

and let X denote the true mean of X within a sampling

domain. In the context of the SeaWiFS data processing,

3
0

200

400

600

800

1000

1200

0.010 0.10 1.0 10 100

0

20

40

60

80

100

120

140

0.01 0.1 1 10 100

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 1. Histograms of chlorophyll concentration derived from in situ measurements. The top panel displays

11,176 measurements from the world ocean collected by C.S. Yentsch, 1956–86. The bottom panel displays 1,047

surface measurements from the northwest Atlantic continental shelf, Marine Resources Monitoring, Assessment,

and Prediction (MARMAP), 1978–82. (Campbell and O’Reilly 1988)

4
0.0

0.4

0.8

1.2

0.010 0.10 1.0 10 100

Mean = Median = Mode = 0

Standard Deviation = 0.4

Normal (Gaussian)

Distribution of log(X)

0.0

0.2

0.4

0.6

0.8

0 2 4 6 8 10

Mode = 0.4

Median = 1.0

Mean = 1.5

Lognormal Distribution of X

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 2. The lognormal distribution: The top panel displays a histogram of log(X), where log(X) is normally

distributed with mean 0 and standard deviation 0.4. The bottom panel shows the corresponding histogram of

the lognormal variable, X.

5
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

“sampling domain” refers to a specific bin and averaging

period; X is the level-2 variable, and X its level-3 equivalent. The question is: what is the best method for estimating X given a sample of n measurements (pixels):

X1,...,Xn?

In the case of a lognormal distribution, the sample

mean (or arithmetic average):

Xavg = 1

n

!n

i=1

Xi (2)

tends to underestimate the true population mean when

sample sizes are small (Baker and Gibson 1987). The

higher the variance of the underlying distribution, the more

this is true. The reason for this is that small samples tend

to miss high values which occur much less frequently than

low values. However, the high values have a significant

influence on the mean of the distribution. For example,

much of the biological production in the ocean occurs in

localized areas such as upwelling zones, and in transient

blooms of relatively short duration. A sample that misses

these areas and blooms would significantly underestimate

global or regional production.

Sample sizes involved in binning GAC data will be

small. Since the GAC data have a 4 km spacing between

pixels, at most 9 pixels from a single orbital pass can fall

into an 9 × 9 km bin. The average sample size will be

closer to four in data sets derived from a single orbital pass.

Although sample sizes will increase with longer averaging

periods, the variance will also increase. Thus, there was

concern that small sample sizes and large variances might

make the arithmetic average a poor estimator for level-3

means.

The practice of transforming data first, computing the

mean, mx, of log-transformed data

mx = 1

n

!n

i=1

ln"Xi

# (3)

and then estimating the mean of X as

Xgeom = emx (4)

gives the geometric mean. In the case of a lognormal variable, the geometric mean is the median of the distribution.

For any distribution that is positively skewed, the geometric mean will underestimate the population mean.

Studies have shown that the maximum likelihood estimator for a lognormal mean

Xmle = e

"

mx+ 1

2 s2

x

#

(5)

performs better than either of the other two when variances

are large and sample sizes small (Baker and Gibson 1987).

In (5), mx is the sample mean of ln(X), given by (3), and

s2

x is the sample variance given by

s2

x = 1

n

!n

i=1

$

ln"Xi

#

− mx

%2

. (6)

Note that this is not the more commonly used unbiased estimator which uses a divisor of n−1 instead of n. However,

this is the maximum likelihood estimator for the variance

of a normal random variable. In order for (5) to be the

maximum likelihood estimator for X, mx and s2

x must be

maximum likelihood estimators for the mean and variance

of ln(X) (Crow and Shimizu 1988).

In the statistical study presented in Section 3, the three

estimators, Xavg, Xgeom, and Xmle, were compared using

CZCS data and a time series of moored fluorometer data

(Medeiros and Wirick 1992). Results obtained for both

time and space averages were:

1. The sample mean, Xavg (2), and the maximum

likelihood estimator, Xmle (5), gave equivalent

results.

2. The geometric mean or median, Xgeom (4), was

systematically less than the other two.

The same results were obtained for other standard CZCS

variables: K490 and normalized water-leaving radiances

LWN (λi). Thus, based on their performance as estimators

of the mean, Xavg and Xmle were regarded as acceptable

estimators for the true population mean, X.

2.2.2 Estimating Level-4 Variables

It is not possible to prescribe a general method for estimating level-4 variables. The appropriate method will

depend on the nature of the relationship involved, i.e.,

whether it is linear or nonlinear, and the form it takes.

Let Y = f(X) be a relationship that defines the variable Y as a function of the level-2 variable X, and let Y be

the level-3 equivalent of Y . That is, Y represents the true

mean of Y within a sampling domain. In general, X may

be a vector of level-2 variables, i.e., Y may be a function

of more than one level-2 variable.

The problem that motivates this issue is that Y is not,

in general, equal to f

"

X#. Substitution of the mean of X

into the function is only legitimate for linear functions. In

general, the mean of a function of several variables is not

equal to the function of the means.

For any general function, the only way to obtain an

accurate estimate of the true mean, Y , would be to compute Yi = f(Xi) at each pixel in the level-2 data, and then

determine its average using either the arithmetic average,

Y avg, or the maximum likelihood estimate, Y mle. In this

case, the function Y = f(X) would be a level-3 variable

computed by averaging over pixels in the level-2 data. An

example is ICK (1) which will be computed in this way.

6
J.W. Campbell, J.M. Blaisdell, and M. Darzi

It is not possible or practical to anticipate the many

functions or mathematical relationships that may be applied to SeaWiFS data. Thus, there needed to be guidelines and methods for using level-3 data to obtain accurate

estimates of the mean of level-4 variables.

The decision was made to use the maximum likelihood estimation (MLE) method instead of the more common arithmetic average (AVG) method because the MLE

method provides a way to estimate the mean (and other

statistics) for a large class of level-4 variables of the form

Y = AXB (7)

where A and B are constants, and X is a single variable,

i.e., not a vector.

For variables in this class, ln(Y ) is linearly related to

ln(X)

ln(Y ) = ln(A) + B ln(X). (8)

Therefore, the mean and variance of ln(Y ) can be estimated as

my = ln(A) + Bmx (9)

and

s2

y = B2s2x (10)

where mx and s2

x are the mean and variance of ln(X) derived from the level-3 statistics saved for X.

According to the MLE method, the mean of Y is then

given by

Y mle = e

!

my+ 1

2 s2

y

"

. (11)

It should be noted that if (5) proves to be an accurate estimator for the mean of X, then (11) will be an accurate estimator for the mean of Y . There is no loss of accuracy since

(8)–(10) are exact relationships (not approximations).

The procedures for estimating the variance and other

statistics of level-3 and level-4 variables are described in

more detail in Sections 2.3 and 2.4 and in Appendix C. The

equations used are based on MLE methods for estimating

parameters of a lognormal distribution, and hence, they

are referred to as MLE estimators. As will be shown, the

MLE estimator is a robust estimator for the mean. That

is, it generally performs well even when the underlying distribution is not lognormal. Indeed, the MLE method was

not selected on the basis of an assumed lognormal distribution, but because it performed well compared with the

arithmetic average (AVG estimator), and because it provided a method for estimating the mean of level-4 variables

of the form given by (7).

An example of such a function is the euphotic depth,

which is commonly defined as the 1% light-penetration

depth (Kirk 1983). Using the level-2 variable K490 and

applying Beer’s Law, this depth may be defined as

Ze ≡ −ln(0.01)

K490

(12)

which represents the 1% light-penetration depth at λ =

490 nm. If the mean of K490 based on level-3 data is used

to estimate the mean euphotic depth, this will yield a biased estimate of the mean euphotic depth. However, the

MLE method allows for an accurate estimate of the mean

Ze based on the saved statistics of ln!

K490".

The equations proposed by Morel and Berthon (1989)

for deriving integral euphotic chlorophyll, #Chl$tot, from

satellite-derived chlorophyll (or pigment) also take the form

of (7). Several algorithms for estimating integral productivity (Smith et al. 1982, Platt 1986, and Morel and Berthon 1989) involve the product of #Chl$tot and photosynthetically available radiation (PAR) at the surface, PAR(0).

The mean of this product can be derived as the product of

the means of #Chl$tot and PAR(0) since the two variables

are uncorrelated. Thus, these algorithms may be applied

to level-3 data using the saved statistics of standard level-2

variables.

2.2.3 Statistics Saved for Each Domain

Another issue that was raised concerned the choice of

statistics to save for each sampling domain. Given that

Xmle (5) is to be used for estimating the mean of the

level-2 data in each domain, the statistics saved must include the sum and sum of squares of the natural logarithm

of each variable. In addition, counts of the number of pixels

contributing to the sums and similar ancillary information

should also be saved.

Beyond this, further questions regarding what statistics to save are motivated by the concern expressed earlier

as to how level-4 variables will be estimated. Two alternatives exist: either a) sufficient information is provided

in the level-3 data to allow estimation of these variables

using saved statistics of other variables or b) the variables

should be computed at each pixel of level-2 data and their

statistics saved as part of the level-3 data set. The latter

is more costly from the standpoint of the storage required

to add additional level-3 variables. As stated earlier, the

MLE method permits the former choice for variables of the

form given in (7).

There are other level-4 variables which cannot be calculated using only the saved statistics of the standard level-2

variables. Any variable that is a function of two or more

level-2 variables would require additional information on

the covariances between level-2 variables. An example of

this is the variable ICK (1) which appears in several primary productivity algorithms (Balch et al. 1992, Platt and

Sathyendranath 1988, Eppley et al. 1985, Smith and Baker

1978, and Bannister 1974). To apply the MLE method,

one must estimate the mean and variance of the natural

logarithm of ICK

ln!ICK

" = ln!

CHL" − ln!K490". (13)

The mean of ln!ICK

" is simply the difference between

the means of ln(CHL) and ln!K490"

, but the variance of

7
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

ln!ICK

"

:

var#ln!ICK

"$

= var#ln!CHL"$+ var#ln!K490"$

− 2 cov#ln!CHL", ln!K490"$ (14)

involves the covariance, denoted by “cov” in (14), between

ln(CHL) and ln!K490"

, as well as their variances. The

CZCS algorithms for CHL and K490 in Section 3 resulted

in a nonlinear relationship between ln(CHL) and ln!K490"

.

Thus, their covariance varied from sample to sample. For

this reason, it was decided to compute the variable ICK

(1) at each pixel in the level-2 data and save statistics of

ln!ICK

"

as part of the level-3 data.

2.2.4 Weighting of Temporal Statistics

After each level-2 scene is generated, valid level-2 data

from individual pixels will be binned. Sums and sums

of squares accumulated at this stage are called spatial

statistics, i.e., no temporal averaging is involved since data

from the same scene are regarded as simultaneous. Spatial

statistics from the same day will be combined into daily

products, from the same 8-day period into weekly products, and so forth. The daily, weekly, monthly, and longerterm products will become the level-3 data, and the spatial

statistics pertaining to individual scenes will be discarded.

On a given day, there may be two sets of spatial statistics for the same bin. Two sets might occur within the

same orbit on different tilt segments, i.e., before and after a change in the sensor’s tilt, or from different orbits

in high-latitude areas where swaths overlap. In the case

of two sets from the same orbit, only one set will be used.

The set having the better sun-target viewing geometry will

be selected. However, two sets of spatial statistics from

different orbits will receive the same treatment as spatial

statistics from different days. The same algorithms, called

temporal binning algorithms, will be used to combine data

separated by time gaps regardless of the size of the time

gap.

Let N be the number of sets of spatial statistics (orbits) contributing to a temporal mean; let ti be the time at

which the ith set was acquired; and let ni be the number

of pixels contributing to the ith set, where i = 1,...,N. In

considering the temporal binning algorithms, a major concern was the fact that the times are unevenly distributed,

and that the sample size (hence precision) varies from one

time to another. Samples sizes will vary between 1 and

9, depending on where the bin lies relative to the ground

track. Time gaps occur because of clouds, sunglint, and

other factors.

The methods used to compensate for unevenly distributed data generally involve a scheme for weighting data.

The alternative is to use simple composite statistics (unweighted data), which was the method used to create

level-3 CZCS data such as the North Atlantic monthly

composites (Feldman et al. 1989 and Esaias et al. 1986).

These monthly composites have served as useful products

for a number of scientific investigations (Campbell and

Aarup 1992, Yentsch 1990, and Lewis et al. 1988), but

some of the spatial patchiness in these data sets is an artifact of the uneven temporal distribution of data.

Chelton and Schlax (1991) have made a strong case for

the superiority of optimal interpolation methods as compared to simple composite averages for deriving temporal

means of irregularly spaced data. Such methods, known

as kriging in the geostatistics literature (Journal 1989), require the use of correlation functions which must be determined a priori. When applied to satellite data, the methods could require both temporal and spatial correlation

functions.

The advantage of optimal interpolation methods is that

they allow estimates to be based on data that lie outside the domain (bin and time interval) being estimated.

The disadvantage is their computational complexity. Data

must be deseasonalized before applying the optimal interpolation method. That is, seasonal trends must be estimated and subtracted from the data. Therefore, at least

a year of data must be collected before optimal interpolation methods can be applied. This is not compatible with

the plan to generate level-3 data products along with the

level-2 data processing.

It was decided not to apply optimal interpolation methods in the level-3 binning process. However, the binned

statistics will be useful in applying optimal interpolation

methods during post-processing. As an example, daily

composite statistics might be used in deriving weekly and

monthly means using optimal interpolation methods.

The question was, therefore, whether to use simple

composite statistics (all data within a given domain are

given equal weight) or to develop a weighting scheme that

could be implemented easily at the time the level-2 data

are processed. In general, a decision to use weighted versus unweighted statistics should depend on the distribution

of the data vis-a-vis any trends that might exist. Simple

unweighted statistics are recommended in the case where

there is no trend (either spatial or temporal), or where the

trend is impractical to estimate. The latter is the case for

the spatial statistics. These will be unweighted sums and

sums of squares of the pixels falling within each bin because it is impractical to estimate spatial trends for each

bin.

In the case of weekly and monthly statistics, there may

be significant trends that call for weighted sums. If simple composite (unweighted) statistics are used, each of the

N sets of spatial statistics will, in effect, be weighted by

its sample size, ni. Thus, for example, a data set having

ni = 9 would be much more heavily weighted than one

with ni = 1. Trends may be lost in this process. Alternatively, a temporal mean might be calculated as the average

of N spatial means, regardless of the number of pixels contributing to the spatial means. However, this would give

8
J.W. Campbell, J.M. Blaisdell, and M. Darzi

too much weight to a data set with ni = 1 compared to one

with ni = 9. This concern reflects the belief that precision

is a function of sample size.

As a compromise to these two alternative approaches,

it was decided to apply a weight of √ni to the spatial mean

at time ti, where ni is the number of pixels falling in the

bin at time ti. This is effected by applying the weight

wi = 1

√ni

(15)

to the sums and sums of squares associated with the spatial

statistics for time ti. Details of the weighting scheme are

given in Appendix B.

2.3 Protocols for Level-3 Statistics

The level-3 data products available for each day, week,

month, and year of the SeaWiFS mission will allow users to

compute the mean, standard deviation, median, and mode

of each level-3 variable in each bin. The level-3 variables

consist of level-2 variables, and in addition, the variable

ICK (1).

For each level-3 variable X, the level-3 data consists

of a pair of sums for each bin

S1 = !N

i=1

1

√ni

!ni

j=1

ln"Xij # (16)

and

S2 = !N

i=1

1

√ni

!ni

j=1

$

ln"Xij #%2(17)

where Xij is the jth observation of X at time ti. Each

observation corresponds to a pixel in the level-2 data. The

number ni is the number of pixels at time ti containing

valid level-2 data.

In addition, the following statistics are saved for each

bin:

b bin index number (range: 1,..., 5, 940, 422),

N total number of orbits contributing data,

n total number of pixels contributing data, and

W sum of weights.

For the latter two quantities, their formulation is as follows:

n = !N

i=1

ni (18)

and

W = !N

i=1

√ni. (19)

In addition to the above variables, there will be a 16-bit

time distribution variable T whose bits indicate whether

data were available (bit = 1) or absent (bit = 0) in time intervals (days, two-day intervals, or months) covered by the

averaging period. That is, each bit of the 16-bit number

represents a time interval within the averaging period, and

if a bit is set to 1, it indicates data were available during

that interval.

2.3.1 The Mean and Variance of ln(X)

To estimate statistics for the variable X, the first step

is to calculate the mean and variance of ln(X). These are

given by

mx = S1

W (20)

and

s2

x = S2

W − m2

x. (21)

2.3.2 The Mean and Other Statistics of X

The mean of X is estimated by

Xmle = emx+ 1

2 s2

x (22)

and the standard deviation by

SDX = Xmle&

es2

x − 1. (23)

The median or geometric mean may be estimated by

Xmed = emx (24)

and the mode (most frequent value) by

Xmod = emx−s2

x . (25)

The above equations are based on the MLE method

which was demonstrated to be valid for means of CZCS

data and moored fluorometer data. Equations (22)–(25)

are based on an assumed lognormal distribution of X within the sampling domain. For a discussion of the underlying assumptions and robustness of the estimators see Section 3.3.

2.4 Protocols for Level-4 Statistics

As defined earlier, a variable, Y = f(X), which is a

function of one or more level-3 variables, is called a level-4

variable. Here, guidelines are given for computing statistics of several classes of level-4 variables. It is not possible

to specify protocols for all level-4 variables, in general, because the procedures depend on the function f(X).

2.4.1 Computing Statistics for Y=A+BX

If Y is a linear function of X, then the mean of Y is

given by the same linear function of the mean of X

Y mle = A + BXmle. (26)

9
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

The same is true for the median and mode of Y . The

standard deviation of Y is scaled by the factor B

SDY = B

!

SDX

"

. (27)

2.4.2 Computing Statistics for Y=AXB

The MLE method was chosen because it provides a

robust method for estimating the mean of level-4 variables

of this form. To use the MLE method, one must first

estimate the mean and variance of ln(Y ). These statistics,

my and s2

y, can then be substituted into (22)–(25), in place

of mx and s2

x, to estimate the mean, standard deviation,

median, and mode of Y .

Let Y = f(X) be a function of this form where X is

a single level-3 variable. Its natural logarithm is a linear function of ln(X) (8). If mx and s2

x are statistics of

ln(X) derived from the level-3 data sets by (20) and (21),

respectively, then the mean and variance of ln(Y ) are, respectively:

my = ln(A) + Bmx (28)

and

s2

y = B2s2x. (29)

Statistics of Y = f(X) can be derived by substituting

my = mx and s2

y = s2x into (22)–(25).

