DOC
EPA
United States
Department of
Commerce
National Oceanic and
Atmospheric Administration
Seattle WA 98115
United States
Environmental Protection
Agency
Office of Environmental
Engineering and Technology
Washington DC 20460
EPA-600/7:80-168
October 1980
Research and Development
A Comparison of the
Mesa-Puget Sound
Oil Spill Model with
Wind and Current
Observations from
August 1978
Interagency
Energy/Environment
R&D Program
Report
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RESEARCH REPORTING SERIES
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tion Service, Springfield, Virginia 22161.
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A COMPARISON OF THE MESA-PUGET SOUND
H OIL SPILL MODEL WITH WIND
AND CURRENT OBSERVATIONS
FROM AUGUST 1978
by
Robert J. Stewart
Carol H. Pease
Pacific Marine Environmental Laboratory
Environmental Research Laboratories
National Oceanic and Atmospheric Administration
3711 15th Ave. N.E.
Seattle, Washington 98105
Prepared for the MESA (Marine Ecosystems Analysis) Puget Sound
Project, Seattle, Washington in partial fulfillment of
EPA Interagency Agreement No. D6-E693-EN
Program Element No. EHE625-A
This study was conducted
as part of the Federal
Interagency Energy/Environment
Research and Development Program
Prepared for
OFFICE OF ENERGY, MINERALS, AND INDUSTRY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
AUGUST 1980
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Completion Report Submitted to
PUGET SOUND ENERGY-RELATED RESEARCH PROJECT
OFFICE OF MARINE POLLUTION ASSESSMENT
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
by
Pacific Marine Environmental Laboratory
Environmental Research Laboratories
National Oceanic and Atmospheric Administration
3711 15th Ave. N.E.
Seattle, Washington 98105
This work is the result of research sponsored by the Environmental
Protection Agency and administered by the National Oceanic and Atmospheric
Administration.
The National Oceanic and Atmospheric Administration does not approve,
recommend, or endorse any proprietary product or proprietary material men-
tioned in this publication. No reference shall be made to the National
Oceanic and Atmospheric Administration or to this publication in any adver-
tising or sales promotion which would indicate or imply that the National
Oceanic and Atmospheric Administration approves, recommends, or endorses
any proprietary product or proprietary material mentioned herein, or which
has as its purpose to be used or purchased because of this publication.
ii
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CONTENTS
Figures iv
Tables v
Abstract vi
Acknowledgements vii
1. Introduction 1
2. Conclusions 2
3. Recommendations 4
4. Tidal Current Simulation 6
Functional Validation of Model 6
Accuracy of the Model's Interpolation Scheme 7
5. Significance of the Current Model Errors 13
CASE 1: No Knowledge of Tides 14
CASE 2: No Knowledge of Nontidal Current 19
CASE 3: Existing Model with Assumed Steady Current 22
6. Wind Field Simulation 23
Significance of the Wind Model Errors 26
7. Drifter Response Studies 28
8. References 32
APPENDICES
A. Calculation of Tidal Phase and Amplitude from Model
Coeficients 34
B. Moments for Randomly Oriented Constituent Displacements .... 36
111
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FIGURES
Number Page
1. Location of Straits 11, 12, and 13, and the weighting
factors applied to Parker's stations 38
2. Dispersion envelopes assuming no knowledge of tidal
currents (Case 1) 39
3. Tide model contribution to reducing position error
relative to Case 1 40
4. Dispersion envelopes associated with ignorance of
high-frequency nontidal current oscillations (Case 2) 41
5. Observed net currents at various locations in the
Strait of Juan de Fuca and the Central Basin 42
6. Dispersion caused by randomly oriented net current
of .66 km/hr (Case 2) 43
7. A comparison of Case 1 and Case 2 errors 44
8. Difference-current dispersion ellipses at 1, 3,
and 10 hours 45
9. Wind Pattern No. 3 46
10. Wind Pattern No. 4 47
11. Wind Pattern No. 5 48
12. Wind station locations 49
13. Principal components of surface wind observations 50
14. Scatter plot of principal component weights 51
15. Scatter plot of wind observations, sorted by
pattern type at four central basin stations 52
16. a. Average residual currents on August 25, 1978
(residual = drifter-model) ..... 53
b. Average residual currents on August 26, 1978
(residual = drifter-model) 54
iv
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TABLES
Number Page
1.
2.
3.
4.
5.
A comparison of a tidal analysis of model output with
an analytical prediction based on model coefficients
A comparison of simulated tidal currents with actual
A comparison of linear interpolation errors with
Parameters for the discrete frequency simulation of
the nontidal oscillatory currents
8
q
1?
18
20
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ABSTRACT
This report compares the winds and currents observed in August 1978 in
the Strait of Juan de Fuca with simulated wind and current fields taken from
the MESA-Puget Sound oil spill model. This model is described in a companion
report, Pease (in press). A method is developed for relating these errors in
velocity to uncertainties in predicted position. The tidal current subprogram
of the oil spill model is shown to reduce the uncertainty in trajectory posi-
tion by an amount that is somewhere in the range of 50% to 90% of the total
uncertainty that can be caused by ignorance of the tides. It is also shown
that the uncertainty in trajectory position is strongly affected by our
inability to predict the baroclinic motions in the region. Over times less
than 10 hours, the dispersion is mainly tidal, and the tidal current subpro-
gram contributes significantly to the prediction of position. After 10 hours,
however, the bulk of the dispersion is due to the low-frequency (periods
longer than a week) baroclinic motions. These baroclinic motions are poorly
understood, and a program of basic research directed at illuminating their
causes and statistical properties is called for, if predictions are to be made
over periods longer than 10 hours. The regional wind model developed by
Overland, Hitchman, and Han (1979) and used as a subprogram in the model is
compared with wind observations from a short period of time. We conclude that
the selection of a master station for use in scaling the pattern strength
cannot be done in an arbitrary fashion. We also find that the repertoire of
patterns presently available in the program library is not sufficiently com-
prehensive to allow reliable modeling of the surface winds.
vi
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ACKNOWLEDGEMENTS
The authors are most grateful to Ms. Rita Chin who has labored diligently
over the computer analysis of the many diverse data sources incorporated in
our study. Ms. Chin also prepared the draft figures and was of general assis-
tance throughout the course of this project. Our thanks also to Ms. Sue Larsen
who prepared this manuscript and Ms. Virginia May who prepared the figures.