2.4.3 Statistics for Other Functions

So far the only considerations were functions of a single variable X. In general, if Y is a function of two or

more level-3 variables, knowledge of the covariances between the level-3 variables is required to derive statistics

for Y . It was initially recommended that a covariance matrix be saved as part of the level-3 statistics, but the storage costs were considered too high. Subsequently, it was

decided to save statistics of ICK because this function appears frequently in primary productivity algorithms.

Another situation involving a function of several level-2

variables occurs when a regional bio-optical algorithm is

applied to derive better estimates of the CZCS-like pigment concentration. For example, suppose the standard

(global) CZCS-like pigment algorithm is

PIG = Ag

#

LWN(λi)

LWN(λj)

$Bg

(30)

where LWN (λi) and LWN (λj ) are the normalized waterleaving radiance in bands i and j, and the wish is to compute pigment according to an alternative algorithm

PIGr = Ar

#

LWN (λi)

LWN (λj )

$Br

(31)

using regionally-derived parameters, Ar and Br. In this

situation, it is possible to use the saved level-3 statistics

for PIG to estimate statistics for PIGr. Substituting the

means of LWN (λi) and LWN (λj ) into (31) is not recommended.

The recommended procedure is, first, to estimate the

mean and variance of ln!PIG"according to (20) and (21).

These statistics can be denoted by mg and s2

g, respectively.

The mean of ln!PIGr

"

is then given by

mr = ln!

Ar

" +

Br

Bg

%

mg − ln!

Ag

"&

(32)

and the variance of ln!PIGr

"

is

s2

r = B2

r s2g

B2

g

. (33)

These statistics can then be substituted into (22)–(25),

replacing mr = mx, and s2

r = s2x, to obtain the statistics

for PIGr.

This flexibility is the primary reason that the MLE

method was chosen over the more commonly used estimation methods, e.g., arithmetic averages, for estimating

spatial and temporal means. As shown in Section 3, the

MLE estimator for the mean proved to be equivalent to

the arithmetic average for spatial averages of CZCS data,

and, in most situations, for temporal averages of moored

fluorometer data. The statistical study detailed in Section 3 provides empirical evidence to support the use of

the MLE method, as well as theoretical results which explain its success and, in some instances, failure for certain

data sets.

3. EMPIRICAL BASIS

In 1992–93, a study was conducted to address statistical questions related to level-3 binning algorithms for

SeaWiFS data. The questions addressed and recommendations derived from this study have been presented in

Section 2 of this report. Here, the actual results of this

study are presented. Results pertaining to spatial binning

algorithms are presented in Section 3.1, followed by results

pertaining to temporal binning algorithms in Section 3.2.

Following the presentation of results, Section 3.3 contains

a discussion of the major conclusions. Questions concerning the equivalence of the MLE and AVG methods are

addressed in this section, and specific situations are described when the two methods would and would not be

equivalent.

3.1 Spatial Statistics

The first step in creating level-3 data involves averaging

data from a single orbital pass. This is considered the spatial binning step, because the data involved are regarded

as simultaneous.

Three questions related to spatial binning were addressed:

10
J.W. Campbell, J.M. Blaisdell, and M. Darzi

1. How should level-2 data be averaged to provide

the best estimate of their mean?

2. How should level-4 means be estimated?

3. What statistics should be saved?

These are the first three questions presented and discussed

in Sections 2.2.1–2.2.3.

3.1.1 Methods

Full-resolution CZCS data were used to address the

aforementioned questions. The procedure was to use the

full-resolution data to define the true mean of each variable

within 9 × 9 km2 bins and to compare other estimates of

the mean against the true mean.

Seven scenes were selected as representative of the full

range of variability in CZCS data. Details of these scenes

are given in Table 1. The level-1 data were processed according to standard algorithms using the DSP ANLY2DBL

code [Rosenstiel School of Marine and Atmospheric Science (RSMAS) 1990]. (The version of ANLY2DBL.EXE used

in processing CZCS data was created 19 April 1990, and

modified 18 September 1991). The resulting level-2 variables involved in this study were:

LWN (λi) normalized water-leaving radiances in bands

i = 1–3,

CHL pigment concentration (chlorophyll), and

K490 diffuse attenuation coefficient at λ = 490 nm.

The normalized water-leaving radiances are radiances corrected for variations in solar zenith angle across the scan.

All radiances are corrected to correspond to a solar zenith

angle of zero. Details of the algorithms used may be found

in Gordon et al. (1988).

The algorithm for K490 was

K490 = 0.022 + 0.088!

LW

"

λ1

#

LW

"

λ3

#

$−1.491

(34)

where LW

"

λi

# is the non-normalized water-leaving radiance in band i. The quantity CHL was derived using a

bifurcated algorithm that involved two ratio formulas:

CHL13 = 1.130!

LW

"

λ1

#

LW

"

λ3

#

$−1.705

(35)

and

CHL23 = 3.327!

LW

"

λ2

#

LW

"

λ3

#

$−2.44

. (36)

According to this algorithm, CHL was equal to CHL13 except when both formula values exceeded 1.5 mg m−3, in

which case, CHL was equal to CHL23. The CHL13 ratio was employed in all of the scenes analyzed, whereas

the CHL23 ratio was employed in only three of the seven

scenes.

After the scenes were processed to standard level-2

data, pixels in each scene were sorted into 9 × 9 km2 bins

oriented in rows perpendicular to the ground track of the

satellite. Based on an instantaneous field-of-view (IFOV)

angle of 0.865×10−3 radians (0.496◦) and a sensor altitude

of 955 km (and ignoring tilt), the spatial resolution of pixels at nadir is 0.825 km. The maximum number of pixels

that fit into a 9 × 9 km2 bin was 121 (11 × 11). This occurred only within ±300 pixels of nadir where pixels have

spatial resolutions ≤ 0.9 km.

3.1.1.1 Estimators of the Mean

Only cloud-free bins containing 121 pixels were used for

the analysis. All estimators were evaluated using both fullresolution (LAC) data and 4 km resolution (GAC) data.

The latter were obtained by subsampling every fifth pixel

on every fifth line (since 5 × 0.825 ≈ 4 km). Thus, LAC

estimators were based on 121 level-2 observations, whereas

for GAC data, the number of observatons (pixels falling in

these bins) ranged from 4–9.

The estimators compared were:

AVG arithmetic average (2) based on LAC data,

AVG4 arithmetic average based on GAC data,

MLE maximum likelihood estimator (5) based on

LAC data,

MLE4 maximum likelihood estimator based on GAC

data,

MED geometric mean or median estimator (4)

based on LAC data, and

MED4 geometric mean or median estimator based

on GAC data.

For each bin, the AVG estimator based on LAC data

(n = 121) is given in (2) and was considered the true mean.

In this equation, Xi is the ith observation or realization of

the variable X %equal to LWN (λ1), LWN (λ2), LWN (λ3),

CHL, or K490&, and n is the number of observations (pixels) falling in a bin. The true mean was computed for

each variable and each bin having n = 121 valid observations. The other estimators of the mean were compared

with Xavg to determine how well they performed.

3.1.1.2 Standard Level -2 Variables

Let X ≡ %LWN (λ1),LWN (λ2),LWN (λ3),CHL,K490& refer to the vector of standard variables, and let Y = f(X)

be any function that is derived from one or more of the

standard variables.

The arithmetic mean of the function based on LAC

data (n=121)

Y avg = 1

n

'n

i=1

Yi (37)

11
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Table 1. CZCS scenes used for the analysis of spatial statistics. The scenes are listed in increasing order

of mean pigment (see Fig. 1). The number of lines listed were for the whole scene, and the number of bins

given is the number of 9 × 9 km3 bins containing data. Time is given in Greenwich Mean Time (GMT) in the

(left-to-right) order of hour, minutes, and seconds. (Note: In Tables 2 and 3, the number of bins listed is the

number of bins containing n = 121 pixels. Only these cloud-free bins were used to define true means in the

images.)

ID Orbit Date Time Tilt Location Lines Bins

1 1,200 19 Jan 79 1:56:27 20◦ Northwestern Pacific 1,023 3,186

2 218 9 Nov 78 0:52:23 0 Northwestern Pacific 1,023 2,964

3 1,029 6 Jan 79 16:38:13 −14 South Atlantic 1,023 3,087

4 1,016 5 Jan 79 18:33:31 6 Eastern Tropical Pacific 2,376 8,266

5 1,452 6 Feb 79 7:19:17 8 Indian Ocean 1,584 7,475

6 971 2 Jan 79 12:31:21 −2 Northwest of Africa 1,023 1,040

7 1,386 1 Feb 79 12:45:19 20 Southwest of Africa 1,584 4,020

was considered its true mean, where Yi = f

!

Xi

" is the

function calculated at pixel i. This defined the AVG estimator for Y . Similarly, the AVG4, MLE and MLE4 estimators for the mean of Y were defined by substituting Yi

for Xi in the appropriate equations. In addition to these

estimators, the FNC (function) estimator was defined as

Y fnc = f

!

Xavg" (38)

where Xavg is the arithmetic average of X. This would be

the result of calculating the function using level-3 means.

It was called FNC when Xavg was the AVG estimator, and

FNC4 when Xavg was the AVG4 estimator.

Functions that were investigated were as follows:

ICK integral pigment (1) within the upper optical

depth,

Ze 1% light depth, and

YA,B pigment algorithm A

!

LWN (λ1)/LWN (λ3)

"B,

where A = 1 and B = −1, −2, and −3.

3.1.1.3 Relative Errors

For each bin, the relative error in an estimate of the

mean, Xest, was defined as a percentage of the true mean

Xavg

ERROR = Xest − Xavg

Xavg

× 100% (39)

where Xest was the estimate based on the MLE, MED,

AVG4, MLE4, or MED4 estimator. Similarly, relative errors in estimates of the mean of a function, Y est, were

defined as a percentage of Y avg, where Y est was the estimate based on the MLE, FNC, AVG4, MLE4, or FNC4

estimator.

3.1.2 Results

In Table 1, the scenes are listed in order of increasing

mean pigment. In presenting results, scenes will be identified by the number (order) found in column 1 of this table.

3.1.2.1 Pigment Distributions

The pigment means and coefficients of variation (CV)

for the seven scenes are compared in Fig. 3. Histograms of

log(CHL) are shown in Fig. 4, where the abscissa is the 8-

bit image value V , which is related to the logarithm (base

10) of pigment as

log!CHL" = −1.4+0.012 V. (40)

The distributions of log(CHL) shown in Fig. 4 appear to

be either single normal distributions, e.g., scene 1, or mixtures of normal distributions, e.g., scene 3. Thus, CHL is

approximately lognormally distributed within each scene

or within portions of each scene.

In scenes 4, 6, and 7, the bifurcated CHL algorithm

resulted in a discontinuity at CHL = 1.5 mg m−3 (V =

132). Values to the left of V = 132 have been calculated

according to CHL13 (35), whereas values to the right were

calculated according to CHL23 (36). This is an artifact of

the CZCS pigment algorithm, which will be avoided when

defining the SeaWiFS CHL algorithm. In scenes 6 and

7, CHL was recalculated using the CHL13 algorithm for

all pixels. The resulting CHL distributions are shown in

Fig. 5.

3.1.2.2 Comparison of Estimators

Representative results for estimators of CHL are shown

in Figs. 6 and 7. Each point in these scatter plots corresponds to a bin in scene 4, the scene with the highest overall variance. The scales are log-log. In Fig. 6, the MLE,

MED, MLE4, and MED4 estimates are plotted against the

AVG estimate. The patterns shown here are typical of

those observed in all the scenes analyzed. In all scenes,

the MLE estimator was nearly identical to the AVG estimator, whereas the MED estimator underestimated AVG.

There was no discernible difference between the MLE4 versus AVG and MED4 versus AVG plots. Both contained

substantially more scatter than the plots involving MLE

and MED estimates.

12
-1.6

-1.2

-0.8

-0.4

0.0

0.4

1234567

0.058 0.061

0.160 0.188

0.335

0.858

1.530

log(CHL)

0

100

200

300

400

1234567

76

30

55

358

39

151

180

CV (%)

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 3. The mean pigment (upper panel) and coefficient of variation (lower panel) for the seven CZCS scenes

used in this analysis. The scenes are ordered from lowest to highest mean pigment. The numbers appearing

above each bar are the mean pigment (mg m−3) and coefficient of variation (standard deviation expressed as a

percentage of the mean).

13
0

5000

10000

15000

20000

25000

30000

0 50 100 150 200

Scene #1.

Northwest Pacific Ocean

January 19, 1979

0

5000

10000

15000

20000

25000

0 50 100 150 200

Scene #2.

Northwest Pacific Ocean

November 7, 1978

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 4. Pigment histograms of seven CZCS scenes used in this analysis. The abscissa is the image value V

which is linearly related to the logarithm of pigment: log(CHL) = −1.4 + 0.012(V).

14
0

5000

10000

15000

20000

25000

0 50 100 150 200

Scene #4.

Eastern Tropical Pacific

January 5, 1979

0

2000

4000

6000

8000

10000

12000

14000

16000

0 50 100 150 200

Scene #3.

South Atlantic

January 6, 1979

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 4. (cont.) Pigment histograms of seven CZCS scenes used in this analysis.

15
0

500

1000

1500

2000

2500

3000

0 50 100 150 200

Scene #6.

Northwest of Africa

January 2, 1979

0

5000

10000

15000

20000

25000

30000

0 50 100 150 200

Scene #5.

Indian Ocean

February 6, 1979

0

5000

10000

15000

20000

25000

30000

0 50 100 150 200

Scene #5.

Indian Ocean

February 6, 1979

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 4. (cont.) Pigment histograms of seven CZCS scenes used in this analysis.

16
0

1000

2000

3000

4000

5000

6000

0 50 100 150 200

Scene #7.

African Coastal Upwelling

February 1, 1979

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 4. (cont.) Pigment histograms of seven CZCS scenes used in this analysis.

In Fig. 7, the AVG4 estimator is compared with the

AVG and MLE4 estimators. In the AVG4 versus AVG plot,

the scatter is strictly the result of sample size differences;

whereas in the AVG4 versus MLE4 plot, the scatter is the

result of differences between the estimators. It is clear from

these comparisons that the errors associated with GAC

estimators were predominantly the result of their reduced

sample size. When two GAC estimators were compared,

e.g., AVG4 versus MLE4 in Fig. 7, the two agreed as well

as the corresponding LAC estimators.

Color Plates 1 and 2 show level-3 mean CHL images for

the seven scenes. That is, each pixel in these images is a bin

in the level-3 data. Plate 1 compares the AVG and MLE

estimators, and Plate 2 compares the AVG4 and MLE4

estimators. Difference images are shown in Plate 3. Differences between the MLE and AVG estimators seemed to

be spatially organized with the largest differences located

along fronts and coastlines. In contrast, there were no obvious spatial patterns in the differences between MLE4 and

AVG4 estimators.

The combined histograms of relative errors (39) in CHL

estimators from all seven scenes are shown in Figs. 8 and

9, and summarized in Table 2a. In all but a few cases,

the MLE estimator differed from the AVG estimator by

less than 1%; whereas, the MED estimator consistently

underestimated the mean CHL. Its bias or average error

was −2.1%, and 95th percentile range was −11% to −1%.

All three GAC estimators had a tendency to underestimate the true mean CHL. Errors associated with the

AVG4 estimator are strictly the result of reducing sample

sizes from n = 121 in the AVG estimator to n ≤ 9 in

the AVG4 estimator. The error histograms for AVG4 and

MLE4 are remarkably similar. Their biases were −0.76%

and −0.75%, respectively, and their 95th percentile range

was −19% to +18%. The MED4 tended to underestimate

the true mean as did the other GAC estimators, but the

MED4 was a poorer estimator indicated by its larger† negative bias (−2.60%).

In the LAC error histograms (Fig. 8), true differences in

the performance of the estimators may be seen; whereas,

in the GAC histograms (Fig. 9), errors associated with

reduced sample size are added to errors or differences between estimators. Differences between GAC estimators

DIFF1 = MLE4 − AVG4

AVG4 × 100% (41)

and

DIFF2 = MED4 − AVG4

AVG4 × 100% (42)

were examined. Here, a distinction is made between errors

(39) which are relative to the true mean (AVG) and differences, (41) and (42), which are relative to AVG4, another

estimate of the mean.

Histograms of DIFF1 and DIFF2 are shown in Fig. 10.

These results for GAC estimators are very similar to the

patterns seen when comparing LAC estimators (compare

Fig. 10 with Fig. 8). The AVG4 and MLE4 estimators

agree, as well as the AVG and MLE estimators; differences

between the two methods of estimating the mean CHL

are negligible. Likewise, differences between the MED4

and AVG4 estimators followed the same pattern as differences between the MED and AVG estimators. In both

cases, the geometric mean underestimated the arithmetic

average. The large errors in AVG4, MLE4, and MED4

(Fig. 9) were clearly dominated by the sample size effect.

The patterns seen in Figs. 8–10 for CHL estimators

are similar to those that are obtained for other variables.

GAC error histograms for the other variables (comparable to Fig. 9) are shown in Figs. 11–14, and summaries

† In referring to biases, the terms larger and smaller refer to

the magnitude or absolute value of the bias.

17
0

500

1000

1500

2000

2500

3000

0 50 100 150 200

Scene #6.

Northwest of Africa

January 2, 1979

0

1000

2000

3000

4000

5000

6000

0 50 100 150 200

Scene #7.

African Coastal Upwelling

February 1, 1979

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 5. CHL histograms for scenes 6 and 7 derived using CHL13 algorithm only.

18
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 6. In these scatter plots, four estimators of the mean are compared with the true mean (AVG) defined

as the arithmetic average of all pixels in a bin (sample size = 121). The level-2 data used were CZCS-derived

pigment values from scene 4 (see Table 1). Like the AVG estimator, the MLE and MED estimators are based

on full-resolution (LAC) data, whereas the MLE4 and MED4 estimators are based on 4 km subsampled (GAC)

data. The scales on each plot are log-log where the range is from 0.04 (V=0) to 45 (V=255), where V is the

8-bit image value [see (40)].

Fig. 7. In these scatter plots, the ordinate (AVG4) is the arithmetic average based on 4 km subsampled (GAC)

data for the same scene as in Fig. 6. The plot on the left compares this estimator with the average based on

full-resolution (LAC) data. The scatter in this plot is strictly the result of sample size differences. The AVG4

has less precision since its sample size is reduced from n = 121 (LAC) to n ≤ 9 (GAC). The plot on the right

compares the AVG4 and MLE4 estimators. Like the MLE and AVG estimators (Fig. 6), the MLE4 and AVG4

are practically identical.

19
0

5000

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4

95%

2% 1% 1%

10000

15000

20000

25000

0

2000

4000

6000

8000

10000

12000

14000

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4

5%

57%

20%

6%

2%3% 1%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 8. Histograms of CHL estimation errors based on 21,290 bins analyzed and full-resolution (LAC) data.

For each bin, the error is defined as the difference between the estimator and the arithmetic average (AVG)

of all data in the bin expressed as a percentage of AVG. The top histogram shows the error calculated as

(MLE − AVG)/ AVG) (%). The bottom histogram shows the error calculated as (MED − AVG)/ AVG (%).

20
0

200

400

600

800

1000

1200

1400

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0 .76%

0

200

400

600

800

1000

1200

1400

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0 .75%

0

200

400

600

800

1000

1200

1400

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -2 .6%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 9. Histograms of CHL estimation errors based on 21,290 bins analyzed and 4 km subsampled (GAC)

data. The top histogram shows the error calculated by (AVG4 − AVG)/ AVG (%). The middle histogram

shows the error calculated by (MLE4 − AVG)/ AVG (%). The bottom histogram shows the error calculated by

(MED4 − AVG)/ AVG (%).

21
0

5000

10000

15000

20000

25000

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4

96%

1% 2%

0

2000

4000

6000

8000

10000

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4

28%

42%

15%

6%

2% 3%

1% 1% 1%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 10. Histograms of DIFF1 and DIFF2 based on 21,290 bins analyzed. The top histogram was calculated

with (MLE4 − AVG4)/ AVG4 (%). The bottom histogram was calculated with (MED4 − AVG4)/ AVG4 (%).

22
0

1000

2000

3000

4000

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50

Bi as = - 0.31%

0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.31%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 11. Histograms of K490 estimation errors based on 20,373 bins analyzed and 4 km subsampled (GAC)

data. The top histogram was calculated using (AVG4 − AVG)/ AVG (%). The bottom histogram was calculated

for (MLE4 − AVG)/ AVG (%).

23
0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.59%

0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.99%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 11. (cont.) Histogram of K490 estimation errors based on 20,373 bins analyzed and 4 km subsampled

(GAC) data was calculated using (MED4 − AVG)/ AVG (%).

Fig. 12. Histogram of LWN (443) estimation errors based on 21,290 bins analyzed and 4 km subsampled (GAC)

data was calculated for (AVG4 − AVG)/ AVG (%).

24
0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.89%

0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -1.66%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 12. (cont.) Histograms of LWN (443) estimation errors based on 21,290 bins analyzed and 4 km subsampled

(GAC) data. The top histogram was calculated for (MLE4 − AVG)/ AVG (%). The bottom histogram was

calculated for (MED4 − AVG)/ AVG (%).

25
0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.94%

0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.92%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 13. Histograms of LWN (520) estimation errors based on 21,290 bins analyzed and 4 km subsampled (GAC)

data. The top histogram was calculated for (AVG4 − AVG)/ AVG (%). The bottom histogram was calculated

for (MLE4 − AVG)/ AVG (%).

26
0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -1.32%

0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -1.25%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 13. (cont.) Histogram of LWN (520) estimation errors based on 21,290 bins analyzed and 4 km subsampled

(GAC) data was calculated for (MED4 − AVG)/ AVG (%).

Fig. 14. Histogram of LWN (550) estimation errors based on 21,290 bins analyzed and 4 km subsampled (GAC)

data was calculated for (AVG4 − AVG)/ AVG (%).

27
0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -1.24%

0

1000

2000

3000

4000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -1.81%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 14. (cont.) Histograms of LWN (550) estimation errors based on 21,290 bins analyzed and 4 km subsampled

(GAC) data. The top histogram was calculated for (MLE4 − AVG)/ AVG (%). The bottom histogram was

calculated for (MED4 − AVG)/ AVG (%).

28
J.W. Campbell, J.M. Blaisdell, and M. Darzi

of relative errors are listed in Tables 2a–e for CHL, K490,

LWN (λ1), LWN (λ2), and LWN (λ3), respectively. The last

two columns on the right in this table give the 95th percentile range for the relative errors. All GAC estimators

had negative biases. AVG4 and MLE4 were nearly identical with average errors on the order of −1%; whereas,

MED4 had average errors of approximately −2%. The

CHL variable had the highest overall errors, with a 95th

percentile range generally around ±20%; whereas, the other variables had errors that were generally within

±10%.

The three scenes that used both CHL13 and CHL23 had

substantially higher errors than those of the other scenes.

These scenes also had the highest variance in CHL and

other variables. Although higher estimation errors would

be expected when sampling from distributions with higher

variance, there was the need to determine whether the

CHL errors were anomalously large due to the bifurcated

CHL algorithm. The higher variance in CHL might have

been an artifact resulting from the discontinuous nature of

the pigment distribution.

To determine whether this was true, the analysis was

repeated for scenes 6 and 7 using CHL13 to derive CHL

for all pixels (Fig. 5). The results were essentially the

same. These images still had large intrabin variances in

CHL (CHL13), and their error distributions (not shown)

were essentially unchanged.

Statistics pertaining to the ICK function estimators are

presented in Table 3, and error histograms for the MLE

and FNC estimators are shown in Fig. 15 and for the

AVG4, MLE4, and FNC4 estimators in Fig. 16. In contrast to the MED estimator, the FNC estimator tended to

overestimate the true mean. In the case of the FNC4 estimator, this tendency (positive bias) was apparently offset

by the underestimation tendency (negative bias) associated with small sample sizes. The result was that the bias

of the FNC4 estimator was nearly zero.

In Section 2.4, a protocol was presented for estimating

the mean of level-4 variables of the form Y = AXB based

on saved statistics of the level-2 variable X. The accuracy

of the prescribed protocol depends strictly on whether the

MLE estimator is a good approximation to the AVG or

true mean of these functions.

Results for the Ze and YA,B functions (not shown) established that the MLE estimator was essentially identical

to the AVG estimator. Root-mean-square (rms) errors for

the MLE4 and AVG4 estimators were within ±5% for Ze.

Errors for YA,B increased as B changed from −1 to −3,

with the highest rms errors being associated with scenes

6 and 7. MLE4 and AVG4 errors were within ±5% for

B = −1, within ±15% for B = −2, and ±30% for B = −3.

These ranges are consistent with the results for the CHL

algorithm where B = −1.7 (CHL13) and −2.4 (CHL23).

As in the case of the ICK function, the FNC4 estimator

was not significantly different from the MLE4 and AVG4

estimators. In all three cases, errors were dominated by

the effects of reduced sample size.

3.2 Temporal Statistics

After the spatial statistics are derived from data on a

single orbital pass, these statistics will be averaged over

time to produce temporal statistics. No further reduction

in spatial resolution takes place, but after being averaged

over time, temporal statistics will have reduced temporal

resolution.

Statistical questions regarding the use of weighted versus unweighted statistics have been discussed above. These

questions were not addressed in this study. This phase of

the study focused on questions concerning the performance

of the estimators studied in the earlier (spatial statistics)

phase of the study. Specifically, the questions addressed

were:

1. Would the MLE estimator continue to be equivalent to the AVG estimator as variance increases

due to temporal variability?

2. Would the MED and FNC estimators diverge

further from the AVG?

In other words, the goal of this phase of the study was

to determine whether the results obtained for spatial statistics would also pertain to temporal statistics. The greatest

differences between the MLE and AVG estimators occurred

in bins having the highest variance. Since temporal statistics, in general, will have increased variance due to temporal variability within bins, it was not known whether

the MLE and AVG estimators would remain equivalent.

Furthermore, it was predicted that the small but systematic errors in the MED and FNC estimators would increase

with increases in variance.

3.2.1 Methods

Ideally, several time series of CZCS images from different geographic regions should be analyzed to address these

questions. However, this approach was not considered feasible. Since CZCS was operated only 10% of the time, its

sampling frequency for any bin was much lower than that

expected for SeaWiFS, which will operate continuously.

To investigate how phytoplankton pigment distributions vary over time at a fixed location, and to answer

the above questions, the Shelf Edge Exchange Program II

(SEEP II) moored fluorometer data (Medeiros and Wirick

1992) was analyzed. These data consisted of temporal

records of chlorophyll fluorescence from six moored fluorometer arrays located along the outer edge of the continental shelf off the Delmarva Peninsula. The mooring arrays were deployed between February 1988 and May 1989.

Details of the SEEP II data are given in Table 4.

At each mooring, a time series of daily satellite-derived

surface chlorophyll measurements was simulated by selecting the SEEP measurement closest to 10 am from the shallowest fluorometer. The depths of these instruments ranged

from 16–39 m (see Table 4).

29
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Table 2a. Summary of relative errors for CHL estimators.

Estimator Scene Number Bias Error 95% Range

Used Number of Bins [%] (rms) [%] Minimum Maximum

MLE 1 2,750 0.00 0.00 −1 0

2 1,850 0.00 0.00 −1 0

3 2,584 −0.03 1.26 −1 0

4 5,773 −0.03 0.42 −1 0

5 4,535 −0.05 1.42 −1 0

6 513 0.07 1.19 −2 2

7 3,285 −0.21 1.13 −3 1

Combined 21,290 −0.05 0.95 −1 0

MED 1 2,750 −1.14 1.20 −3 −1

2 1,850 −1.16 1.25 −3 −1

3 2,584 −1.08 2.15 −6 0

4 5,773 −1.98 2.95 −8 −1

5 4,535 −1.37 2.27 −3 −1

6 513 −4.67 7.32 −24 0

7 3,285 −5.13 7.38 −21 −1

Combined 21,290 −2.11 3.74 −11 −1

AVG4 1 2,750 1.12 7.28 −13 15

2 1,850 0.68 7.07 −13 15

3 2,584 −2.00 6.07 −14 8

4 5,773 −3.12 9.35 −21 13

5 4,535 0.00 7.31 −14 14

6 513 0.71 11.79 −22 25

7 3,285 0.71 16.72 −26 42

Combined 21,290 −0.76 9.86 −19 18

MLE4 1 2,750 1.12 7.29 −13 15

2 1,850 0.69 7.06 −13 15

3 2,584 −1.99 6.06 −14 8

4 5,773 −3.10 9.26 −21 13

5 4,535 −0.01 7.25 −14 14

6 513 0.88 11.86 −22 25

7 3,285 0.69 16.51 −26 41

Combined 21,290 −0.75 9.77 −19 18

MED4 1 2,750 0.17 7.14 −14 14

2 1,850 −0.26 6.95 −14 14

3 2,584 −2.96 6.50 −16 7

4 5,773 −4.81 9.95 −24 11

5 4,535 −1.04 7.21 −15 13

6 513 −3.50 12.52 −30 18

7 3,285 −4.07 14.02 −32 24

Combined 21,290 −2.60 9.38 −22 14

30
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Table 2b. Summary of relative errors for K490 estimators.

Estimator Scene Number Bias Error 95% Range

Used Number of Bins [%] (rms) [%] Minimum Maximum

MLE 1 2,757 0.00 0.00 −1 0

2 1,875 0.00 0.00 −1 0

3 2,584 0.00 0.02 −1 0

4 5,773 0.00 0.03 −1 0

5 4,562 0.00 0.02 −1 0

6 424 0.00 0.16 −1 0

7 2,398 0.00 0.05 −1 0

Combined 20,373 0.00 0.04 −1 0

MED 1 2,757 0.00 0.00 −1 0

2 1,875 0.00 0.00 −1 0

3 2,584 −0.05 0.24 −2 0

4 5,773 −0.16 0.55 −2 0

5 4,562 −0.12 0.36 −2 0

6 424 −1.09 2.08 −8 0

7 2,398 −0.93 1.41 −4 0

Combined 20,373 −0.21 0.67 −3 0

AVG4 1 2,757 0.26 1.63 −4 3

2 1,875 0.20 1.71 −4 3

3 2,584 −0.69 2.12 −5 3

4 5,773 −1.02 3.45 −8 5

5 4,562 0.01 3.46 −7 7

6 424 0.37 5.39 −12 11

7 2,398 0.06 6.54 −12 16

Combined 20,373 −0.31 3.59 −8 7

MLE4 1 2,757 0.26 1.63 −4 3

2 1,875 0.20 1.71 −4 3

3 2,584 −0.68 2.12 −5 3

4 5,773 −1.03 3.44 −8 5

5 4,562 0.01 3.45 −7 7

6 424 0.37 5.39 −12 11

7 2,398 0.05 6.52 −12 16

Combined 20,373 −0.31 3.58 −8 7

MED4 1 2,757 0.21 1.62 −4 3

2 1,875 0.14 1.71 −4 3

3 2,584 −0.80 2.16 −6 3

4 5,773 −1.27 3.51 −9 4

5 4,562 −0.24 3.41 −7 6

6 424 −0.75 5.37 −14 8

7 2,398 −0.81 6.09 −13 13

Combined 20,373 −0.59 3.50 −8 6

31
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Table 2c. Summary of relative errors for LWN (λ1) estimators.

Estimator Scene Number Bias Error 95% Range

Used Number of Bins [%] (rms) [%] Minimum Maximum

MLE 1 2,750 0.00 0.00 −1 0

2 1,850 0.00 0.00 −1 0

3 2,584 0.00 0.00 −1 0

4 5,773 0.01 0.31 −1 0

5 4,535 0.01 0.32 −1 0

6 513 0.00 0.04 −1 0

7 3,285 0.75 3.39 −1 8

Combined 21,290 0.12 1.35 −1 0

MED 1 2,750 0.00 0.00 −1 0

2 1,850 0.00 0.02 −1 0

3 2,584 −0.03 0.19 −1 0

4 5,773 −0.20 0.74 −2 0

5 4,535 −0.08 0.41 −2 0

6 513 −0.60 1.13 −4 0

7 3,285 −3.97 9.10 −31 0

Combined 21,290 −0.70 3.60 −6 0

AVG4 1 2,750 0.35 1.57 −3 3

2 1,850 0.46 1.73 −4 3

3 2,584 −0.76 1.93 −5 2

4 5,773 −1.58 3.53 −9 4

5 4,535 0.73 3.05 −6 6

6 513 0.08 4.18 −9 8

7 3,285 −4.63 12.01 −36 11

Combined 21,290 −0.99 5.39 −12 5

MLE4 1 2,750 0.35 1.57 −3 3

2 1,850 0.46 1.73 −4 3

3 2,584 −0.76 1.93 −5 2

4 5,773 −1.58 3.53 −9 4

5 4,535 0.73 3.04 −6 6

6 513 0.09 4.18 −9 8

7 3,285 −3.95 12.32 −33 15

Combined 21,290 −0.89 5.50 −11 5

MED4 1 2,750 0.30 1.57 −3 3

2 1,850 0.39 1.73 −4 3

3 2,584 −0.88 2.01 −5 2

4 5,773 −1.85 3.78 −10 3

5 4,535 0.52 3.09 −6 6

6 513 −0.50 4.36 −10 7

7 3,285 −7.94 16.22 −54 6

Combined 21,290 −1.66 6.93 −16 5

32
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Table 2d. Summary of relative errors for LWN (λ2) estimators.

Estimator Scene Number Bias Error 95% Range

Used Number of Bins [%] (rms) [%] Minimum Maximum

MLE 1 2,750 0.01 0.12 −1 0

2 1,850 0.04 0.29 −1 0

3 2,584 0.00 0.03 −1 0

4 5,773 0.03 0.22 −1 0

5 4,535 0.02 0.20 −1 0

6 513 0.03 0.22 −1 0

7 3,285 0.03 0.36 −1 0

Combined 21,290 0.02 0.23 −1 0

MED 1 2,750 −0.04 0.24 −1 0

2 1,850 −0.15 0.55 −2 0

3 2,584 −0.09 0.33 −2 0

4 5,773 −0.27 0.73 −3 0

5 4,535 −0.11 0.47 −2 0

6 513 −1.26 2.45 −9 0

7 3,285 −0.90 2.23 −7 0

Combined 21,290 −0.30 1.07 −3 0

AVG4 1 2,750 0.43 3.09 −6 6

2 1,850 0.49 3.61 −7 7

3 2,584 −1.29 2.80 −7 3

4 5,773 −2.30 4.55 −11 5

5 4,535 0.57 3.24 −6 6

6 513 0.37 5.41 −13 11

7 3,285 −2.50 6.08 −14 7

Combined 21,290 −0.94 4.19 −10 6

MLE4 1 2,750 0.44 3.09 −6 6

2 1,850 0.49 3.59 −7 7

3 2,584 −1.29 2.79 −7 3

4 5,773 −2.27 4.51 −11 5

5 4,535 0.59 3.21 −6 6

6 513 0.41 5.40 −13 11

7 3,285 −2.48 6.02 −14 7

Combined 21,290 −0.92 4.16 −10 6

MED4 1 2,750 0.25 3.13 −7 6

2 1,850 0.22 3.71 −8 7

3 2,584 −1.48 2.94 −7 3

4 5,773 −2.68 5.00 −12 4

5 4,535 0.34 3.40 −7 6

6 513 −0.78 5.80 −16 9

7 3,285 −3.38 6.71 −17 5

Combined 21,290 −1.32 4.53 −11 6

33
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Table 2e. Summary of relative errors for LWN (λ3) estimators.

Estimator Scene Number Bias Error 95% Range

Used Number of Bins [%] (rms) [%] Minimum Maximum

MLE 1 2,750 0.01 0.09 −1 0

2 1,850 0.03 0.22 −1 0

3 2,584 0.00 0.03 −1 0

4 5,773 0.02 0.19 −1 0

5 4,535 0.00 0.07 −1 0

6 513 0.06 0.35 −1 1

7 3,285 0.01 0.25 −1 0

Combined 21,290 0.01 0.17 −1 0

MED 1 2,750 −0.60 0.79 −2 0

2 1,850 −0.77 0.95 −2 0

3 2,584 −0.19 0.50 −2 0

4 5,773 −0.67 1.02 −3 0

5 4,535 −0.13 0.38 −2 0

6 513 −2.09 3.50 −11 0

7 3,285 −1.02 2.01 −6 0

Combined 21,290 −0.59 1.19 −3 0

AVG4 1 2,750 1.07 4.75 −9 10

2 1,850 0.94 4.88 −9 10

3 2,584 −1.92 3.75 −9 4

4 5,773 −3.43 6.18 −15 6

5 4,535 0.77 3.91 −8 8

6 513 0.57 6.90 −15 14

7 3,285 −3.16 6.37 −15 7

Combined 21,290 −1.25 5.26 −13 8

MLE4 1 2,750 1.07 4.74 −9 10

2 1,850 0.95 4.86 −9 10

3 2,584 −1.92 3.74 −9 4

4 5,773 −3.40 6.14 −15 6

5 4,535 0.77 3.90 −8 8

6 513 0.65 6.93 −14 14

7 3,285 −3.14 6.36 −15 7

Combined 21,290 −1.24 5.24 −13 8

MED4 1 2,750 0.67 4.73 −9 9

2 1,850 0.49 4.98 −10 9

3 2,584 −2.24 3.95 −10 3

4 5,773 −4.07 6.78 −16 5

5 4,535 0.47 3.92 −8 8

6 513 −1.26 7.42 −20 11

7 3,285 −4.08 7.05 −17 6

Combined 21,290 −1.81 5.63 −14 8

34
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Table 3. Summary of relative errors for ICK estimators.