We are indebted to Dr. James E. Overland, Mr. James R. Holbrook, Mr. Carl A.
Pearson, and Dr. Harold 0. Mofjeld, all of PMEL, for their contributions of
data, computer programs, and meteorological and oceanographic expertise.
Mr. Pearson was most helpful in providing his time and computer programs for
the tidal and spectral analysis of the various current velocity time series.
Mr. Holbrook provided us with the current meter observations from Straits 11,
12, 13 and further gave us his own analyses of these data which we have used
in several places. Mr. Holbrook's interpretations of the data and his review
of portions of our manuscript were of great value to us. Dr. Mofjeld has pro-
vided advice on a number of items, and his review of our manuscript resulted
in several changes which improved the report.
vii
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SECTION 1
INTRODUCTION
In early 1978, an oil spill trajectory model was completed at the Pacific
Marine Environmental Laboratory (PMEL) for the MESA-Puget Sound project of-
fice. This model is described in Pease (in press). The model was developed
under the direction of Dr. Jerry A. Gait, then of PMEL. The model was devel-
oped for two main purposes. In the event of an actual oil spill, it is to be
used to assist in the cleanup operations. In this mode the model would pre-
dict locations and times of arrival of portions of the spill. This informa-
tion could then be used to improve the deployment of cleanup equipment. The
model is also intended for use in a simulation capacity. The site selection
of new petroleum transshipment facilities, refineries, or storage facilities
involves the consideration of a number of factors, including probable environ-
mental impacts. It was anticipated that the model predictions would be of
value in comparing alternative facility locations.
Given the oceanographic complexity of the region the creators of the
model did not aspire to comprehensiveness. However, they did wish to exploit
the substantial literature on tidal currents that was available as a result of
a long-term program of current meter studies by the National Ocean Survey
(NOS) (Parker, 1977; NOS, 1973a; NOS, 1973b; NOS, 1979; and by Parker within
Cannon (ed.), 1978). It was felt that a fine-scale model based on interpola-
tion and extrapolation of the NOS tidal harmonic analyses would be a good
start toward the development of a capability to predict and simulate current
behavior.
The details of the surface winds were also little understood. But the
concurrent development of a regional wind model (Overland, Hitchman, and Han
1979), offered hope of simulating this complicated variable at a level of
detail commensurate with the intended resolution of the model.
Thus, the model development was undertaken not as the final step in a
definitive summary of regional oceanography and meteorology, but rather as a
first step at attempting to integrate these phenomena. The philosophy under-
lying this effort rested mainly on the engineer's empiricism, "let's do what
we can," rather than a more scientific appraisal of the possibilities.
In late August of 1978, a cooperative oceanographic experiment was under-
taken in the region just north of Dungeness Spit (see Frisch, Holbrook, and
Ages, 1980). The experiment consisted of drifter motion studies, CODAR and
current meter measurements, CTD casts, and surface wind measurements. The
results of this experiment provided a good opportunity to examine the wind and
current simulation portions of the oil spill trajectory model, and so this
study was begun.
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SECTION 2
CONCLUSIONS
Two questions were posed: is the current simulation accurate enough to
be useful, and is the wind properly modelled using the pattern method? Both
questions can be addressed from two viewpoints. First, is the technique
properly implemented in the existing model, and secondly, is the modelling
concept valid? The implementation question is simply an independent check
that the model operates on its input data and coefficient arrays in a fashion
consistent with the equations that comprise the model. The modelling concept
question is deeper and more philosophical. The model was created to exploit
the availability of both the tidal current data and the regional wind model.
Little thought has been given to the adequacy of these representations. We
have attempted to quantify the contributions these submodels make to reducing
the uncertainty of a trajectory prediction, and we have made recommendations
for both the interpretation of existing model results and subsequent modelling
efforts.
A major difficulty we encountered in addressing the modelling concept
question was the determination of what constituted an acceptable error. This
problem included both the definition of the error in terms of the various
parameters that enter the problem and the interpretation of the error param-
eter. We have attempted a novel analysis of this problem for the current
simulation subprograms. We have shown that a useful measure of trajectory
dispersion is the variance of the time-integrated difference between the
actual and simulated velocities. This integrated difference-velocity is a
displacement, and the variance of this displacement is a measure of the area
surrounding a predicted position in which a drifting object is likely to be
found.
In order to put into perspective the tidal current subprogram's contri-
bution to trajectory prediction calculations, we have analyzed the displace-
ment errors for three illustrative cases. Case 1 considers trajectory pre-
dictions made with perfect knowledge of nontidal currents but completely
ignoring tidal currents. The area of the resultant dispersion is shown in
Figure 2. Figure 3 depicts the reduction in the tidal dispersion that should
be achieved through use of the model, based on our estimates of the model's
errors. Figures 4, 5, and 6 treat the converse case, case 2, in which tra-
jectory predictions are made with perfect knowledge of tidal currents and no
knowledge of the nontidal currents. Figure 6 graphically illustrates the
importance of the nontidal, slowly varying currents caused by density vari-
ations within the region.