Estimator Image Number Bias RMS 95% Range

Used Number of Bins [%] Error [%] Minimum Maximum

MLE 1 2,750 0.00 0.00 −1 0

2 1,850 0.00 0.00 −1 0

3 2,584 −0.01 0.45 −1 0

4 5,752 −0.01 0.10 −1 0

5 4,533 −0.02 0.48 −1 0

6 423 −0.01 0.23 −1 0

7 2,379 −0.04 0.20 −1 0

Combined 20,271 −0.01 0.30 −1 0

FNC 1 2,750 0.14 0.38 −1 1

2 1,850 0.23 0.48 −1 1

3 2,584 0.39 2.61 −1 2

4 5,752 0.86 1.45 −1 3

5 4,533 0.80 1.32 −1 1

6 423 2.57 4.52 −1 13

7 2,379 1.84 2.53 0 6

Combined 20,271 0.78 1.76 −1 3

AVG4 1 2,750 0.87 5.74 −11 12

2 1,850 0.49 5.44 −11 11

3 2,584 −1.34 3.98 −10 5

4 5,752 −2.19 6.09 −15 8

5 4,533 0.07 3.68 −8 7

6 423 0.61 7.96 −15 16

7 2,379 −0.30 6.96 −13 16

Combined 20,271 −0.64 5.45 −12 10

MLE4 1 2,750 0.87 5.74 −11 12

2 1,850 0.49 5.45 −11 11

3 2,584 −1.34 3.98 −10 5

4 5,752 −2.18 6.06 −15 8

5 4,533 0.07 3.68 −8 7

6 423 0.62 7.97 −16 17

7 2,379 −0.33 6.87 −-13 15

Combined 20,271 −0.64 5.43 −12 10

FNC4 1 2,750 1.19 5.83 −11 12

2 1,850 0.82 5.52 −10 12

3 2,584 −0.91 3.92 −9 6

4 5,752 −1.45 6.10 −14 9

5 4,533 0.59 3.83 −7 8

6 423 2.86 10.05 −15 24

7 2,379 1.28 8.66 −12 24

Combined 20,271 0.05 5.84 −11 11

35
0

5000

10000

15000

20000

25000

-6 -4 -2 0 2 4 6 8 10 12 14

99%

1%

0

2000

4000

6000

8000

10000

-6 -4 -2 0 2 4 6 8 10 12 14

5%

45% 46%

1% 2%1%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 15. Histograms of CHL/K490 estimation errors based on 20,271 bins analyzed and full resolution (LAC)

data. For each bin, the error is defined as the difference between the estimator and the arithmetic average

(AVG) of all data in the bin expressed as a percentage of AVG. The FNC estimator is the AVG estimator of

CHL divided by the AVG estimator of K490. The top histogram was calculated for (MLE − AVG)/ AVG (%).

The bottom histogram was calculated for (FNC − AVG)/ AVG (%).

36
0

500

1000

1500

2000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.64%

0

500

1000

1500

2000

-50 -40 -30 -20 -10 0 10 20 30 40 50

Bias = -0.64%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 16. Histograms of CHL/K490 estimation errors based on 20,271 bins analyzed and 4 km subsampled (GAC)

data. The top histogram was calculated for (AVG4 − AVG)/ AVG (%). The bottom histogram was calculated

for (FNC4 − AVG)/ AVG (%).

37
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Table 4. Location and depth of SEEP II moored fluorometers and the period covered by the time series data

used in the analysis of temporal statistics.

Latitude Longitude Depth Time Series

ID Deployment [deg. min.] [deg. min.] [m] Start Finish

1 Spring 37 52.60 74 43.90 39 7 Feb 88 8 Apr 88

Summer 37 52.49 74 43.90 18 25 Jun 88 19 Oct 88

Winter 37 47.62 74 44.60 19 12 Nov 88 17 Mar 89

2 Spring 37 46.11 74 29.50 16 8 Feb 88 9 Apr 88

Winter 37 34.69 74 35.13 24 15 Nov 88 28 Jan 89

3 Spring 37 41.99 74 20.35 19 8 Feb 88 9 Jun 88

Summer 37 41.98 74 20.37 21 25 Jun 88 17 Oct 88

Winter 37 41.96 74 20.27 19 11 Nov 88 8 May 89

5 Spring 37 39.80 74 15.85 21 8 Feb 88 7 Jun 88

Summer 37 39.78 74 15.72 22 26 Jun 88 17 Oct 88

Winter 37 39.73 74 15.78 21 15 Nov 88 2 May 89

6 Spring 37 37.91 74 12.86 20 12 Feb 88 7 Jun 88

Summer 37 37.90 74 12.87 20 25 Jun 88 19 Oct 88

Winter 37 37.95 74 12.77 35 13 Nov 88 6 May 89

8 Spring 36 52.63 74 39.04 22 13 Feb 88 8 Jun 88

K490 was derived from the chlorophyll measurement by

the formula

K490 = 0.022 + 0.079 CHL0.875. (43)

This is the relationship between K490 (34) and CHL13 (35).

In the CZCS imagery analyzed, this relationship would

hold for most of the data since CHL equals CHL13 in most

pixels.

Weekly and monthly means of CHL and K490 were derived using the AVG, MLE, and MED estimators. When

sample sizes are small (e.g., n ≤ 7), the effect of sample

size dominates the error statistics. To control for this effect in weekly means, only weeks having 7 days, i.e., no

missing data, were analyzed. However, because there were

fewer months, all months were analyzed, regardless of their

sample size. The AVG estimator was regarded as the true

mean. Errors for the MLE and MED estimators were expressed as a percentage of the AVG estimator.

Weekly and monthly means of the function ICK (1)

were also derived. Estimators compared with the AVG or

true mean were the MLE and MED estimators, and an

FNC estimator defined in two ways:

FNC(AVG) = AVG estimator of CHL

AVG estimator of K490

(44)

and

FNC(MLE) = MLE estimator of CHL

MLE estimator of K490

. (45).

The FNC(MLE) estimator would be applicable if, as recommended, spatial statistics are derived according to the

MLE estimator.

To investigate the behavior of the AVG, MLE, MED,

and FNC estimators as samples sizes increase over time,

cumulative means were obtained as follows:

AVG(n) arithmetic average of all data from days 1

to n,

MLE(n) MLE estimate based on data from days 1

to n,

MED(n) MED estimate based on data from days 1

to n, and

FNC(n) FNC estimate based on data from days 1

to n.

The cumulative means began day 1 at the start of each

deployment. Since each mooring had up to three separate

deployments (see Table 4), there were 1–3 sets of cumulative means for each mooring. These were plotted against

n to observe how the estimators behaved as a function of

sample size.

In a similar manner, the behavior of the estimators

as functions of area were investigated using CZCS data.

Beginning at one or two selected locations in each CZCS

scene (Table 1), the estimators were calculated for bins of

increasing area L×L, with L increasing from 9 km to the

size of the image. The maximum value of L was 480 km.

Increases in area may be regarded as analogous to increases

in time. To the extent that this is true, these results would

pertain to the estimation of temporal means.

3.2.2 Results

Histograms of log(CHL) from each mooring are shown

in Fig. 17. Based on the normal (Gaussian) appearance

of these histograms, the distribution of chlorophyll over

time at a single location is approximately lognormal, or a

mixture of lognormals.

38
0

20 0

40 0

60 0

80 0

10 00

12 00

0.0 1 0.1 1 10 10 0 10 00

spring

summer

wint er

Chl, mg/ m3

Mooring #1

Deployment

0

20 0

40 0

60 0

80 0

10 00

12 00

0.0 1 0.1 1 10 10 0 10 00

spring

summer

wint er

Chl, mg/ m3

Mooring #3

Deployment

0

20 0

40 0

60 0

80 0

10 00

12 00

0.0 1 0.1 1 10 10 0 10 00

spring

summer

wint er

Chl, mg/ m3

Mooring #6

Deployment

0

20 0

40 0

60 0

80 0

10 00

12 00

0.0 1 0.1 1 10 10 0 10 00

spring

wint er

Chl, mg/ m3

Mooring #2

Deployment

0

20 0

40 0

60 0

80 0

10 00

12 00

0.0 1 0.1 1 10 10 0 10 00

spring

summer

wint er

Chl, mg/ m3

Mooring #5

Deployment

0

20 0

40 0

60 0

80 0

10 00

12 00

0.0 1 0.1 1 10 10 0 10 00

spring

Chl, mg/ m3

Mooring #8

Deployment

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 17. Histograms of CHL from SEEP II moored fluorometer data. Data are from the shallowest fluorometer

at each mooring. All data from moorings 1, 2, 3, 5, 6, and 8 are included in these histograms.

39
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Histograms of relative errors in weekly and monthly

means are shown in Figs. 18–25 and summarized in Table 5.

The upper panel in each figure is the error histogram for

weekly means, and the lower panel is for monthly means.

The patterns seen in Fig. 18 for the MLE estimator of CHL

were similar to those obtained for the MLE estimators of

the other variables. The MLE and AVG estimates agreed

within ±5% most of the time, with a slight tendency for

MLE to exceed AVG, as indicated by the small positive

biases (usually much less than 1%) in all cases. As in the

case of the CZCS data, the K490 and ICK variables had

much smaller MLE errors than the CHL variable.

The MED estimator had relatively large negative errors for all three variables. That is, the MED estimator

underestimated the arithmetic average by 40% or more in

some cases, and monthly mean errors were about a factor

of 2 greater than weekly mean errors.

Errors that are associated with the FNC(AVG) and

FNC(MLE) estimators are shown in Figs. 24 and 25, respectively, and are summarized in Table 5d. Both distributions are positively skewed, with errors as high as 30%

or more. The FNC(MLE) estimator had larger errors than

the FNC(AVG) estimator.

Results for cumulative means provided important insight concerning the behavior of the estimators, in particular when the MLE and AVG estimators were substantially

different. These insights will be illustrated here with results from moorings 3 and 6. Figure 26 shows cumulative

mean CHL estimates from the spring deployments of mooring 3 (upper panel) and mooring 6 (lower panel). In the

case of mooring 3, MLE(n) and AVG(n) remained approximately equal over the entire averaging period, whereas

MED(n) was always less than the other two and gradually

diverged as the averaging period increased. These results

are typical of what was obtained for the majority of the

cases.

The lower panel in Fig. 26 illustrates a case where

MLE(n) and AVG(n) diverged. The two cumulative means

showed an abrupt divergence at about day 70; prior to that

day, they had been nearly equal. Inspection of the data

(Fig. 27, upper panel) revealed that there were a number

of anomalously low values beginning after day 60. The

dark squares in Fig. 27 were data that were missing from

the original records. These had been set to zero and were

ignored when calculating cumulative means. However, the

open squares lying near the horizontal axis were small positive values (e.g., 0.01, 0.02, etc.) which may have also been

bad data. If these are eliminated from the record, then

MLE(n) and AVG(n) agree (bottom panel of Fig. 27).

This suggests that the MLE estimator can be sensitive to outliers, particularly outliers that are close to zero.

When a data value approximately equal to zero is included

in the arithmetic average of n values, the effect is to reduce the AVG estimator by a factor of (n−1)/n. However,

the logarithm of a number approximately equal to zero is

a large negative number, and its effect on the statistics of

the logarithm can be extreme. Including this value will reduce the mean of the logarithm but increase the variance,

somewhat offseting effects on the MLE estimator. In general, however, the net effect will be to increase the MLE

estimator since the variance of the logarithm is increased

substantially by the inclusion of a large negative value.

Another case in which MLE(n) and AVG(n) diverged

was the summer deployment of mooring 3. The simulated satellite data from this record are shown in the upper

panel of Fig. 28, and the cumulative means in the lower

panel. Like the previous example, there were a number

of low values in the record. However, it is not obvious

that these are bad data, and so there is no justification

for removing them to make MLE(n) and AVG(n) agree.

MLE(n) was approximately 10% higher than AVG(n) for

n ≥ 35 days. The cumulative means of K490 and ICK

for this mooring are shown in Fig. 29. Differences between MLE(n) and AVG(n) for these variables were much

smaller than those for CHL. However, the two FNC estimates were consistently higher than AVG(n) and MLE(n),

with differences approaching 30% by the end of the record.

Cumulative means starting at two locations in CZCS

scene 4 are illustrated in Fig. 30 (LAC means) and Fig. 31

(GAC means). In these figures, the cumulative mean CHL

within areas of size L2 is plotted against L. In the northern portion of scene 4 (off the west coast of Mexico), the

MLE and AVG cumulative means diverged at length scales

larger than 50 km. However, in the southern region of this

scene, the MLE and AVG means remained nearly equal for

areas up to 460×460 km2. Results for all the CZCS scenes

are summarized in Table 6. Whenever the MLE and AVG

estimators diverged for CZCS cumulative means, the AVG

estimator was greater than the MLE estimator. This occurred in the scenes that had high chlorophyll levels and/or

high variances. In contrast, when the MLE and AVG estimators in SEEP data diverged, the MLE estimator was

usually greater than the AVG estimator.

3.3 Discussion

From the study of CZCS and SEEP II data, it was

concluded that the AVG and MLE estimators are equivalent with respect to their accuracy as estimators of means

within sampling domains. The MED and FNC estimators are not considered acceptable as estimators of the

mean. The MED estimator systematically underestimated

the mean, and the magnitude of its error increased with

increasing intrabin variance. The FNC estimator, i.e., the

result of substituting a mean into a function to derive a

level-4 variable, also had systematic errors that increased

with increasing variance.

In the case of satellite data from the same scene (spatial statistics), the MLE estimator proved to be nearly

identical to the AVG estimator when sample sizes were

large (n = 121). The same was true for the MLE4 and

40
0

40

80

120

160

200

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Bias = 0.14%

0

10

20

30

40

50

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Bias = 1.37%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 18. Histograms of the relative error in MLE estimates of mean CHL at SEEP moorings. The top histogram

is for the weekly means (n = 213), calculated with 100% × (MLE − AVG)/ AVG. The bottom panel is for the

monthly means (n = 74), also calculated with 100% × (MLE − AVG)/ AVG.

41
0

10

20

30

40

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10

Bias = -5.46%

0

3

6

9

12

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10

Bias = -10.7%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 19. Histograms of the relative error in MED estimates of mean CHL at SEEP moorings. The top histogram

is for the weekly means (n = 213), calculated with 100% × (MED − AVG)/ AVG. The bottom panel is for the

monthly means (n = 74), also calculated with 100% × (MED − AVG)/ AVG.

42
0

40

80

120

160

200

-30 -20 -10 0 10 20 30

Bias = -0.04%

0

15

30

45

60

75

-30 -20 -10 0 10 20 30

Bias = 0.01%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 20. Histograms of the relative error in MLE estimates of mean K490 at SEEP moorings. The top histogram

is for the weekly means (n = 213), calculated with 100% × (MLE − AVG)/AV G. The bottom panel is for the

monthly means (n = 74), also calculated with 100% × (MLE − AVG)/ AVG.

43
0

20

40

60

80

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20

Bias = -2.38%

0

5

10

15

20

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20

Bias = -4.38%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 21. Histograms of the relative error in MED estimates of mean K490 at SEEP moorings. The top histogram

is for the weekly means (n = 213), calculated with 100% × (MED − AVG)/ AVG. The bottom panel is for the

monthly means (n = 74), also calculated with 100% × (MED − AVG)/ AVG.

44
0

50

100

150

200

250

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Bias = 0.06%

0

15

30

45

60

75

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Bias = 0.35%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 22. Histograms of the relative error in MLE estimates of mean CHL/K490 at SEEP moorings. The top

histogram is for the weekly means (n = 213), calculated with 100% × (MLE − AVG)/ AVG. The bottom panel

is for the monthly means (n = 74), also calculated with 100% × (MLE − AVG)/ AVG.

45
0

40

80

120

160

-40 -30 -20 -10 0 10 20

Bias = -0.89%

0

10

20

30

40

-40 -30 -20 -10 0 10 20

Bias = -2.28%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 23. Histograms of the relative error in MED estimates of mean CHL/K490 at SEEP moorings. The top

histogram is for the weekly means (n = 213), calculated with 100% × (MED − AVG)/ AVG. The bottom panel

is for the monthly means (n = 74), also calculated with 100% × (MED − AVG)/ AVG.

46
0

20

40

60

80

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40

Bias = 2.72%

0

5

10

15

20

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40

Bias = 5.47%

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 24. Histograms of the relative error in FNC(AVG) estimates of mean CHL/K490 at SEEP moorings. The

top histogram is for the weekly means (n = 213), calculated with 100% × (FNC − AVG)/ AVG. The bottom

panel is for the monthly means (n = 74), also calculated with 100% × (FNC − AVG)/ AVG.

47
0

20

40

60

80

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40

Bias = 2.92%

0

5

10

15

20

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40

Bias = 6.99%

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 25. Histograms of the relative error in FNC(MLE) estimates of mean CHL/K490 at SEEP moorings. The

top histogram is for the weekly means (n = 213), calculated with 100% × (FNC − AVG)/ AVG. The bottom

panel is for the monthly means (n = 74), also calculated with 100% × (FNC − AVG)/ AVG.

48
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Table 5a. Summary of relative errors for weekly means derived from SEEP mooring data: results for CHL.

Estimator Mooring Number Bias Error Range

Used Reference of Weeks [%] (rms) [%] Minimum Maximum

MLE 1 39 −0.33 2.64 −13 4

2 18 0.02 0.06 0 0

3 53 0.18 1.28 −2 9

5 53 0.24 1.21 −2 6

6 37 0.47 2.27 −3 12

8 13 0.26 2.33 −3 7

Combined 213 0.14 1.79 −13 12

MED 1 39 −5.32 10.83 −41 0

2 18 −1.17 1.44 −3 0

3 53 −4.01 7.26 −27 0

5 53 −6.07 9.06 −42 0

6 37 −7.15 11.07 −34 0

8 13 −10.51 12.81 −22 −1

Combined 213 −5.46 9.24 −42 0

Table 5a. (cont.) Summary of relative errors for monthly means derived from SEEP mooring data: results

for CHL.