These slowly varying currents appear as net flows in the 15- and 29-day
current meter observations that are used to estimate the amplitude and phase
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of the tidal-current constituents. These baroclinic motions are not well
understood, and modelling them appears to be a topic of state-of-the-art
research. There seems to be no way of dealing with these currents except to
hypothesize a particular steady-current field. Figure 8 depicts the trajec-
tory dispersion ellipses we can anticipate assuming it is possible to select
the correct steady-current field.
The study plan for the wind field analysis was developed around the idea
of taking a detailed look at the wind field during a short, five-day time
interval. This approach provided us with good insights regarding the qualita-
tive performance of the model. We found that for the particular conditions of
the test, the model did not reproduce the fine-scale features seen in the wind
observations. This was particularly true in the region immediately off Port
Angeles. We also examined a method of simulating a time history of the wind
and found that the arbitrary selection of Race Rocks as a master station was
not supported by the data. A principal-component analysis of a portion of the
data was performed.
An attempt was also made to reconcile the Evans-Hamilton drifter data
(Cox, Ebbesmeyer, and Helseth, 1979) using the combined wind and current
fields. The effort was unsuccessful mainly because of the effects of model-
ling errors. It was possible to detect a current reversal off New Dungeness
on 26 August 1978 by calculating the average of the difference between the
observed drifter velocities and the model's predicted tidal-current veloc-
ities.
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SECTION 3
RECOMMENDATIONS
The procedural necessity for making a best-effort analysis of the poten-
tial environmental effects of a proposed development has led to the creation
of oil-spill models not just for Puget Sound, but for a wide variety of loca-
tions. The model examined here is representative, and we believe our conclu-
sions regarding the accuracy of the wind and current simulations have general
application. Our primary recommendation is thus that the users of an oil-
spill model should be alert to the possible inaccuracies. If the model has
been developed for a hydrodynamically complex region such as the one studied
here, then one must remain skeptical of any simplistic interpretation of the
model's results. In the Strait of Juan de Fuca, for example, the growth over
time of the position error exhibits a strong dependency on the slowly varying,
baroclinic components of the current; and these current phenomena are not
simulated by the model. Thus, the model contributes beneficially to the
prediction of a drifter's location only over brief time intervals.
We found that for prediction intervals of of 1 to 5 hours, the model can
reduce position uncertainty due to tides by a factor of about two, as compared
to predictions using no tidal-current model. During this time, the position
uncertainty caused by unknown baroclinic currents is small, but growing, and
in the time interval from 5 to 10 hours, uncertainties due to baroclinic
currents grow to overwhelming importance. These results are, of course,
specific to the region studied, but the principle can be applied to other
locations.
Techniques for using the model in an actual spill need to be developed to
accommodate these short-comings if the model is to be of any assistance. If
the tide model had only very small errors, then in the event of a spill it
would be feasible to simply subtract the simulated current from an instan-
taneously measured current, and so to estimate the important slowly varying
component. However, as shown in Table 4, the amplitudes of the model errors
are in the range of 10 cm/sec to 30 cm/sec for the important M2 and Kx con-
stituents, thus even a 20-cm/sec steady current will be masked by the model
error. Nor can we avoid this difficulty by using some other simple scheme to
interpolate between the tidal-current reference stations, as demonstrated in
Table 3. Therefore, our second recommendation is that additional research be
devoted to either improving the accuracy of the interpolation scheme or devel-
oping a practicable observational scheme that will allow the determination of
the slowly varying currents over short time intervals, such as 1 to 6 hours.
Figures I6a and I6b suggest that it might be possible to develop a hybrid
system for this purpose based on the use of 20 to 40 drifters coupled with the
existing tidal-current model. These drifters should be released on a 2-km
grid in the subject area and positions would have to be determined hourly,
-------
irrespective of the visibility. Alternatively, a mobile CODAR with complete
data reduction and analysis systems might be developed. A research program of
this type is essential if we are to develop any long-term predictive capabili-
ties for use in a spill situation in the Strait of Juan de Fuca or the central
basin.
The current subprogram's shortcomings are, of course, caused by its
failure to model or otherwise account for the slowly varying baroclinic cur-
rents. These currents have only recently been appreciated for their impor-
tance to pollutant transport in the region. Topics that might profitably be
explored include baroclinic effects on the tides; analytical or numerical
simulation of the coastal intrusion phenomena; and further field studies using
CODAR and current meter arrays. It would be premature to expect that these
studies would immediately lead to an improved modelling capability, but it
might be helpful to use modelling as a focus for the work.
The wind field simulation was also found to be rather inaccurate. How-
ever, this problem continues to be studied at PMEL as part of the Marine
Services program, and progress has been made over the past two years. The
wind field simulation model should be updated in the near future to take
advantage of this work. There are some difficulties, however, that will
require special attention. Foremost is the problem of establishing a suitable
statistical framework on which to perform the comparison of observation and
simulation. Recent work in nonparametric tests of multivariate processes
should be examined to determine whether they can help solve this difficulty
(cf. Friedman and Rafsky, 1979). An automated procedure for pattern selec-
tion is also required so that statistical comparisons can be made using suffi-
ciently large numbers of samples.
Finally, we must point out that this study has dealt with winds and
currents. These parameters, in conjunction with topographical considerations,
are undoubtedly the factors that cause oil spill transport. However, the
equations that combine these phenomena and produce an oil transport prediction
are still little known. It is of vital importance that the direction of
future research be compatible with this appreciation of the problem. Specif-
ically, studies of winds and currents should be limited to wind and current
phenomena. Combining them in an "oil spill" model seems to be pointless. The
requirement that the wind and current models be computerized and of known
accuracy should suffice to ensure their ultimate compatibility in an oil spill
model.
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SECTION 4
TIDAL CURRENT SIMULATION
Three current meter arrays were deployed in the central basin region on
July 16, 1978. These stations are referred to as Straits 11, 12, and 13.
Each station included a VACM current meter at a depth of 4 m, and each current
meter acquired enough data to allow at least a 29-day tidal harmonic analysis.
We have used this current meter data as the principal standard for judging the
accuracy of our tidal-current simulation routine. The current meter data was
obtained from Mr. James Holbrook of the PMEL Coastal Physics Group. It was
subsequently analyzed using the R2SPEC computer program which is maintained by
Mr. Carl Pearson also of the Coastal Physics Group.
The output of the R2SPEC analysis includes the amplitude, direction of
flood (ebb), and phase lag of the various tidal constituents. The analysis is
done both in east-west, north-south components, and in terms of components
oriented along the major and minor axis of the tidal ellipse. The major axis
of this ellipse determines the flood/ebb direction. The major-axis represen-
tation of the motion was most suitable for a comparison with the tide model,
since the tide model assumes all motions lie along the major axis of the M2
constituent (Pease, in press).
FUNCTIONAL VALIDATION OF MODEL
Although the model had been extensively debugged and tested in previous
projects, the availability of R2SPEC gave us another means of verifying the
proper operation of the model. Further, the R2SPEC test was not redundant
with earlier efforts to validate the correct functioning of the model. We
therefore proceeded with the test.
The model generates its simulated currents by summing as many as three
reference station currents, each weighted by a factor lying between 0 and 1.0.
The current at the reference station is in turn generated by the summation of
the five tidal constituents analyzed and tabulated by Parker (1977). To
perform the test we generated (U,V), (east-west, north-south), time series for
a 29-day period at the three current meter station locations. The simulated
time series at Strait 11 was created with the sum of a .6 weighting on
Parker's Station 21 and a .4 weighting on Station 30. The Strait 12 current
was generated with the sum of a .3 weighting on Parker's Stations 28 and 42,
and a .4 weighting on Station 29. Strait 13 was created with a weighting of
.2 on Parker's Stations 31 and 38, and .6 on Station 30. These synthetic
currents were then analyzed by R2SPEC, and the amplitude and phase of the five
tidal constituents of the simulated tidal currents were determined. Figure 1
shows the location of Straits 11, 12, and 13, and the locations and weights of
the reference stations used in the summation.
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An alternative method for determining the amplitude and phase at the
three stations is, using trigonometry, to determine the amplitude and phase as
analytic functions of the weighting factors and the reference stations' ampli-
tudes and phases. This derivation is outlined in appendix A.
The comparison for the 5 tidal constituents used in the model is shown in
Table 1. The calculated phase and amplitude, equations (A4a), (A4b), and the
equivalent R2SPEC estimates are very similar for the M2 and Oa constituents at
all three locations, and they are reasonably close for the S2, N2, and Kx
constituents. The reason for the slight disagreement in the latter constitu-
ent estimates is that R2SPEC includes corrections for errors normally caused
by the presence of the host of other tidal constituents. These other
constituents are not present in the model's simulated tidal velocity,
equation (A5), and so these "corrections" are deleterious. If they were
nullified, the analysis would compare even more closely with the calculated
phase and amplitudes. In any event, the comparison is sufficiently close to
state that the model's simulation of tidal currents accurately reflects the
data taken from Parker (1977) and stored in the various arrays. There is no
time base error and no indication of transformation errors in going from the
major-axis coordinates to east-west, north-south coordinates. We note in
passing that the minor-axis velocity estimates from R2SPEC were uniformly zero
for all constituents and locations, reflecting exactly the model's simulation
process.
ACCURACY OF THE MODEL'S INTERPOLATION SCHEME
We know that the model operates consistently on the data stored in its
coefficient arrays. It is not certain, however, whether these coefficients
have been selected to create an accurate simulation of tidal currents at an
arbitrary point. In order to judge the merits of the model from this second
viewpoint, we compared the three model time series with currents observed at
Straits 11, 12, and 13. Through a minor communication failure uncovered in
the final writing of this report, the model locations for Straits 11 and 12
were taken to be 48°13'N, 126°6'W and 48°20'N, 122°58'W, respectively. These
locations are in error by about 2 km from the sites of the July 16th deploy-
ment. It was judged that the comparison was still valid, however, since tidal
phase, amplitude, and direction of flood change very little over distances as
small as 2 km in this area.
Table 2 shows the flood direction, major-axis amplitude, GMT phase, and
minor-axis amplitude for the M2, N2, S2, Kt, and Ox constituents, as estimated
from the 29-day records obtained at Straits 11, 12, and 13. Immediately
beneath these values are the equivalent parameters as calculated from the
simulated time series obtained from the model.
In general, we can see that the model's simulated tidal currents cor-
respond reasonably well with the observed currents. The errors in flood
direction range from 1° or 2° to 42°, with a typical value being approximately
10°. The simulated major-axis amplitudes are uniformly low, with the discrep-
ancy ranging from a few cm/sec, to 15 cm/sec in the K! constituent at Strait
11. The percentage error of this deficiency ranges from 15% or 20% for the
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TABLE 1
A COMPARISON OF A TIDAL ANALYSIS OF MODEL OUTPUT WITH AN ANALYTICAL PREDICTION
BASED ON MODEL COEFFICIENTS
00
H2
FLOOD
LOCATION
STRAIT
**
11
STRAIT
***
12
STRAIT
13
TYPICAL
CAL'C'D
R2SPEC
CAL'C'D
R2SPEC
CAL'C'D
R2SPEC
DIFFERENCE
DIR
OT
80°
80°
42°
42°
95°
95°
0°
amp.