Estimator Mooring Number Bias Error Range

Used Reference of Months [%] (rms) [%] Minimum Maximum

MLE 1 13 −1.69 4.90 −1 16

2 6 0.06 0.18 0 0

3 17 0.91 4.09 −3 14

5 16 0.98 2.53 −2 8

6 17 2.29 6.02 −12 12

8 5 1.75 2.88 0 5

Combined 74 1.37 4.18 −12 16

MED 1 13 −7.59 13.75 −42 −1

2 6 −2.27 2.70 −5 0

3 17 −10.07 15.02 −37 −1

5 16 −9.24 11.10 −28 −1

6 17 −17.55 21.92 −39 −1

8 5 −11.92 15.18 −24 −3

Combined 74 −10.66 15.24 −42 0

49
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Table 5b. Summary of relative errors for weekly means derived from SEEP mooring data: results for K490.

Estimator Mooring Number Bias Error Range

Used Reference of Weeks [%] (rms) [%] Minimum Maximum

MLE 1 39 −0.18 0.89 −4 1

2 18 −0.03 0.15 0 0

3 53 0.03 0.29 −1 1

5 53 −0.01 0.41 −1 1

6 37 −0.11 0.57 −1 1

8 13 0.06 0.70 −1 2

Combined 213 −0.04 0.54 −4 2

MED 1 39 −2.52 4.72 −17 0

2 18 −0.72 0.90 −2 0

3 53 −1.65 3.05 −15 0

5 53 −2.62 3.81 −17 0

6 37 −2.92 4.95 −18 1

8 13 −4.69 5.75 −10 0

Combined 213 −2.38 4.02 −18 1

Table 5b. (cont.) Summary of relative errors for monthly means derived from SEEP mooring data: results

for K490.

Estimator Mooring Number Bias Error Range

Used Reference of Months [%] (rms) [%] Minimum Maximum

MLE 1 13 0.27 0.75 0 3

2 6 −0.09 0.26 0 0

3 17 −0.14 0.70 −2 2

5 16 0.20 0.47 0 1

6 17 −0.20 1.36 −5 2

8 5 0.15 0.56 −1 1

Combined 74 0.01 0.82 −5 3

MED 1 13 −3.94 7.19 −22 0

2 6 −1.42 1.68 −3 −1

3 17 −4.49 6.01 −15 −1

5 16 −4.06 5.06 −13 0

6 17 −5.79 7.88 −20 0

8 5 −4.98 6.43 −10 −2

Combined 74 −4.38 6.25 −22 0

50
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Table 5c. Summary of relative errors for weekly means derived from SEEP mooring data: results for ICK.

Estimator Mooring Number Bias Error Range

Used Reference of Weeks [%] (rms) [%] Minimum Maximum

MLE 1 39 0.00 0.13 −1 0

2 18 −0.01 0.03 0 0

3 53 0.09 0.55 0 4

5 53 0.04 0.11 0 1

6 37 0.16 0.69 0 4

8 13 0.05 0.23 0 1

Combined 213 0.06 0.40 −1 4

MED 1 39 −0.73 1.93 −9 0

2 18 −0.08 0.10 0 0

3 53 −0.74 2.09 −11 0

5 53 −0.95 1.90 −11 0

6 37 −1.40 2.71 −12 0

8 13 −1.44 1.96 −5 0

Combined 213 −0.89 2.03 −12 0

Table 5c. (cont.) Summary of relative errors for monthly means derived from SEEP mooring data: results

for ICK.

Estimator Mooring Number Bias Error Range

Used Reference of Months [%] (rms) [%] Minimum Maximum

MLE 1 13 0.12 0.33 0 1

2 6 −0.03 0.03 0 0

3 17 0.25 0.70 0 2

5 16 0.11 0.29 0 1

6 17 0.95 1.51 0 3

8 5 0.40 0.86 0 2

Combined 74 0.35 0.84 0 3

MED 1 13 −1.15 2.66 −9 0

2 6 −0.17 0.20 0 0

3 17 −1.91 3.89 −11 0

5 16 −1.40 1.91 −6 0

6 17 −5.09 7.13 −14 0

8 5 −2.26 3.12 −6 −1

Combined 74 −2.28 4.17 −14 0

51
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Table 5d. Summary of relative errors for weekly means derived from SEEP mooring data: results for FNC

estimators of ICK.

Estimator Mooring Number Bias Error Range

Used Reference of Weeks [%] (rms) [%] Minimum Maximum

FNC(AVG) 1 39 2.90 7.25 0 30

2 18 0.40 0.52 0 1

3 53 1.90 3.63 0 13

5 53 3.01 5.16 0 28

6 37 3.53 5.71 0 19

8 13 5.32 6.62 0 11

Combined 213 2.72 5.24 0 30

FNC(MLE) 1 39 2.65 5.85 0 24

2 18 0.45 0.56 0 1

3 53 2.07 4.32 0 18

5 53 3.29 6.06 0 35

6 37 4.16 7.28 0 28

8 13 5.53 7.23 1 17

Combined 213 2.92 5.66 0 35

Table 5d. (cont.) Summary of relative errors for monthly means derived from SEEP mooring data: results

for FNC estimators of ICK.

Estimator Mooring Number Bias Error Range

Used Reference of Months [%] (rms) [%] Minimum Maximum

FNC(AVG) 1 13 3.62 7.23 0 23

2 6 0.70 0.90 0 1

3 17 5.09 8.59 0 25

5 16 4.46 5.43 1 14

6 17 9.76 12.62 1 26

8 5 5.96 7.85 1 13

Combined 74 5.47 8.45 0 26

FNC(MLE) 1 13 5.28 12.14 0 40

2 6 0.86 1.02 0 2

3 17 6.28 11.26 0 31

5 16 5.32 7.42 0 23

6 17 12.52 16.30 0 29

8 5 7.72 10.60 2 18

Combined 74 6.99 11.46 0 40

52
1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 20 40 60 80 100 120 140

MED

AVG

MLE

Averaging Interval (days)

Mean CHL

(a) Mooring 3: Feb-Jun 1988

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140

MED

AVG

MLE

Averaging Interval (days)

Mean CHL

(b) Mooring 6: Feb-Jun 1988

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 26. Cumulative mean CHL estimates for data from the spring 1988 deployment of SEEP moorings 3 and

6.

53
0

1

2

3

4

5

6

7

0 20 40 60 80 100 120 140

CHL

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140

MED

AVG

MLE

Averaging Interval (days)

Mean CHL

 Mooring 6: Feb-Jun 1988

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 27. Results of correcting for bad data in the time series at mooring 6 (Fig. 26). In the upper panel, which

displays a record of the 10 am CHL measurements versus time at mooring 6, the dark squares are data missing

from the original record. The open squares near zero are probably bad data. Cumulative means derived after

removing these low values are shown in bottom panel.

54
0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 20 40 60 80 100 120

CHL

0.0

0.4

0.8

1.2

1.6

0 20 40 60 80 100 120

MED

AVG

MLE

Averaging Interval (days)

Mean CHL

Mooring 3: Jun-Oct 1988

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 28. Simulated satellite CHL time series and cumulative mean CHL for summer deployment of mooring 3

(Jun.–Oct. 1988). The upper panel displays 10 am CHL measurements, and the lower panel displays cumulative

means for the data shown in the upper panel.

55
0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100 120

Mean K490

MED

MLE

AVG

Mooring 3: Jun-Oct 1988

6

7

8

9

10

11

12

13

0 20 40 60 80 100 120

Mean CHL/K490

MLE

AVG

FNC(MLE)

FNC(AVG)

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 29. Cumulative means of K490 (upper panel) and CHL/K490 (lower panel) for summer deployment of

mooring 3 (Jun.–Oct. 1988).

56
0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32

0 50 100 150 200 250

AVG

MLE

MED

Scene 4: Northern Region

(Southwest of Mexico)

Mean CHL

Length, km

0.12

0.13

0.14

0.15

0.16

0 100 200 300 400 500

AVG

MLE

MED

Scene 4: Southern Region

(Tropical Pacific)

Mean CHL

Length, km

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Fig. 30. Cumulative mean CHL in CZCS scene 4 based on LAC data within boxes of increasing area (L×L)

plotted against length, L. Results for boxes in the northern nearshore region (upper panel) and for the southern

offshore region (lower panel).

57
0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32

0 50 100 150 200 250

AVG4

MLE4

MED4

Scene 4: Northern Region

(Southwest of Mexico)

Mean CHL

Length, km

0.12

0.13

0.14

0.15

0.16

0 100 200 300 400 500

AVG4

MLE4

MED4

Scene 4: Southern Region

(Tropical Pacific)

Mean CHL

Length, km

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 31. Cumulative mean CHL in CZCS scene 4, in this case based on GAC data, within boxes of increasing

area (L×L) plotted against length, L. Results for boxes in the northern nearshore region (upper panel) and for

the southern offshore region (lower panel).

58
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Table 6. Comparison of cumulative means derived from CZCS data for largest areas (L×L). The length (L)

is shown in column 2.

CZCS Length Number Estimator % Error

Scene [km] of Pixels AVG AVG4 AVG AVG4

1 479 336,494 0.059 0.059 0.0 1.0

2 463 314,643 0.061 0.061 0.0 0.7

3 441 286,176 0.166 0.163 0.0 −1.5

4-N 222 72,381 0.297 0.302 0.0 1.4

4-S 460 310,518 0.143 0.140 0.0 −2.4

5-N 480 339,127 0.298 0.297 0.0 −0.4

5-S 291 124,475 0.384 0.382 0.0 −0.4

6 224 73,390 0.629 0.634 0.0 0.1

7-N 361 191,092 0.919 0.933 0.0 1.5

7-S 238 82,714 1.052 1.072 0.0 1.9

CZCS Length Number Estimator % Error

Scene [km] of Pixels MLE MLE4 MLE MLE4

1 479 336,494 0.058 0.059 −0.2 1.0

2 463 314,643 0.061 0.061 −0.2 0.5

3 441 286,176 0.166 0.164 0.4 −1.0

4-N 222 72,381 0.276 0.276 −7.1 −7.3

4-S 460 310,518 0.144 0.140 0.3 −2.2

5-N 480 339,127 0.297 0.297 −0.5 −0.4

5-S 291 124,475 0.386 0.385 0.5 0.2

6 224 73,390 0.531 0.537 −15.5 −14.7

7-N 361 191,092 0.883 0.908 −3.9 −1.2

7-S 238 82,714 0.973 0.992 −7.4 −5.7

CZCS Length Number Estimator % Error

Scene [km] of Pixels MED MED4 MED MED4

1 479 336,494 0.057 0.058 −2.05 −0.85

2 463 314,643 0.059 0.059 −3.93 −3.28

3 441 286,176 0.157 0.155 −5.01 −6.46

4-N 222 72,381 0.229 0.224 −22.97 −24.71

4-S 460 310,518 0.136 0.132 −5.31 −7.75

5-N 480 339,127 0.287 0.287 −3.99 −3.79

5-S 291 124,475 0.335 0.335 −12.76 −12.87

6 224 73,390 0.192 0.194 −69.47 −69.17

7-N 361 191,092 0.488 0.490 −46.86 −46.72

7-S 238 82,714 0.582 0.583 −44.70 −44.56

AVG4 estimators which were based on much smaller samples (n ≤ 9). In both cases, differences were less than

±2% (Fig. 8 and Fig. 10). These results differ somewhat

from those of Baker and Gibson (1987) who found that

the arithmetic average underestimated the true mean of

a lognormal variate, and that the maximum likelihood estimator was a better estimator of the mean when sample sizes were small. In the small samples that resulted

from using GAC data, both the MLE4 and AVG4 estimators had a slight tendency to underestimate the true mean

(AVG), as indicated by their small negative biases (Fig. 9,

and Figs. 11–14), but no significant difference was found

between the two estimators.

In the case of weekly and monthly means derived from

the SEEP II data, the MLE and AVG estimators again

proved to be nearly identical. The AVG estimator was

nominally the true mean, but since it was based on small

samples (7 days for weekly means and 31 or fewer days

for monthly means), it is not necessarily better than other

estimators of the mean.

Although the MLE and AVG estimators are equivalent

with respect to accuracy, it was recommended that the

MLE estimator be used because of its flexibility in allowing

the estimation of level-4 variables from saved statistics of

level-3 variables. In the remainder of this discussion, two

questions are raised regarding the equivalence of the MLE

and AVG estimators, and the answers discussed.

The first question is: How important is the assumption

59
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

that the variable is lognormally distributed? If the variable

being sampled is lognormally distributed, then the MLE

estimator (5) is the maximum likelihood estimator of the

mean, and the estimators for the standard deviation (23),

median (24), and mode (25) are also maximum likelihood

estimators of these parameters. But what if the underlying

distribution is not lognormally distributed? How robust

will the estimator be if the lognormal assumption is not

valid?

The empirical evidence based on CZCS data and the

SEEP II time series data supports the use of the MLE

estimator. These data sets taken as a whole, i.e., a whole

CZCS scene or a 16-month record from a single moored

fluorometer, were approximately lognormal or mixtures of

lognormal distributions. This was demonstrated for both

satellite and in situ CHL distributions (Figs. 4 and 17),

and observed for the other variables, but not shown. It

is not surprising, therefore, that small subsets drawn from

the whole data set behave as random samples drawn from

a lognormal distribution.

However, the binned data were not random samples.

Instead, they consisted of measurements made close together in space or time, and thus, they were correlated.

To the extent that the binned data are positively correlated, the intrabin variance will be less than the variance

of a random sample of the same size drawn from the whole

data set.

It is possible to show that the MLE estimator will be a

good approximation to the mean of any distribution with

s2 ≤ 0.5, where s2 is the variance of the natural logarithm

of the variable. This result is derived from the series expansion for the exponential function

ex =1+ x + x2

2! + x33! + ... (46)

Let X be any random variable whose distribution is

unknown. Define x = ln(X), and let m and s2 be the

mean and variance of x. Then

X = ex = eme(x −m)

, (47)

and the expected value of X is

E[X] = em E

!

1+(x −m) + (x −m)2

2!

+ (x −m)3

3! + ... "

= em E

!

1 + m2

2! + m33! + ... "

(48)

where mi denotes the ith central moment of x, defined by

mi ≡ E

#

(x −m)

i

$

. (49)

It is also noted that m1 = 0, and m2 = s2.

If the variance is less than or equal to 0.5, then the

terms involving higher order central moments will be a

rapidly decreasing series. In fact, the series in brackets in

(48) can be approximated by its first two terms

E[X] ≈ em

%

1 + s2

2

&

≈ eme

1

2 s2

. (50)

The term on the right is the MLE estimator of the mean.

Thus, there are two situations when the MLE estimator is

valid: 1) when the underlying distribution is lognormal, or

2) when the variance of the natural logarithm of the variable is less than or equal to 0.5 (or the standard deviation

of the base-10 logarithm is less than or equal to 0.3).

Figure 32 is a plot of the average variance of the logarithm of CHL within bins of size L×L plotted as a function

of L for the CZCS scenes 1–5. It is noted that the variance

within 9 × 9 km2 bins was less than 0.5 for all five scenes,

and the variance remained less than 0.5 as L increased up

to the maximum length of 480 km. In scene 4, variances

exceeded 0.5 at L greater than about 100 km.

The second question is: Under what circumstances do

the MLE and AVG estimators disagree, and is it possible

to predict the nature and magnitude of their differences?

In the study of cumulative means (Figs. 26–31), there

were examples shown where the MLE and AVG estimators began to diverge as the size of the sampling domain

increased. In one example (Figs. 26–27), the divergence

could be associated with bad data, and the conclusion

was that the MLE estimator was sensitive to anomalously

low values. The possibility that similar errors might affect

level-3 SeaWiFS data should be considered.

The discussion related to the first question suggests

another circumstance in which the MLE and AVG estimators might disagree: the situation where the variance of

the logarithm is large and the variable is not lognormally

distributed. A situation such as this would occur when the

sampling domain contains a mixture of lognormal distributions. In the case of spatial statistics, this would occur

in frontal areas between sharply contrasting water types,

e.g., high-chlorophyll waters mixing with low-chlorophyll

waters. It is likely to be more common in sampling domains covering longer time periods.

Most of the CZCS scenes can be modeled as mixtures

of lognormal distributions. Table 7 lists the means and

variances of lognormal distributions that were fit to modes

of the histograms shown in Fig. 4. Values of CHL derived

according to the CHL23 formula were excluded from the

fits. Note that within all modes, the variance was less

than 0.5. However, when two or more modes are mixed,

the variance of the mixture distribution will be increased

due to differences between modes.

It is possible to quantify errors associated with the

MLE estimator in the case of mixture distributions. An

60
0.0

0.2

0.4

0.6

0.8

1.0

1 10 100 1000

Scene #4

#2

#3

#5

#1

s2

L, km

9 km x 9 km

square bins

Average variance of ln(CHL)

within bins of size L x L

J.W. Campbell, J.M. Blaisdell, and M. Darzi

Table 7. Results of fitting normal distributions to the modes of the histograms of log(CHL) in the CZCS scenes

analyzed.

Scene Mode 1 Mode 2 Mode 3

Number m s2 m s2 m s2

1 −2.82 0.04

2 −2.98 0.02 −2.68 0.06

3 −2.41 0.04 −1.70 0.04

4 −2.35 0.06 −1.81 0.06 −1.06 0.16

5 −2.38 0.07 −1.25 0.08 −0.63 0.04

6 −2.65 0.09 −1.58 0.45 0.22 0.09

7 −1.78 0.15 −1.00 0.18 −0.07 0.15

Fig. 32. Average variance of ln(CHL) within areas of size L×L, as a function of L for CZCS scenes 1–5. Results

for scenes 6 and 7 were not obtained because of the discontinuity in the CHL distributions in these scenes, which

is an artifact of the bifurcated CZCS algorithm.

example is the case where there are two modes mixing in

a sampling domain. Let each mode be a lognormal distribution with parameters mi and s2

i where i = 1 or 2. If P

is the proportion of the distribution that is mode 1, then

the mean of the distribution is

Xavg = Pe

!

m1+ 1

2 s2

1

"

+ (1 − P)e

!

m2+ 1

2 s2

2

"

, (51)

and the MLE estimator is

Xmle = exp#P

$

m1 + s2

1

2

%

+ (1 − P)$m2 + s2

2

2

%

+ P(1 − P)(m1 − m2)2

2

&

. (52)

Relative errors for pair-wise mixtures of the modes listed in Table 7 are plotted against P in Fig. 33, where m1 <

m2, and P is the proportion of the lower-chlorophyll mode.

There are 14 curves shown in this figure, but only 5 have

errors that are significantly different from zero. The largest

errors (differences between MLE and AVG) occurred when

modes from scene 6 were mixed, and especially when mode

1 (mean CHL = 0.07 mg m−3) was mixed with mode 3

(mean CHL = 1.3 mg m−3). Of all the cases considered

here, the highest positive error (40%) occurred when 30%

of mode 1 was mixed with 70% of mode 3 in scene 6, and

the highest negative error (−30%) occurred when 90% of

mode 1 was mixed with 10% of mode 3.

The patterns shown here indicate that the MLE can either under or over estimate the true mean when there are

mixtures of lognormal distributions within the sampling

domain. The MLE estimator tended to exceed AVG for

low values of P, whereas AVG exceeded MLE for high val61
-30

-20

-10

0

10

20

30

40

50

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Scene 6:

Modes 1 & 3

Scene 6:

Modes 2 & 3

Scene 6:

Modes 1 & 2

Error (%)

Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Fig. 33. Relative errors (%) in the MLE estimator resulting from mixtures of two lognormal distributions,

plotted against P, the proportion of the mixture derived from the lower-chlorophyll mode. The 14 cases depicted

in this figure represent all pair-wise combinations of modes within the seven CZCS scenes (see Table 7).

ues of P. The former was the situation with the SEEP data

where there were a few low values in the data record that

caused MLE(n) > AVG(n) to diverge. Likewise, the opposite seemed to be the case with CZCS data, where there

were relatively few high values, e.g., values derived using

the CHL23 algorithm, that brought about divergences between MLE(L) < AVG(L).

The situations depicted in Fig. 33 may not be inclusive

of all possible mixtures that would occur in nature, but

they do span the range in the seven scenes analyzed. It is

clear that patterns are complex, and yet, reassuring that

with very few exceptions, errors were within ±10%.

3.4 Conclusions

The MLE estimator is a reasonably accurate estimator for the mean of CHL and other satellite-derived variables within sampling domains. It behaves as well as the

arithmetic average, and yet it has an advantage over the

AVG estimator in that it can be used to estimate means of

a large class of level-4 variables derived from the level-3

data. There were two situations that assure agreement between the AVG and MLE estimators: 1) if the variable is

lognormally distributed within the sampling domain, or 2)

if its variance is low. If the variance of its natural logarithm is less than 0.5, then AVG and MLE should agree

regardless of the underlying distribution.

Two circumstances were identified where the MLE and

AVG estimators are expected to disagree. One is the case

where there are anomalously low values in the data (presumably bad data), and the other is where the sampling domain contains a mixture of lognormal distributions. Based

on mixtures found in seven CZCS scenes spanning a wide

variety of ocean environments, relative errors would typically be within ±10%.

Acknowledgments

The editors wish to acknowledge the editorial assistance of

Barbara A. Leroux, who worked on this document during her

participation in the NASA/Goddard Prince George’s County

Teacher Intern Program.

Editorial Note

This document is presented as submitted with minor modifications to correct typographical or obvious clerical errors and to

maintain the established style of the SeaWiFS Technical Report

Series.

62
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Appendices

A. Equal-area Gridding Scheme for SeaWiFS Binned

Data

B. Scheme for Weighting Data

C. Algorithms for Binning and Interpreting SeaWiFS

Binned Data

Appendix A

Equal-area Gridding Scheme for SeaWiFS Binned Data

Introduction: This appendix† describes the equal-area gridding

scheme developed by the RSMAS Remote Sensing Group for

binned ocean fields. The same approach has been adopted for

AVHRR Ocean Pathfinder SST products and is proposed for

MODIS. The gridding scheme is based on that adopted by the

International Satellite Cloud Climatology Project (ISSCP).

This document does not motivate the need for an equal area

grid for SeaWiFS or other oceanographic products. Such motivation can be found in a paper by W. Rossow and L. Gardner

(1984). Furthermore, this document describes only the design

of the proposed equal-area grid, and does not discuss other related topics such as rules for spatially or temporally combining

observations into the equal-area bins.

Overview: The gridding scheme proposed consists of rectangular bins or tiles, arranged in zonal rows. A compromise

between data processing and storage capabilities, on one hand,

and the potential geophysical applications of satellite data, on

the other hand, suggest that a suitable minimum bin size would

be approximately 8–10 km on a side.

In the scheme proposed here, the tiles are approximately

9.28 km on a side. This size (9.28 km) was chosen because (a)

it has approximately the desired minimum resolution, and (b)

it results in 2,160 zonal rows of tiles from pole to pole, i.e., 1,080

in each hemisphere. This particular number of rows (2,160) has

some advantages which will be discussed in more detail below.

Because the total number of rows is even, the bins will never

straddle the equator, i.e., there will be an equal number of rows

above and below the equator. This avoids possible situations

where the Coriolis factor is zero, a characteristic that numerical

modellers expect from any gridding scheme adopted.

The total number of approximately 9 km bins is 5,940,422.

The bins or tiles are arranged in a series of zonal rows; the number of tiles per row varies. The rows immediately above and

below the equator have 4,320 tiles. This number is derived by

dividing the perimeter of the Earth at the equator by the standard tile size, i.e., 2πRe/9.28, where Re is the equatorial radius

of the Earth (Re = 6378.145 km). The number of tiles per row

decreases approximately as a cosine function as the rows get

closer to each pole (rigorously, there should be an adjustment

for ellipticity of the Earth, as the equatorial radius decreases

progressively to the smaller polar radius; this adjustment is

not applied in the current implementation). At the poles, the

number of tiles is always three. This special situation will be

discussed in detail below. The number of tiles per row as a

function of latitude is shown on Fig. A-1.

The number of bins in each zonal row is always an integer.

To ensure an integer number of bins, the width of each bin (the

size of a bin along a parallel, or x-length) must vary slightly

† This text is courtesy of the Remote Sensing Group, Rosenstiel School of Marine and Atmospheric Science, University

of Miami.

from row to row. The bins, however, are always 9.28 km long

along the meridians. That is, only one of the bin dimensions

changes. The size of the bins at each zonal row is established in

the following manner. First, a preliminary value for the number

of tiles (Np) at a given latitude (L) is computed as

Np = 2πr/X,

where X is the x-size of a bin at the equator (9.28 km) and r

is the radius of the circle produced by slicing the Earth with a

plane parallel to the equator at latitude L. The radius r can be

calculated as

r = Re cos(L),

where Re is the equatorial radius of the Earth. If the fractional

part of Np is greater than or equal to 0.5, then Np is rounded up

to the nearest integer, i.e., the final number of tiles will be the

integer portion of Np plus one; otherwise, Np is rounded down.

The final number of tiles is the integer portion of Np. Once

the final integer number of tiles along a row is calculated, the

x-size of the tiles must be adjusted. This is done by dividing

the perimeter of the row (2πr) by the integer number of tiles.

The result is the x-length (width) of a tile for a given row.

Because the x-length of the tiles is adjusted to ensure an integer number at each row, the equal area characteristics of this

binning scheme are not rigorously preserved. However, variations in tile size are negligible throughout most of the globe

and only become relevant at very high latitudes, where there

are fewer tiles per row, and any adjustments are more noticeable. As the number of tiles increases with distance from the

poles, the difference between tile sizes rapidly becomes practically unnoticeable. To provide an idea of the magnitude of the

fluctuations in tile size, the worst possible case occurs when half

a tile remains uncovered after filling a zonal row with an integer number of tiles. Once a row has 100 bins (approximately 16

rows, or 148 km from the poles), the worst possible difference

between the actual tile x-length and the standard x-length is of

the order of 0.5%, i.e., half a tile’s length redistributed among

about 100 tiles. For a tile of about 9 km a side, this represents

a difference in the x-length of about 45 m. Through a similar

calculation, a row with 50 bins (about 80 km away from the

poles) has a 1% variation with respect to the standard bin size.

The gridding scheme described here has an extremely useful

feature. The number of 9.28 km tiles in each hemisphere (1,080)

is divisible by many numbers (e.g., 2,3,4,5,6); and therefore, it is

extremely easy to generate an integer number of rows at many

useful spatial resolutions. For instance, 12 rows of approximately 9.28 km tiles can be combined to generate zonal bands

of 1◦ (1◦ of latitude is equal to 111.12 km; 12 bins would form

a band 111.20 km wide). Another example is the use of 30 rows

to generate zonal bands of 2.5◦, a typical output resolution of

atmospheric circulation models.

The poles: Both the North and South Poles are special cases

in the gridding scheme presented here. The pole areas are always covered by three tiles shaped like pie sectors. While the

meridional size of the polar bins (the y-length) will be the usual

9.28 km, the length of the bins along the arc of the sectors will

be slightly larger. Neglecting sphericity, the area encompassed

by the last row of tiles is πX2, where X = 9.28 km. If the

area of the circle is expressed as a rectangle of height X, the

remaining dimension is πX. If the perimeter is divided by three

(to yield three tiles), each tile will have dimensions X by πX/3

(approximately 1.05X). Thus, the bases of the triangular polar

63
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

tiles are about 5% larger than the x-length of the equatorial

tiles.

Binning software: Several routines have been developed to

perform the principal transformations required for binning and

mapping data, such as converting latitudes and longitudes into

bin numbers. Other routines perform the inverse transformation, i.e., given a bin number they return a latitude and longitude corresponding to the centroid of that bin. These routines

use a common initialization routine that must be executed prior

to calling the conversion routines.

Two numbering schemes are used internally, corresponding

to one- and two-dimensional (1-D and 2-D, respectively) accessing schemes. The 1-D scheme numbers all bins consecutively, beginning with 1 at the southernmost row and working

eastward from −180◦ around each circle of latitude. The 2-D

scheme uses a row number, from 1 to 2,160, and a number to

indicate its location within the row, beginning at 1 for each

row.

Variable Dictionary: The variables and their definitions for the

pseudocode are presented below.

NUMROWS The (integer) number of rows in the grid (equal to

2,160 for SeaWiFS).

BASEBIN An integer*4 array of size NUMROWS that contains

the index number of the first bin in each row.

NUMBIN An integer array of size NUMROWS containing the

total number of possible bins in each row.

LATBIN A real*4 array of size NUMROWS that contains the

center latitudes (decimal degrees) of the corresponding BASEBINs.

TOTBINS The (integer*4) number of possible bins in the grid

(equal to 5,940,422 for NUMROWS=2,160).

ROW The row number (integer); range is 1 to NUMROWS.

COL The bin number (integer); the range is from 1 to

NUMBIN(ROW).

IDX The bin index number (integer*4); range is 1 to

TOTBINS.

LAT The input latitude (real*4) for obtaining the corresponding bin’s ROW and COL, or IDX; or the output

latitude for a bin specified by ROW and COL, or IDX.

(The range for LAT is −90 to +90 decimal degrees.)

LON The input longitude (real*4) for obtaining the

corresponding bin’s ROW and COL, or IDX; or the

output longitude for a bin specified by ROW and

COL, or IDX. (The range for LAT is −180 to 180

decimal degrees.)

Pseudocode: The following pseudocode demonstrates the generation of the grid and the calculations for determining the

center latitude and longitude for a given bin and for identifying

a bin given a latitude and a longitude. The algorithms are illustrative in purpose and do not necessarily represent an optimal

implementation. They are based on software developed by J.

Brown, University of Miami.

#

# Set up NUMBIN and BASEBIN arrays

#

BASEBIN(1) = 1

do from ROW = 1 to NUMROWS

LATBIN(ROW) = ((ROW-0.5)*180.0/NUMROWS) - 90.0

NUMBIN(ROW) =

int((2*NUMROWS*cos_dbl_deg(LATBIN(ROW)))+0.5)

if ROW>1 then BASEBIN(ROW) = BASEBIN(ROW-1) + NUMBIN(ROW-1)

end do

TOTBINS = BASEBIN(NUMROWS) + NUMBIN(NUMROWS) - 1

#

# Identify bin from lat (-90 to +90) and lon (-180 to 180)

#

ROW = integer((90.0+LAT)*(NUMROWS/180.0)) + 1

ROW = minimum(ROW,NUMROWS)

LON = LON + 180.0

COL = integer(LON*NUMBIN(ROW)/360.0) + 1

COL = minimum(COL,NUMBIN(ROW))

IDX = BASEBIN(ROW) + COL - 1

#

# Get bin center lat/lon for given bin index

#

ROW = NUMROWS

IDX = maximum(IDX,1)

do while IDX<BASEBIN(ROW)

ROW = ROW - 1

end do

LAT = LATBIN(ROW)

LON = 360.0*(IDX-BASEBIN(ROW)+0.5)/NUMBIN(ROW)

LON = LON - 180.0

#

# Get bin center lat/lon for given bin row/column

#

LAT = LATBIN(ROW)

LON = 360.0*(COL-0.5)/NUMBIN(ROW)

LON = LON - 180.0

Appendix B

Scheme for Weighting Data

This appendix describes the scheme used to weight data from

different times (orbits) in producing temporal means and variances. The level-2 SeaWiFS data will be log-transformed before

the following schemes are applied. Note that the lower case letter x is used to denote the natural logarithm of the variable

X, that is, x = ln(X). The MLE estimator for the mean of

a lognormal variable X requires that the maximum likelihood

estimators of the mean and variance of x be obtained first.

The Textbook Case for Unweighted Data: If the data within a

sampling domain, xi, i=1,..., n, are independent and identically distributed normal random variables with a true mean μ

and variance σ2, then the sample mean

x = 1

n

!n

i=1

xi (B1)

is the maximum likelihood estimator of μ. The sample variance

is defined as

s

2 = 1

n

!n

i=1

(xi − x)

2 (B2)

and computed as

s

2 = 1

n

!n

i=1

xi

2 − x2 (B3)

is the maximum likelihood estimator of the variance, σ2. Note

that s2 is not the more common unbiased estimator of the variance which is obtained by multiplying (B3) by n/(n − 1). For

the specific case of SeaWiFS spatial statistics, i.e., for data

64
J.W. Campbell, J.M. Blaisdell, and M. Darzi

within a bin obtained during the same orbital pass, equations

(B1) and (B3) will be used to compute the mean and variance

of x = ln(X), for each variable X.

The General Case for Weighted Data: Let wi be the weight

given to the ith observation (data point). The weighted mean

and variance analogous to (B1) and (B3) are

x = 1

W

!n

i=1

wixi (B4)

s

2 = 1

W

!n

i=1

wix2

i − x2 (B5)

where W is the sum of the weights

W =

!n

i=1

wi. (B6)

The Specific Case for Weighted Data: How this applies to the

weighting of spatial statistics as they are binned over time are

considered here. In general, there will be N sets of spatial

statistics, each corresponding to a time ti, i=1,..., N, and each

set of spatial statistics will be based on ni observations from

the same orbital pass. To be obtained is a weighted mean and

variance of the data over observation xij where j refers to the

jth observation at time ti and i = 1,..., N, and j = 1,..., ni.

(Recall that xij = ln(Xij).

One approach would be to compute a mean, xi, and variance,

s2

i , for each set of spatial statistics, and then simply average

the means and variances over all times, ti = 1,..., N. If this

approach is used, the weights applied to each observation would

be

wij = 1

ni

. (B7)

It was decided that this gave too much weight to data sets

having few observations. The alternative is to weight all data

equally, but this gives too much weight to the data sets with

numerous observations. The compromise was to use

wij = 1

√ni

. (B8)

This is equivalent to weighting xi and s2

i by √ni.

The weighted mean and variance are

x = 1

W

!N

i=1

1

√ni

!ni

j=1

xij (B9)

s

2 = 1

W

!N

i=1

1

√ni

!ni

j=1

x2

ij − x2 (B10)

where the sum of the weights is

W = !N

i=1

!ni

j=1

1

√ni

=

!N

i=1

√ni. (B11)

Equations (B9) and (B10) will be used to obtain the temporal

statistics of ln(X) in each sampling domain.

Appendix C

Algorithms for Binning and Interpreting

SeaWiFS Binned Data

Three algorithms are described and their pseudocodes presented

in this appendix: the Space Binner algorithm bins data from a

single scene; the Time Binner algorithm bins output from the

Space Binner (or from the Time Binner) to accumulate sums

over binning periods; and the Bin Data Interpreter is used to

interpret binned data products to derive the mean, standard deviation, median and mode of level-3 data. Only GAC data will

be binned operationally to generate archived level-3 products.

Spatial Binning Algorithm: The spatial binning algorithm is

applied to the level-2 GAC scenes. In general, there will be

one set of spatial statistics created for each scene. The only

exception will be when an orbit crosses 180◦ longitude, in which

case there will be two sets of spatial statistics corresponding to

different days.

Let Xji be an acceptable observation of the variable Xj in pixel

i, and let LON(i) and LAT(i) be the longitude and latitude at the

center of pixel i. (A pixel will be considered to have acceptable level-2 data if it passes screens for sun glint, clouds and

other masks, in which case all of the variables will be considered

acceptable.) From these coordinates, the bin index number b

will be determined according to known relationships (see Appendix A).

Then for each variable j, the natural logarithm LOGX = ln(Xji)

is obtained, and the following sums incremented

SUMX(b,j) = SUMX(b,j) + LOGX (C1)

and

SUMXX(b,j) = SUMXX(b,j) + LOGX × LOGX . (C2)

In addition, the number of pixels contributing to the sums in

bin b is incremented

N(b) = N(b)+1, (C3)

and a binary-valued variable is set to 1 to indicate bin b contains

data

NSEG(b)=1. (C4)

After processing all valid data from this scene, the total weight

for each bin is computed

W(b) = "N(b), (C5)

and the variable sums are weighted as per (B9) and (B10)

SUMX(b,j) = SUMX(b,j)

W(b) (C6)

and

SUMXX(b,j) = SUMXX(b,j)

W(b) . (C7)

Finally, a 16-bit number TT(b) is defined for each bin. This

number will be used in subsequent stages of temporal binning

to indicate the temporal distribution of the data. In the spatial

65
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

binning algorithm, all bits in TT(b) will be 0 except the lowest

bit which will be set to 1 if there are data in bin b.

The output from the spatial binning algorithm consists of the

spatial statistics for each bin: b, N(b), NSEG(b), W(b), TT(b),

and a pair of weighted sums, SUMX(b,j) and SUMXX(b,j), for

each variable j.

Space Binner Code: This program takes one level-2 scene as

input and bins it into one (or two, if the level-2 scene crosses

180◦ longitude) level-3 binned data product as output. This

is called spatial binning since the bins are of lower resolution

and the level-2 product is considered to represent a snapshot,

i.e., no time averaging occurs, of the Earth’s surface. Products

generated by this program are not archived but are used as

input to the time binner.

Variable and Constant Dictionary: The variables and their definitions for the pseudocode are presented below.

Constants

MAXBINS The maximum number of bins (5,940,422).

NVARS The number of derived level-3 geophysical variables whose observational values are stored in the

associated SUMX and SUMXX pairs.

Level-2 Variables

NPIXELS The number of pixels in a scan line of the input

level-2 product.

NSCANS The number of scan lines in the input level-2 product.

PXLAT A real*4 1-D array of size NPIXELS; represents the

latitude for a given pixel I of a given scan line L.

PXLON A real*4 1-D array of size NPIXELS; represents the

longitude for a given pixel I of a given scan line L.

OBS A real*4 2-D array of size NPIXELS×NVARS; represents the derived level-2 values that are to be

binned into the level-3 product for a given pixel I

of a given scan line L.