cm/sec
36.7
36.8
36.5
36.1
33.5
33.6
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TABLE 2
A COHPARISON OF SIMULATED TIDAL CURRENTS WITH ACTUAL OBSERVATIONS AT STRAITS 11, 12, AND 13
Station Location
48°14'I123°6' Observed
Strait
11 48°13',U3°6' Simulated
4802r,122057' Observed
Strait
12 48°20',122<>58' Simulated
48° 19' ,122059' Parker 29
48°14' ,122°57' Observed
Strait
13 48°14, 122°S7' Simulated
Major Epoch Minor
°T cm/sec GMT cm/sec
92 43.9 296 1.4
80 36.8 249 0
65 45.6 319 21.9
42 36.6 311 0
60 31.2 312 .7
96 41.2 302 8.5
95 33.6 281 0
Major Epoch Minor
°T cm/sec GMT cm/sec
89 10.9 265 .6
80 6.3 249 0
75 10.6 294 4.4
42 6.9 307 0
62 6.0 298 1.1
96 8.1 269 .3
95 5.4 274 0
Major Kpoth Minor
°T cm/ sec GMT cm/sec
82 11.5 231 3.8
80 9.9 252 0
64 11.0 352 4.2
A2 10.8 323 0
74 9.3 14.1 3.3
109 17.9 252 3.H
95 8.8 298 0
Major Epoch Minor
°T cm/ sec GMT cm/sec
78 30.3 219 3.3
80 15.5 120 0
84 15.4 '238 4.1
42 13.7 188 0
4li 16.3 222 3.7
109 26.5 225 8.7
<)r, 13.5 170 0
Major Epoch Minor
"T cm/si-c GMT cm/sec
85 18.2 186 3.3
80 15.4 164 0
56 11.9 199 1.7
42 8.9 185 0
52 11.6 187 .3
105 15.0 178 2.7
95 9.3 161 0
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M2, S2, and Oj constituents, to 40% or 50% for the N2 and Kt constituents.
The error in the GtfT Epoch value ranges from 5° or 10° to 40° or 50° (with one
error of 100° in the Kx component of Strait 11), but there appears to be no
pattern in this error except that the Kj constituent is uniformly bad. The
neglect of the minor-axis velocity in the model leads to errors that are
typically 5 cm/sec or less, with the exception of the M2 component of Strait
12.
A portion of the discrepancy between the observed currents and the
model's simulated currents may be due to the differing depths of Parker's
reference stations and the current meters we are using for the comparison.
Parker's stations vary in depth from 5 m to 22 m, whereas Straits 11, 12, and
13 are at 4 m. Over such a range of depths we know that the direction and
amplitude of a tidal constituent can change as a result of baroclinic effects
on the tide. Directional changes of 10°-15° and amplitude changes of 5 cm/sec
are common in this region in the top 20 m to 30 m. Because there is as yet no
known simple, proven method for predicting this discrepancy, we shall ignore
it for the present analysis.
It seemed possible that the particular selection of weights is solely
accountable for the errors described above. It can be seen from Figure 1 that
the Strait 12 position and Parker's Station 29 are close together. Thus, one
would think it possible to use Parker's Station 29 as a basis for predicting
Strait 12 currents. The consequence of this selection may be seen in Table 2.
The major-axis amplitude deficiency is exacerbated in the M2 and S2 compo-
nents, and partially corrected in the remaining constituents. On the other
hand, errors in direction of flood are uniformly reduced, and the Epoch values
are significantly better for the Kj, S2, and N2 components. However, errors
still remain, and it appears that we cannot expect to achieve perfection from
a simple revision of the coefficent arrays.
The determination of the weighting factors and the station indexing were
done manually by the authors of the model in a somewhat subjective way. A
simple objective approximation to the process behind the station indexing and
weighting is that of a linear interpolation. Are the errors seen at Straits
11, 12, and 13 consistent with this interpretation? If not, can we expect to
diminish the errors seen in Table 2 by using a more objective weighting scheme
based on linear interpolation between stations?
Table 3 shows the results of a simple analysis of Parker's station data
which suggests an answer to these questions. Parker's station locations
were studied and eight station triplets were found where all three stations
fell nearly on a straight line. The outermost stations were then used to pre-
dict the M2 amplitude and phase of the middle station. The weighting factor
applied to a predictor station was simply the fractional distance from the
middle station. The distance between the outermost stations varied from
12.6 to 39.6 kilometers, and the weighting factors were typically in the
range of .45 to .55, reflecting the fact that the middle station was usually
about halfway between the predictor stations. Because the predicted current
exhibits an error in both phase and magnitude, we require two measures of
10
-------
the error. The least ambiguous way to write this error is as the sum of a
component that is in phase with the reference signal and a component that lags
by 90° Thus:
cos at - a cos (at + 6) = bl cos at + b2 sin at
(1)
where a is the linear interpolation estimate of a, and 6 is the phase error of
the estimate. We do not consider the effect of the statistical uncertainty
that clouds our knowledge of the amplitude and the phase of the reference
signal, as well as our knowledge of the amplitude and phase of the predictors.
Equation (1) may be solved for bl and b2 with the result:
= a cos (6) - a
(2a)
b2 = a sin (6)
(2b)
These bj and b2 values for the eight triplets are shown in Table 3 in the two
far-right columns.
The bj and b2 values for the Straits 11, 12, and 13 are also shown in
Table 3, and we can see that their range of values is not unlike that seen in
the linear interpolation scheme. There is not a sufficient number of data
points to allow us the luxury of a statistical test of the hypothesis that the
errors of both the model and the linear interpolation scheme are from the same
distribution. However, as a practical matter it appears very likely that
there is no significant difference between the methods.
11
-------
TABLE 3
A COMPARISON OF LINEAR INTERPOLATION ERRORS WITH OBSERVED MODEL ERRORS
ro
Predictor Stations
Reference St. St.