Output Variables

SUMX A real*4 2-D array of size MAXBINS×NVARS; represents the sum of the natural logarithm of the

level-3 geophysical variable’s values divided by the

square root of the number of those values for a

given bin IDX; saved in the output product if, and

only if, N(IDX) is greater than zero.

SUMXX A real*4 2-D array of size MAXBINS×NVARS; represents the sum of squares of the natural logarithm

of the level-3 geophysical variable’s values divided

by the square root of the number of those values

for a given bin IDX; saved in the output product if,

and only if, N(IDX) is greater than zero.

N An integer*2 1-D array of size MAXBINS; represents the number of values summed into SUMX and

SUMXX for all variables (Js) and for a given bin IDX;

saved in the output product if, and only if, N(IDX)

is greater than zero.

NSEG An integer*2 1-D array of size MAXBINS; represents the number of level-2 scenes which contribu

ted to SUMX and SUMXX for all Js for a given bin IDX;

saved in the output product if, and only if, N(IDX)

is greater than zero. For this program, since only

one scene is input, all saved values of NSEG will be

1.

W A real*4 1-D array of size MAXBINS; represents the

weight factor for all Js for a given bin IDX; calculated as the square root of N(IDX); saved in the

output product if, and only if, N(IDX) is greater

than zero.

TT An integer*2 1-D array of size MAXBINS; the bit

values of TT represent the time trend of the values

summed into SUMX and SUMXX for all Js for a given

bin IDX; saved in the output product if, and only

if, N(IDX) is greater than zero. For this program,

since only one scene is input, all saved values of TT

will have the lowest bit only set to 1.

IDX An integer*4 word representing the index number

of each bin with a value ranging from 1 to MAXBINS;

saved in the output product if, and only if, N(IDX)

is greater than 0.

Note: For each N(IDX)> 0, 8×NVARS+14 bytes of information will be output.

Other Variables

I Counter index of pixels on a scan line. Range is

from 1 to NPIXELS.

J Counter index of geophysical variables to be binned.

Range is from 1 to NVARS.

L Counter index of scan lines. Range is from 1 to

NSCANS.

XLOG Natural logarithm (real*4) of OBS for a given I

and J.

#

# Initialize

#

do from IDX=1 to MAXBINS

do from J=1 to NVARS

SUMX(IDX,J) = 0.0

SUMXX(IDX,J) = 0.0

end do

N(IDX) = 0

NSEG(IDX) = 0

TT(IDX) = 0

end do

read from level-2 scene: NPIXELS, NSCANS

#

# Input level-2 scene and accumulate stats for each bin

#

#

do from L=1 to NSCANS

read arrays PXLAT, PXLON, OBS for scan line L

do from I=1 to NPIXELS

if sample I passes screen flags then

IDX = get_bin_index(PXLAT(I),PXLON(I))

do from J=1 to NVARS

XLOG = natural_log(OBS(I,J))

SUMX(IDX,J) = SUMX(IDX,J) + XLOG

SUMXX(IDX,J) = SUMXX(IDX,J) + XLOG*XLOG

end do

N(IDX) = N(IDX) + 1

NSEG(IDX) = 1

end if

end do

end do

#

66
J.W. Campbell, J.M. Blaisdell, and M. Darzi

# Divide sums by weight and output space binned product

#

do from IDX=1 to MAXBINS

if N(IDX) > 0 then

set lowest bit of TT(IDX)

W(IDX) = square_root(N(IDX))

do from J=1 to NVARS

SUMX(IDX,J) = SUMX(IDX,J)/W(IDX)

SUMXX(IDX,J) = SUMXX(IDX,J)/W(IDX)

end do

write to space binned level-3 product:

IDX, N(IDX), NSEG(IDX), W(IDX), TT(IDX)

SUMX(IDX,J), SUMXX(IDX,J), for J=1 to NVARS

end write

end if

end do

Temporal Binning Algorithm: The temporal binning algorithm

combines the appropriate spatial statistics within each sampling

domain. The sampling domain for a particular bin will be either

a day, week, month, or year.

For each set of spatial statistics there is an associated time

t. The output from the spatial algorithm at time t will be the

input for the temporal binning algorithm. Let this input be

indexed by the time t: N(b)t, NSEG(b)t, W(b)t, TT(b)t, and

the pairs of weighted sums, SUMX(b,j)t and SUMXX(b,j)t, for

each variable Xj.

If N(b)t > 0, then the temporal sums

SUMX(b,j) = SUMX(b,j) + SUMX(b,j)t (C8)

and

SUMXX(b,j) = SUMXX(b,j) + SUMXX(b,j)t (C9)

are incremented for each variable j. In addition, the number of

pixels contributing to the sums is counted

N(b) = N(b) + N(b)t (C10)

and the number of spatial data sets (orbits) contributing to the

sums

NSEG(b) = NSEG(b) + NSEG(b)t. (C11)

The sum of weights is computed

W(b) = W(b) + W(b)t. (C12)

and the appropriate bit of the time distribution variable TT(b)

is set to 1 to reflect that data were present at time t in bin b.

Output from the temporal binning algorithm consists of the

level-3 data for each bin: b, N(b), NSEG(b), W(b), TT(b), and

a pair of weighted sums, SUMX(b,j) and SUMXX(b,j), for each

variable j. Note that the output from the temporal binning

algorithm is in the same form as its input. In fact, daily binned

products can serve as input to the temporal binning algorithm

to produce weekly, monthly, or longer-term products.

Time Binner Code: This program takes as input level-3 binned

segment products produced by the space binner and combines

them into a binned product representing one day or takes binned

products produced by the time binner (this program) and combines them into longer-term binned products. This process is

called temporal binning since it combines data over a certain

time period while not changing their spatial resolution.

Variable and Constant Dictionary: The variables and their definitions for the pseudocode are presented below.

Constants

MAXBINS The maximum number of bins (5,940,422).

NVARS The number of derived level-3 geophysical variables whose observational values are stored in the

associated SUMX and SUMXX pairs.

Input Variables

NBINS The number of bins to read from an input level-3

product.

SUMX INPUT A real*4 1-D array of size NVARS; represents

SUMX as output by the space or time binner for

all level-3 geophysical variables (Js) of a given bin

IDX being read.

SUMXX INPUT A real*4 1-D array of size NVARS; represents SUMXX

as output by the space or time binner for all level-3

geophysical variables (Js) of a given bin IDX being

read.

N INPUT An integer*2 word; represents N as output by the

space or time binner for a given bin IDX being read.

NSEG INPUT An integer*2 word; represents NSEG as output by

the space or time binner for a given bin IDX being

read.

W INPUT A real*4 word; represents W as output by the space

or time binner for a given bin IDX being read.

TT INPUT An integer*2 word; represents TT as output by

the space or time binner for a given bin IDX being

read.

IDX An integer*4 word representing the index number of the bin being read from an input level-3

product.

Output Variables

SUMX A real*4 2-D array of size MAXBINS×NVARS; represents the sum of the SUMX_INPUT for the level-3

geophysical variables (Js) from all input products

for a given bin IDX; saved in the output product if,

and only if, N(IDX) is greater than zero.

SUMXX A real*4 2-D array of size MAXBINS×NVARS; represents the sum of the SUMXX_INPUT for the level-3

geophysical variables (Js) from all input products

for a given bin IDX; saved in the output product if,

and only if, N(IDX) is greater than zero.

N An integer*2 1-D array of size MAXBINS; represents the sum of the N_INPUT from all input products for a given bin IDX; saved in the output product if, and only if, N(IDX) is greater than zero.

NSEG An integer*2 1-D array of size MAXBINS; represents the sum of the NSEG_INPUT from all input

products for a given bin IDX; saved in the output

product if, and only if, N(IDX) is greater than zero.

W A real*4 1-D array of size MAXBINS; represents the

sum of the W_INPUT from all input products for a

given bin IDX; saved in the output product if, and

only if, N(IDX) is greater than zero.

TT An integer*2 1-D array of size MAXBINS; the bit

sequence of TT represent the time trend of the values summed into SUMX and SUMXX for all Js for a

given bin IDX; saved in the output product if, and

67
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

only if, N(IDX) is greater than zero. The bits represent consecutive time in the binning period, the

lowest bit being the earliest time. For daily binned

products, the bits correspond to the relative sequence of orbits binned. For 8-day products, each

bit represents one day; for monthly products, each

bit represents two days; and for yearly products,

each bit represents one month. A TT(IDX) bit will

be set to 1 only if data, for the time corresponding

to that bit, were binned in bin IDX.

IDX An integer*4 word representing the index number

of each bin with a value ranging from 1 to MAXBINS;

saved in the output product if, and only if, N(IDX)

is greater than 0.

Note: For each N(IDX)> 0, 8×NVARS+14 bytes of information will be output.

Other Variables

J Counter index of geophysical variable to be binned.

Range is from 1 to NVARS.

B Counter index of bins read from input product.

Range is from 1 to NBINS.

#

# Initialize

#

do from IDX=1 to MAXBINS

do from J=1 to NVARS

SUMX(IDX,J) = 0.0

SUMXX(IDX,J) = 0.0

end do

N(IDX) = 0

NSEG(IDX) = 0

W(IDX) = 0.0

end do

#

# Input space or time binned products and accumulate

# statistics for each bin

#

do for each binned input product

read from metadata of binned input products: NBINS

do from B=1 to NBINS

read from bin B:

IDX, N_INPUT, NSEG_INPUT, W_INPUT, TT_INPUT

SUMX_INPUT(J), SUMXX_INPUT(J), for J=1 to NVARS

end read

do from J=1 to NVARS

SUMX(IDX,J) = SUMX(IDX,J) + SUMX_INPUT(J)

SUMXX(IDX,J) = SUMXX(IDX,J) + SUMXX_INPUT(J)

end do

N(IDX) = N(IDX) + N_INPUT

NSEG(IDX) = NSEG(IDX) + NSEG_INPUT

W(IDX) = W(IDX) + W_INPUT

use TT_INPUT, date, or orbit of input to set TT(IDX)

end do

end do

#

# Output time binned product

#

do from IDX=1 to MAXBINS

if N(IDX) > 0 then

write to time binned level-3 product

IDX, N(IDX), NSEG(IDX), W(IDX), TT(IDX)

SUMX(IDX,J), SUMXX(IDX,J), for J=1 to NVARS

end write

end if

end do

Algorithms for Calculating Statistics of Level-3 Variables: The

means, variances, medians, and modes can be estimated using

the level-3 data as described in Section 2.3. Here the same

equations are described in terms of the pseudocode logic used

in this Appendix. The level-3 data provided for each bin are:

b, N(b), NSEG(b), W(b), TT(b), and a pair of weighted sums,

SUMX(b,j) and SUMXX(b,j), for each level-3 variable j.

For each variable Xj, the mean and variance of its natural logarithm are calculated

mx = SUMX(b,j)

W(b) (C13)

and

s

2

x = SUMXX(b,j)

W(b) − m2

x. (C14)

The MLE estimator for the mean of Xj in bin b is

X(b,j) = e

(mx+ 1

2 s2

x) (C15)

and the standard deviation of Xj is estimated by

SD(b,j) = X(b,j)

!

e

s2

x − 1

" 1

2

, (C16)

and [SD(b,j)]

2 is the estimated variance.

Assuming the distribution of Xj is approximately lognormal,

then the median can be estimated by

Xmed(b,j) = e

mx , (C17)

and the mode (most common value) by

Xmod(b,j) = e

(mx−s2

x). (C18)

Bin Data Interpreter Code: This program interprets the geophysical data from binned products created by the space binner

or the time binner. It will calculate the maximum likelihood

estimate (MLE) of the mean, standard deviation, median, and

mode for each level-3 binned geophysical variable.

Variable and Constant Dictionary: The variables and their definitions for the pseudocode presented are below.

Constants

NVARS The number of derived level-3 geophysical variables whose observational values are stored in the

associated SUM_INPUT and SUMXX_INPUT pairs.

Level-3 Input Variables

NBINS The number of bins to read from an input level-3

product.

SUMX INPUT A real*4 1-D array of size NVARS; represents SUMX

as output by the space or time binner for all

level-3 geophysical variables (Js) of a given bin

IDX being read.

SUMXX INPUT A real*4 1-D array of size NVARS; represents SUMXX

as output by the space or time binner for all

level-3 geophysical variables (Js) of a given bin

IDX being read.

68
J.W. Campbell, J.M. Blaisdell, and M. Darzi

N INPUT An integer*2 word; represents N as output by

the space or time binner for a given bin IDX being

read.

NSEG INPUT An integer*2 word; represents NSEG as output

by the space or time binner for a given bin IDX

being read.

W INPUT A real word; represents W as output by the space

or time binner for a given bin IDX being read.

TT INPUT An integer*2 word; represents TT as output by

the space or time binner for a given bin IDX being

read.

IDX An integer*4 word representing the index number of the bin being read from the input level-3

product.

Output Variables

XMEAN A real*4 1-D array of size NVARS; represents the

mean of the weighted cumulative values of the

level-3 geophysical variables (Js).

SIGMA A real*4 1-D array of size NVARS; represents the

standard deviation for the weighted cumulative

values of the level-3 geophysical variables (Js).

XMEDN A real*4 1-D array of size NVARS; represents the

median of the weighted cumulative values of the

level-3 geophysical variables (Js).

XMODE A real*4 1-D array of size NVARS; represents the

mode of the weighted cumulative values of the

level-3 geophysical variables (Js).

N INPUT An integer*2 word; represents N as output by

the space or time binner for a given bin IDX being

output.

NSEG INPUT An integer*2 word; represents NSEG as output

by the space or time binner for a given bin IDX

being output.

TT INPUT An integer*2 word; represents TT as output by

the space or time binner for a given bin IDX being

output.

IDX An integer*4 word representing the index number of the bin being output.

Other Variables

B Counter index of bins read from input product.

Range is from 1 to NBINS.

J Counter index of geophysical variables that have

been binned. Range is from 1 to NVARS.

AVLOGS A real*4 word that represents the mean of the

weighted logs for a geophysical variable J of bin B

being processed. Used to calculate XMEAN, SIGMA,

XMEDN, and XMODE.

VRLOGS A real*4 word that represents the variance of the

weighted logs for a geophysical variable J of bin B

being processed. Used to calculate XMEAN, SIGMA,

and XMODE.

#

# Input information for each bin

#

read from metadata of binned input products: NBINS

do from B=1 to NBINS

read from bin B:

IDX, N_INPUT, NSEG_INPUT, W_INPUT, TT_INPUT

SUMX_INPUT(J), SUMXX_INPUT(J), for J=1 to NVARS

end read

#

# Calc. mean, std.dev., median and mode, and then output

#

do from J=1 to NVARS

AVLOGS = SUMX_INPUT(J)/W_INPUT

VRLOGS = (SUMXX_INPUT(J)/W_INPUT) - (AVLOGS*AVLOGS)

XMEAN(J) = exponential(AVLOGS + (VRLOGS/2.))

SIGMA(J) = XMEAN(J)*sqroot(exponential(VRLOGS) - 1)

XMEDN(J) = exponential(AVLOGS)

XMODE(J) = exponential(AVLOGS - VRLOGS)

end do

write to screen or file useful info for bin IDX

IDX, N_INPUT, NSEG_INPUT, TT_INPUT

XMEAN(J), SIGMA(J), XMEDN(J), XMODE(J), for J=1 to NVARS

end write

end do

Glossary

AVHRR Advanced Very High Resolution Radiometer

CZCS Coastal Zone Color Scanner

DSP Not an acronym; the name of a software package

developed at RSMAS.

GAC Global Area Coverage

GMT Greenwich Mean Time

HRPT High Resolution Picture Transmission

IFOV Instantaneous Field-of-View

ISCCP International Satellite Cloud Climatology Project

LAC Local Area Coverage

MARMAP Marine Resources Monitoring, Assessment, and Prediction

MODIS Moderate Resolution Imaging Spectroradiometer

RSMAS Rosenstiel School of Marine and Atmospheric Science

SeaWiFS Sea-viewing Wide Field-of-view Sensor

SEEP Shelf Edge Exchange Program

SST Sea Surface Temperature

TDI Time Delay and Integration

Symbols

Ag CZCS pigment algorithm constant (global).

Ar CZCS pigment algorithm constant (regional).

AVG Arithmetic average based on LAC data.

AVG4 Arithmetic average based on GAC data.

b Bin index number.

Bg CZCS pigment algorithm constant (global).

Br CZCS pigment algorithm constant (regional).

!Chl"tot Integral euphotic chlorophyll.

CHL Chlorophyll concentration.

CHL13 Pigment concentration calculated from CZCS bands

1 and 3.

CHL23 Pigment concentration calculated from CZCS bands

2 and 3.

69
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

DIFF1 Relative difference between MLE4 and AVG4.

DIFF2 Relative difference between MED4 and AVG4.

E[X] Expected value of x.

ERROR Relative error, in percent, of the estimated mean

from the arithmetic mean.

FNC Function of vector variable X using LAC data.

FNC4 Function of vector variable X using GAC data.

ICK Integrated chlorophyll concentration over the first

optical depth.

ID Mooring identification number.

K490 The diffuse attenuation coefficient at 490 nm.

L Bin dimension in kilometers.

LW N (λi) Normalized water-leaving radiances in i bands (1–

5).

La(λi) Atmospheric aerosol radiances in i bands (6–8).

LW Water-leaving radiance.

mi Central moment of x.

mx Sample mean of the natural logarithm of x.

my Sample mean of the natural logarithm of y.

mr Sample mean of regional ln (pigment).

MLE Maximum likelihood estimator of LAC data.

MLE4 Maximum likelihood estimator of GAC data.

MED Geometric mean or median of LAC data.

MED4 Geometric mean or median of GAC data.

n Sample size.

ni The number of pixels per bin on orbit i.

n The number of days used for temporal averaging.

N The number of orbits contributing to the temporal

mean.

P The proportion of the distribution that is mode 1.

PIG CZCS pigment-like concentration.

PIGr Pigment calculated with regionally-derived parameters.

s2

r The sample variance of regional ln (pigment).

s2

x The sample variance of the natural logarithm of x.

s2

y The sample variance of the natural logarithm of y.

S1 The weighted sum of variable x.

S2 The weighted sum of variable y.

SDx The standard deviation of x.

SDy The standard deviation of y.

ti The time at which orbit i was acquired.

T A 16-bit time distribution variable.

V The 8-bit image value of a pixel, i.e., gray level.

wi The weight factor for orbit i.

W The sum of the weighting factors.

x The natural logarithm of X.

X Any random variable whose distribution is unknown.

X A level-2 variable.

X The true mean of a level-2 variable.

Xavg The arithmetic average of X.

Xgeom The geometric mean of X.

Xmle The maximum likelihood estimator of X.

Xmed The median of X.