Station No. Wt. No. Wt.
4 2 .47 5 .53
5 4 .51 8 .49
7 6 .50 8 .50
12 11 .49 14 .51
20 16 .39 30 .61
21 22 .44 20 .56
22 17 .54 26 .46
25 26 .52 78 .48
Strait 11
Strait 12
Strait 13
Reference St
Dist. Amp
(Km) cm/sec
38.0 41.8
39.6 42.7
13.0 47.3
12.6 72.7
28.2 47.3
16.7 48.9
20.0 63.7
13.9 67.0
43.9
45.6
41.2
. Analysis
M2
Phase
(LT)
39°
61°
54°
70°
74°
67°
84°
67°
296°*
319°*
302°*
Predicted Coefficients
M2 M2
Amp . Phase
cm/sec (LT)
31.4
43.2
43.0
80.5
41.8
54.3
64.5
68.6
36.8
36.6
33.5
63.5
49.8°
62.3°
59.0°
54.1°
79.1°
69.8°
90.4
249°*
311°*
281°*
Error
bl b2
(inphase) (90° Lag)
cm/sec cm/sec
-13.2
-.3
-4.8
6.3
-7.8
4.2
-1.2
-4.0
-17.9
-9-5
-9.5
13.0
-8.4
6.2
-15.3
-13.7
11.4
-15.8
27.2
25.9
5.1
10.9
t
*GMT PHASE LAGS
-------
SECTION 5
SIGNIFICANCE OF THE CURRENT MODEL ERRORS
As we have noted, tidal currents created by the model have errors that
are a sizable fraction of the observed tidal currents. It is not clear, how-
ever, whether these errors are important when viewed in light of the model's
intended use. In fact, the problem of what is an "acceptable" error has not
been discussed previously. This is a critical deficiency in our thinking on
the subject, and to remedy this deficiency, we offer the following analysis
both to help critique the existing model and to guide subsequent efforts.
We have found that one parameter useful as an error criterion is the mean
squared error in predicted position. This parameter is an indication of the
area that ought to be searched if the model is used to locate a drifting
object some period of time after it has been released. If this number is
small, say on the order of 1 (km)2, then we can go to the location predicted
by the model and expect the drifter to be nearby, perhaps within sight of the
search vessel. Alternatively, if the number is large, say 100 (km)2, then we
should anticipate substantial difficulties in locating the object. Inter-
preted another way, if we run the model in a simulation mode, then the growth
of the mean squared error with time makes it more and more unlikely that a
model prediction is a truly representative trajectory. This is particularly
true if the trajectory lies near a beach or some other geographic feature that
might catch the drifter or channel it into a particular region.
Let us consider three illustrative cases.
Case 1. Trajectory predictions are made with no knowledge of tidal
currents, but with perfect knowledge of all nontidal currents.
Case 2. Trajectory predictions are made with no knowledge of the non-
tidal currents, but with perfect knowledge of the tides.
Case 3. Trajectory predictions are made with the present model with
the addition that we have perfect knowledge of the very-low-
frequency (weekly and lower) currents.
The first case is unreasonable in any practical sense because it supposes
that we have a perfect understanding of wind-driven and baroclinic current
phenomena that we are only now learning to identify and categorize and are far
from describing in a statistically useful sense. However, this case offers a
simplified view of the problem that isolates and helps define the role a
tidal-current model plays in trajectory calculations. Specifically, we can
use this case to relate errors in tidal-current amplitude and phase to mean
squared position errors in locating drifting objects.
13
-------
CASE 1; No Knowledge of Tides
In addition to perfect knowledge of nontidal currents, we assume for case
1 that amplitude and phase functions are independent of position. We assume
also that the trajectory calculation is to begin at a random time. With these
qualifications, the displacement due to one tidal constituent is given by the
time integration of the velocity. This has a particularly simple form when we
are dealing with sinusoidal tidal constituents:
Udt =
cos
9£) dt,
which can also be written as:
*• v A *
_* sin(a0t
or- *
where
and
T2 _
"
• a*
TO = t2 -
(3)
Because TI is random, £_ is also random. With trigonometry, we can transform
equation (3) into the following:
= £ V2 (1-cos
where
We let
= cos
sina^Tp
V2 (1- cos
cos(|
(4)
(5)
and note that this parameter is simply the maximum excursion possible in time
period T for a particle being advected by a sinusoidal current. The random
phase variate, |., is uniformly distributed over (0,2n).
Based on equation (4), the distribution of D.do) is:
1 , -6«(tJ S D S 60(T ) (6)
The first two moments of this distribution may be shown to be:
1 = 0
og
= %6J(T0)
(7)
(8)
-------
The first result (7) is a consequence of the symmetry of equation (6); the
second result (8) is the parameter we use to characterize the dispersion of
the case 1 assumption.
These results are for the major-axis excursions of one tidal constituent
only. If we neglect all minor-axis motions, the displacement of a particle
that is acted on by a number of tidal constituents will be given by a summa-
tion of the form
M
D(T0) = I D (T0,4 ) (9)
£=1 * *
The random phase variates, £„, are related to one another by the starting
time, Tj, via the modulo operation,
4k = mod (akTi + 6k> 2n) (10)
There is some uncertainty on our part as to the nature of the statistical
relationships that exist between the random phase variates defined by (10).