Xmod The mode of X.

Xest The estimated mean of X.

X The vector of standard level-2 variables.

Y Any function of a level-2 variable X.

Y The true mean of Y .

Y avg The arithmetic average of Y .

Y mle The maximum likelihood estimator of Y .

Y fnc The arithmetic mean of FNC.

Ze The euphotic depth (depth to 1% light level).

λ1 Wavelength of 440 nm.

λ2 Wavelength of 520 nm.

λ3 Wavelength of 550 nm.

τa(865) The aerosol optical thickness at 865 nm.

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J.W. Campbell, J.M. Blaisdell, and M. Darzi

, D.K. Clark, J.W. Brown, O.B. Brown, R.H. Evans, and

W.W. Broenkow, 1983: Phytoplankton pigment concentrations in the Middle Atlantic Bight: Comparison of ship

determinations and CZCS estimates. Appl. Opt., 22, 20–

36.

, and D.J. Casta ̃no, 1987: Coastal Zone Color Scanner atmospheric correction algorithm: multiple scattering effects.

Appl. Opt., 26, 2,111–2,122.

, O.B. Brown, R.H. Evans, J.W. Brown, R.C. Smith, K.S.

Baker, and D.K. Clark, 1988: A semianalytic radiance model of ocean color. J. Geophys. Res., 93, 10,909–

10,924.

Journal, A.G., 1989: Fundamentals of Geostatistics in Five

Lessons, Short Course. Geology: Vol. 8, American Geophysical Union, Washington, D.C., 40 pp.

Kirk, J.T.O., 1983: Light and Photosynthesis in Aquatic Ecosystems. Cambridge University Press, Cambridge, 401 pp.

Lewis M.R., N. Kuring, and C.S. Yentsch, 1988: Global patterns of ocean transparency: Implications for the new production of the open ocean. J. Geophys. Res., 93, 6,847–

6,856.

McClain, C.R., E-n. Yeh, and G. Fu, 1992: An Analysis of GAC

Sampling Algorithms: A Case Study. NASA Technical

Memorandum 104566, Vol. 4, S.B. Hooker and E.R. Firestone, Eds., NASA Goddard Space Flight Center, Greenbelt, Maryland, 21 pp.

Medeiros, W.H., and C.D. Wirick, 1992: SEEP II: Shelf Edge

Exchange Processes II. Chlorophyll a Fluorescence, Temperature, and Beam Attenuation Measurements from

Moored Fluorometers. BNL 47211 Informal Report,

Brookhaven National Laboratory, Upton, New York,

205 pp.

Morel, A., and J.-F. Berthon, 1989: Surface pigments, algal

biomass profiles, and potential production of the euphotic

layer: Relationships reinvestigated in view of remotesensing applications. Limnol. Oceanogr., 34, 1,545–1,562.

Platt, T., 1986: Primary production of the ocean water column as a function of surface light intensity: Algorithms

for remote sensing. Deep-Sea Res., 33, 1–15.

, and S. Sathyendranath, 1988: Oceanic primary production: estimation by remote sensing at local and regional

scales. Science, 241, 1,613–1,620.

RSMAS, 1990: DSP User’s Manual. Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami,

Florida, 255 pp.

Smith, R.C., R.W. Eppley, and K.S. Baker, 1982: Correlation

of primary production as measured aboard ship in southern

California coastal waters and as estimated from satellite

chlorophyll images. Mar. Biol., 66, 281–288.

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waters and remote sensing. Limnol. Oceanogr., 23, 247–

259.

Yentsch, C.S., 1990: Estimates of ‘new production’ in the MidNorth Atlantic. J. Plankton Res., 12, 717–734.

The SeaWiFS Technical Report Series

Vol. 1

Hooker, S.B., W.E. Esaias, G.C. Feldman, W.W. Gregg, and

C.R. McClain, 1992: An Overview of SeaWiFS and Ocean

Color. NASA Tech. Memo. 104566, Vol. 1, S.B. Hooker

and E.R. Firestone, Eds., NASA Goddard Space Flight

Center, Greenbelt, Maryland, 24 pp., plus color plates.

Vol. 2

Gregg, W.W., 1992: Analysis of Orbit Selection for SeaWiFS:

Ascending vs. Descending Node. NASA Tech. Memo.

104566, Vol. 2, S.B. Hooker and E.R. Firestone, Eds.,

NASA Goddard Space Flight Center, Greenbelt, Maryland, 16 pp.

Vol. 3

McClain, C.R., W.E. Esaias, W. Barnes, B. Guenther, D. Endres, S.B. Hooker, G. Mitchell, and R. Barnes, 1992: Calibration and Validation Plan for SeaWiFS. NASA Tech.

Memo. 104566, Vol. 3, S.B. Hooker and E.R. Firestone,

Eds., NASA Goddard Space Flight Center, Greenbelt,

Maryland, 41 pp.

Vol. 4

McClain, C.R., E. Yeh, and G. Fu, 1992: An Analysis of GAC

Sampling Algorithms: A Case Study. NASA Tech. Memo.

104566, Vol. 4, S.B. Hooker and E.R. Firestone, Eds.,

NASA Goddard Space Flight Center, Greenbelt, Maryland, 22 pp., plus color plates.

Vol. 5

Mueller, J.L., and R.W. Austin, 1992: Ocean Optics Protocols

for SeaWiFS Validation. NASA Tech. Memo. 104566, Vol.

5, S.B. Hooker and E.R. Firestone, Eds., NASA Goddard

Space Flight Center, Greenbelt, Maryland, 43 pp.

Vol. 6

Firestone, E.R., and S.B. Hooker, 1992: SeaWiFS Technical

Report Series Summary Index: Volumes 1–5. NASA Tech.

Memo. 104566, Vol. 6, S.B. Hooker and E.R. Firestone,

Eds., NASA Goddard Space Flight Center, Greenbelt,

Maryland, 9 pp.

Vol. 7

Darzi, M., 1992: Cloud Screening for Polar Orbiting Visible

and IR Satellite Sensors. NASA Tech. Memo. 104566, Vol.

7, S.B. Hooker and E.R. Firestone, Eds., NASA Goddard

Space Flight Center, Greenbelt, Maryland, 7 pp.

Vol. 8

Hooker, S.B., W.E. Esaias, and L.A. Rexrode, 1993: Proceedings of the First SeaWiFS Science Team Meeting. NASA

Tech. Memo. 104566, Vol. 8, S.B. Hooker and E.R. Firestone, Eds., NASA Goddard Space Flight Center, Greenbelt, Maryland, 61 pp.

Vol. 9

Gregg, W.W., F.C. Chen, A.L. Mezaache, J.D. Chen, J.A.

Whiting, 1993: The Simulated SeaWiFS Data Set, Version 1. NASA Tech. Memo. 104566, Vol. 9, S.B. Hooker,

E.R. Firestone, and A.W. Indest, Eds., NASA Goddard

Space Flight Center, Greenbelt, Maryland, 17 pp.

Vol. 10

Woodward, R.H., R.A. Barnes, C.R. McClain, W.E. Esaias,

W.L. Barnes, and A.T. Mecherikunnel, 1993: Modeling

of the SeaWiFS Solar and Lunar Observations. NASA

Tech. Memo. 104566, Vol. 10, S.B. Hooker and E.R. Firestone, Eds., NASA Goddard Space Flight Center, Greenbelt, Maryland, 26 pp.

71
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

Vol. 11

Patt, F.S., C.M. Hoisington, W.W. Gregg, and P.L. Coronado,

1993: Analysis of Selected Orbit Propagation Models for

the SeaWiFS Mission. NASA Tech. Memo. 104566, Vol.

11, S.B. Hooker, E.R. Firestone, and A.W. Indest, Eds.,

NASA Goddard Space Flight Center, Greenbelt, Maryland, 16 pp.

Vol. 12

Firestone, E.R., and S.B. Hooker, 1993: SeaWiFS Technical Report Series Summary Index: Volumes 1–11. NASA Tech.

Memo. 104566, Vol. 12, S.B. Hooker and E.R. Firestone,

Eds., NASA Goddard Space Flight Center, Greenbelt,

Maryland, 28 pp.

Vol. 13

McClain, C.R., K.R. Arrigo, J. Comiso, R. Fraser, M. Darzi,

J.K. Firestone, B. Schieber, E-n. Yeh, and C.W. Sullivan, 1994: Case Studies for SeaWiFS Calibration and Validation, Part 1. NASA Tech. Memo. 104566, Vol. 13,

S.B. Hooker and E.R. Firestone, Eds., NASA Goddard

Space Flight Center, Greenbelt, Maryland, 52 pp., plus

color plates.

Vol. 14

Mueller, J.L., 1993: The First SeaWiFS Intercalibration RoundRobin Experiment, SIRREX-1, July 1992. NASA Tech.

Memo. 104566, Vol. 14, S.B. Hooker and E.R. Firestone,

Eds., NASA Goddard Space Flight Center, Greenbelt,

Maryland, 60 pp.

Vol. 15

Gregg, W.W., F.S. Patt, and R.H. Woodward, 1994: The Simulated SeaWiFS Data Set, Version 2. NASA Tech. Memo.

104566, Vol. 15, S.B. Hooker and E.R. Firestone, Eds.,

NASA Goddard Space Flight Center, Greenbelt, Maryland, 42 pp., plus color plates.

Vol. 16

Mueller, J.L., B.C. Johnson, C.L. Cromer, J.W. Cooper, J.T.

McLean, S.B. Hooker, and T.L. Westphal, 1994: The Second SeaWiFS Intercalibration Round-Robin Experiment,

SIRREX-2, June 1993. NASA Tech. Memo. 104566, Vol.

16, S.B. Hooker and E.R. Firestone, Eds., NASA Goddard

Space Flight Center, Greenbelt, Maryland, 121 pp.

Vol. 17

Abbott, M.R., O.B. Brown, H.R. Gordon, K.L. Carder, R.E.

Evans, F.E. Muller-Karger, and W.E. Esaias, 1994: Ocean

Color in the 21st Century: A Strategy for a 20 -Year Time

Series. NASA Tech. Memo. 104566, Vol. 17, S.B. Hooker

and E.R. Firestone, Eds., NASA Goddard Space Flight

Center, Greenbelt, Maryland, 20 pp.

Vol. 18

Firestone, E.R., and S.B. Hooker, 1995: SeaWiFS Technical Report Series Summary Index: Volumes 1–17. NASA Tech.

Memo. 104566, Vol. 18, S.B. Hooker and E.R. Firestone,

Eds., NASA Goddard Space Flight Center, Greenbelt,

Maryland, 47 pp.

Vol. 19

McClain, C.R., R.S. Fraser, J.T. McLean, M. Darzi, J.K. Firestone, F.S. Patt, B.D. Schieber, R.H. Woodward, E-n. Yeh,

S. Mattoo, S.F. Biggar, P.N. Slater, K.J. Thome, A.W.

Holmes, R.A. Barnes, and K.J. Voss, 1994: Case Studies

for SeaWiFS Calibration and Validation, Part 2. NASA

Tech. Memo. 104566, Vol. 19, S.B. Hooker, E.R. Firestone,

and J.G. Acker, Eds., NASA Goddard Space Flight Center,

Greenbelt, Maryland, 73 pp.

Vol. 20

Hooker, S.B., C.R. McClain, J.K. Firestone, T.L. Westphal,

E-n. Yeh, and Y. Ge, 1994: The SeaWiFS Bio-Optical

Archive and Storage System (SeaBASS), Part 1. NASA

Tech. Memo. 104566, Vol. 20, S.B. Hooker and E.R. Firestone, Eds., NASA Goddard Space Flight Center, Greenbelt, Maryland, 40 pp.

Vol. 21

Acker, J.G., 1994: The Heritage of SeaWiFS: A Retrospective on the CZCS NIMBUS Experiment Team (NET) Program. NASA Tech. Memo. 104566, Vol. 21, S.B. Hooker

and E.R. Firestone, Eds., NASA Goddard Space Flight

Center, Greenbelt, Maryland, 43 pp.

Vol. 22

Barnes, R.A., W.L. Barnes, W.E. Esaias, and C.R. McClain,

1994: Prelaunch Acceptance Report for the SeaWiFS Radiometer. NASA Tech. Memo. 104566, Vol. 22, S.B. Hooker, E.R. Firestone, and J.G. Acker, Eds., NASA Goddard

Space Flight Center, Greenbelt, Maryland, 32 pp.

Vol. 23

Barnes, R.A., A.W. Holmes, W.L. Barnes, W.E. Esaias, C.R.

McClain, and T. Svitek, 1994: SeaWiFS Prelaunch Radiometric Calibration and Spectral Characterization. NASA

Tech. Memo. 104566, Vol. 23, S.B. Hooker, E.R. Firestone,

and J.G. Acker, Eds., NASA Goddard Space Flight Center,

Greenbelt, Maryland, 55 pp.

Vol. 24

Firestone, E.R., and S.B. Hooker, 1995: SeaWiFS Technical Report Series Summary Index: Volumes 1–23. NASA Tech.

Memo. 104566, Vol. 24, S.B. Hooker and E.R. Firestone,

Eds., NASA Goddard Space Flight Center, Greenbelt,

Maryland, 36 pp.

Vol. 25

Mueller, J.L., and R.W. Austin, 1995: Ocean Optics Protocols

for SeaWiFS Validation, Revision 1. NASA Tech. Memo.

104566, Vol. 25, S.B. Hooker and E.R. Firestone, Eds.,

NASA Goddard Space Flight Center, Greenbelt, Maryland, 66 pp.

Vol. 26

Siegel, D.A., M.C. O’Brien, J.C. Sorensen, D.A. Konnoff, E.A.

Brody, J.L. Mueller, C.O. Davis, W.J. Rhea, and S.B.

Hooker, 1995: Results of the SeaWiFS Data Analysis

Round-Robin (DARR-94), July 1994. NASA Tech. Memo.

104566, Vol. 26, S.B. Hooker and E.R. Firestone, Eds.,

NASA Goddard Space Flight Center, Greenbelt, Maryland, 58 pp.

Vol. 27

Mueller, J.L., R.S. Fraser, S.F. Biggar, K.J. Thome, P.N. Slater,

A.W. Holmes, R.A. Barnes, C.T. Weir, D.A. Siegel, D.W.

Menzies, A.F. Michaels, and G. Podesta, 1995: Case Studies for SeaWiFS Calibration and Validation, Part 3. NASA

Tech. Memo. 104566, Vol. 27, S.B. Hooker, E.R. Firestone,

and J.G. Acker, Eds., NASA Goddard Space Flight Center,

Greenbelt, Maryland, 46 pp.

Vol. 28

McClain, C.R., K.R. Arrigo, W.E. Esaias, M. Darzi, F.S. Patt,

R.H. Evans, J.W. Brown, C.W. Brown, R.A. Barnes, and

L. Kumar, 1995: SeaWiFS Algorithms, Part 1. NASA

Tech. Memo. 104566, Vol. 28, S.B. Hooker, E.R. Firestone,

and J.G. Acker, Eds., NASA Goddard Space Flight Center,

Greenbelt, Maryland, 38 pp., plus color plates.

72
J.W. Campbell, J.M. Blaisdell, and M. Darzi

Vol. 29

Aiken, J., G.F. Moore, C.C. Trees, S.B. Hooker, and D.K.

Clark, 1995: The SeaWiFS CZCS-Type Pigment Algorithm. NASA Tech. Memo. 104566, Vol. 29, S.B. Hooker

and E.R. Firestone, Eds., NASA Goddard Space Flight

Center, Greenbelt, Maryland, 34 pp.

Vol. 30

Firestone, E.R., and S.B. Hooker, 1995: SeaWiFS Technical Report Series Summary Index: Volumes 1–29. NASA Tech.

Memo. 104566, Vol. 30, S.B. Hooker and E.R. Firestone,

Eds., NASA Goddard Space Flight Center, Greenbelt,

Maryland, (in production).

Vol. 31

Barnes, R.A., A.W. Holmes, and W.E. Esaias, 1995: Stray

Light in the SeaWiFS Radiometer. NASA Tech. Memo.

104566, Vol. 31, S.B. Hooker, E.R. Firestone, and J.G.

Acker, Eds., NASA Goddard Space Flight Center, Greenbelt, Maryland, 76 pp.

Vol. 32

Campbell, J.W., J.M. Blaisdell, and M. Darzi, 1995: Level-3

SeaWiFS Data Products: Spatial and Temporal Binning

Algorithms. NASA Tech. Memo. 104566, Vol. 32, S.B.

Hooker, E.R. Firestone, and J.G. Acker, Eds., NASA Goddard Space Flight Center, Greenbelt, Maryland, 73 pp.,

plus color plates.

73
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

COLOR PLATES
J.W. Campbell, J.M. Blaisdell, and M. Darzi

PLATE 1. Mean CHL images derived from the AVG and MLE estimators for the seven CZCS scenes listed in Table 1.
Level-3 SeaWiFS Data Products: Spatial and Temporal Binning Algorithms

PLATE 2. Mean CHL images derived from the AVG4 and MLE4 estimators for the seven CZCS scenes listed in Table 1.
J.W. Campbell, J.M. Blaisdell, and M. Darzi

PLATE 3. Differences between level-3 means derived from MLE and AVG estimators (upper images) and difference

between MLE4 and AVG4 estimates (lower images).
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11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The level-3 data products from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) are statistical data sets derived

from level-2 data. Each data set will be based on a fixed global grid of equal-area bins that are approximately 9x9 km2

.

Statistics available for each bin include the sum and sum of squares of the natural logarithm of derived level-2 geophysical

variables where sums are accumulated over a binning period. Operationally, products with binning periods of 1 day, 8

days, 1 month, and 1 year will be produced and archived. From these accumulated values and for each bin, estimates of the

mean, standard deviation, median, and mode may be derived for each geophysical variable. This report contains two major

parts: the first (Section 2) is intended as a users' guide for level-3 SeaWiFS data products. It contains an overview of level0 to level-3 data processing, a discussion of important statistical considerations when using level-3 data, and details of how

to use the level-3 data. The second part (Section 3) presents a comparative statistical study of several binning algorithms

based on CZCS and moored fluorometer data. The operational binning algorithms were selected based on the results of this

study.

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 SeaWiFS Technical Report Series

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 Spatial and Temporal Binning Algorithms Code 970.2

 July 1995 Technical Memorandum

6. AUTHOR(S)

 Janet W. Campbell, John M. Blaisdell, and Michael Darzi

 Series Editors: Stanford B. Hooker and Elaine R. Firestone

 Technical Editor: James G. Acker

Elaine R. Firestone, John M. Blaisdell, and Michael Darzi: General Sciences Corporation, Laurel, Maryland; Janet W.

Campbell: University of New Hampshire, Durham, New Hampshire; and James G. Acker: Hughes STX, Lanham, Maryland