On one hand, equation (10) is somewhat analogous to computer algorithms that
generate random numbers. On this basis, we believe that an argument could be
made that the phases are statistically independent provided the frequencies in
(10) are incommensurate. Unfortunately we know that the frequencies selected
to represent tidal phenomena are not independent, and that there are a number
of simple relationships that link them. For example, the sum of the 0} and Kx
frequencies exactly equals the M2 frequency. These linkages result in charac-
teristic tidal waveforms. In Seattle, for example, the M2, Oj, and Kj tides
often combine so that consecutive diurnal high waters are of nearly equal mag-
nitude while the intervening low waters are of much different size. Because
of the linkage in frequencies, the opposite pattern occurs only rarely. So
from this standpoint, the phase variates are to be expected to have some very
complicated, multidimensional, statistical dependencies. We have not been
able to resolve this problem, so we have opted to assume that the phase vari-
ates for the different tidal constituents are independent.
The importance of this assumed independence is that we may use the cen-
tral limit theorem (CLT) to deduce the approximate form of the distribution of
D(to)- Tne simplest form of the CLT says that the sum of n independent,
identically distributed, random variates with finite mean and variance con-
verges to a normal (or Gaussian) distribution in the limit as n goes to infin-
ity. The theorem also holds for variates distributed according to a scale
factor transformation of a common distrubution function. Equation (6) is of
this type with 6(T0) being the scale factor.
It is well known that the percentile ranges of a normal distribution are
related to the square root of the variance, the standard deviation. Thus, if
a2 is the variance of the distribution on x, then 68% of the distribution
Ills within ± a and 95% lies between ± 2o . In analogy, and with reference
to the CLT, wexshould expect the variance ^f our summed variate to be func-
tionally related to the percentiles of that distribution. Because the distri-
butions of the summands have square-root singularities at the extremes of
their range, and because the number of summands is small (only five), the CLT
15
-------
provides only a rough approximation to the actual distribution of the summed
variate D(TQ). Thus, we cannot be certain that precisely 68% of our observa-
tions will fall within one standard deviation of the mean. However, exper-
ience with the CLT suggests that the bulk of the observations will fall within
one standard deviation, and it is in this loose sense that we assert that the
standard deviation is related to a "characteristic" error of position. Reit-
erating, we assume that the summands are independent, thus the summed variate
D(TO) will be distributed in an approximately Gaussian form, and this allows
us to put a frequency interpretation on the standard deviation that supports
our contention that the standard deviation of D(TQ) is an appropriate measure
of dispersion. Given the standard deviation as the radius of the circle of
dispersion, the variance is thus proportional to the a-rea in which the drift-
ing object is likely to be found.
Also because of the assumed independence of the summed variates, the
variance of the sum may be related to the variances of the summands, equation
(8). This follows because of the well-known result that the variance of the
distribution of the sum of two independent random variables is given by the
sum of the variances of the summands,
a2 = a2 + a2 ,
ss xx yy
where s = x + y.
This result is exact and not dependent on the CLT approximation. Thus, extend-
ing this result to five summands and applying it to our problem,
a2 = I %6|(T0) = Z/Vf (1 - cosa-to)
DD £ £
This completes our argument that the standard deviation (the square root of
the variance) of D(TQ) provides a useful and readily calculated geometric
measure of the dispersion.
Case 1 assumes we make our predictions with no knowledge of the tidal
currents. We interpret this to mean that the major axis of tidal-current
motion has no favored angular orientation. Under the assumption that all five
tidal constituents share a common major axis, the radius of dispersion is
determined from the square root of (lla):
V
(lib)
Figure 2 shows this R(TQ) variate at Straits 11, 12, and 13. The ampli-
tude values used in the presentation are taken from the July 16th analysis
(see Table 2). The circumference of the circle of dispersion is shown at
hourly intervals for TO varying from 1 to 12 hours. This half-day limita-
tion on the use of equation (lib) has been chosen so as to accommodate the
restrictions that are implicit in our assumption that the phase and amplitude
16
-------
parameters are constant with respect to displacement. In an actual realiza-
tion, as a particle is transported away from the release point, the amplitude
and phase parameters will change. To illustrate this advection effect, we
have sketched the dispersion circle about mean-flow streamlines taken from the
CODAR data over the period 23 to 25 August 1978. Notice that the dispersion
curve from Strait 13 encompasses the Strait 11 site at hours 5 through 12.
Particles advected into the vicinity of Strait 11 would no longer have tidal
amplitudes and phases like those at Strait 13 which was their release site,
but rather would behave like particles released at Strait 11. These effects
would play an important and perhaps dominant role in long-term tidal displace-
ment calculations.
Figures 19 and 20 of Parker (1977) show that the M2 and Kj tidal ellipses
are nearly always in close alignment in this region. Thus the assumption of a
shared major axis is not a bad one. It is possible, however, to calculate the
variance of the distribution of a sum of random variates in which each tidal
current variate can assume an arbitrary spatial orientation with respect to a
reference axis. That is,
5
D(TO) = 2 60(T0)cos0 cos(40 + d)
£=1 s, a a
where <(). is the angle between the reference axis and the variate's major axis.
We assume that
-------
TABLE 4
TIDAL CONSTITUENT ANALYSIS OF MODEL ERRORS
00
STATION ,
STRAIT 11
STRAIT 12
STRAIT 13
Constituent
M2
N2
S2
Kl
°1
M2
N2
S2
Kl
°1
M2
N2
S2
Kl
0,
Major Amp. (cm/sec)
30.5
9.7
6.3
4.6
28.1
9.6
1.8
3.8
1.8
6.8
28.9
13.7
2.1
8.8
17.5
Minor Amp. (cm/sec)
1.9
1.0
4.4
2.6
.9
1.1
.9
1.0
.3
1.8
.3
2.2
.3
.9
5.8
Flood Dir. (°T) Model Dir (°T)
84
111
18 80
58
79
90
148
112 42
84
119
98
125
155 95
98
116
-------
Although the assumptions of case 1 are too coarse to provide a practical
guide to the dispersion problem, they are simple to understand, and therefore
they serve a useful conceptual purpose. Moreover, case 1 lays the foundation
for an interesting comparison between the no-knowledge conditions of case 1
and the uncertain knowledge situation we face when using the tidal-current
model. In this real-life situation, errors in trajectory position will be
caused by inaccuracies in the model's tidal current simulation, as well as by
phenomena not included in the model. The analysis above may be used to esti-
mate the relationship between the tidal-current errors and the error in pre-
dicted position. Table 4 shows the R2SPEC analysis of the tidal-current
signal formed from the difference between the observed tidal currents and the
simulated currents. We consider these difference currents to be random errors
associated with the model's predictions. The source of these error signals
can be traced back to the phase and amplitude errors of Table 2 already dis-
cussed, and to the error in the alignment of the major, or flood, axis.
Further, these errors are augmented by the model's explicit neglect of the
minor-axis tidal velocities.
The last column in Table 4 shows the tidal-current flood direction speci-
fied in the model. Notice in the Strait 12 simulation that both the major-
axis amplitude errors are small, and the direction of the difference veloc-
ity's flood is randomly scattered relative to the model's flood direction.
This scattering in the difference velocity flood direction is what we expect
if the model had no systematic errors. As we can readily see from either
Table 2 or the correlation of the difference velocity flood direction and the
simulated velocity flood direction in Table 4, some sort of systematic error
is present in the model's simulation of the Straits 11 and 13 velocities.
Nevertheless, the uniform scattering assumption provides a useful tool for
analyzing errors associated with the model.
Figure 3 shows the dispersion associated with tidal-constituent errors
of Table 4 under the assumption that the major axis of motion is randomly
oriented. The diagram is rigorously correct only for. Strait 12, but the
features of the dispersion envelopes of Straits 11 and 13 are qualitatively
correct if we allow for a moderate (up to 33%) elongation of the circles in
the flood direction, and an equally sizeable reduction in the direction per-
pendicular to the flood. This diagram illustrates the contribution that the
model makes to reducing uncertainty in the predicted position of a drifting
object, assuming all nontidal currents are known exactly. At best, and
after a few hours, the model predicts the location to within about 1 km at
Strait 12. At Strait 13, the radius of the circle of uncertainty is about
3 km after 5 hours, and at Strait 11 the radius is about 4.5 km after 5 hours.
This corresponds to a reduction in search area of 90%-95% at Strait 12, and
(perhaps) 50% at Straits 11 and 13.
CASE 2
Case 2 is the converse of case 1. We now assume that drifter position
predictions are made in ignorance of all nontidal motions, but with complete
knowledge of the tidal currents. Again we are forced to limit the duration of
the trajectory prediction to small times, say 12 hours, to prevent out-running
our localized knowledge.
19
-------
Unlike the tidal currents, the physical mechanisms behind the nontidal
currents are diverse and poorly documented. The simplest way to categorize
these motions is in terms of the manner in which they appear in our current
observations. One class of motion has periods shorter than one hundred hours,
and these motions can be resolved as frequency components in Fourier analyses
of the current records now available. As a general rule, these motions are
weak; a typical root-mean-square amplitude for all the motions falling in the
frequency band between the daily and semidaily tides is only 3 or 4 cm/sec.
The other class of motion is comprised of the long period oscillations that
appear as mean values in our 15- and 29-day current observations. These
motions are not truely steady currents. In all likelihood, these slowly
varying currents are associated with changes in the density structure of the
region. The density structure is principally dependent upon fresh-water
runoff from the Fraser River and also from the onshore advection of low-
density surface waters by winds acting on the region lying west of the coast
of the state of Washington. These motions are generally of large amplitude,
and the intrusions of low-density surface lenses have a sporadic appearance in
the current records.
The estuarine runoff is perhaps the most predictable feature of the
region, and in the Strait of Juan de Fuca at midchannel, the average
outflowing current varies from a minimum value of 10 cm/sec to 30 cm/sec in
late winter to a high of 30 cm/sec to AO cm/sec in late summer (Cannon, 1978,
Figure 27). This flow is also seen in CTD sections across the Strait and in
satellite images of surface-water properties. Superimposed on this outward
flow are intrusions of low-density lenses of water advected to the mouth of
the Strait by winds acting offshore. These intrusions may occur rather fre-
quently, and they can result in eastwardly (up-channel) flow along the south
shore of the Strait of Juan de Fuca with speeds of up to 50 cm/sec and dura-
tions of many days. The CODAR images of the Port Angeles region on 26 August
1978 document the complex behavior of the upstream front of one of these
intrusions (Fris-ch, Holbrook, and Ages, 1980).
The dispersion caused by the weak, high-frequency motions that are re-
solved in our current records may be estimated using a procedure that is en-
tirely analogous to that used on the tides in case 1. We group the current
speed energy in the various frequency bands into five discrete frequencies and
model these motions as though each discrete frequency was the equivalent of a
tidal constituent in case 1. Table 5 shows the parameters we have used for
this representation of the nontidal motions. The values in this table reflect
a Fourier analysis of the kinetic energy at Straits 11, 12, and 13 as seen in
the 16 July 1978 deployment (Holbrook, personal communication).* We assume
for this case that the orientation of this motion is random and so use equa-
tion (12b) to calculate the variance of the drift position. Fiqure A shows
the dispersion envelopes for these representations. Notice that the error
associated with the neglect of these motions is a very small fraction of the
case 1 error, and that it is comparable with the tide model dispersion enve-
lope at Strait 12 (Figure 3). These currents are probably caused by the
oscillating wind stress that acts on the region. These stresses result in
highly localized surface boundary layers, and in weak barotropic current
* Mr. J. R. Holbrook, NOAA/PMEL, Seattle, Washington.
20
-------
TABLE 5
PARAMETERS FOR THE DISCRETE FREQUENCY SIMULATION OF
THE NONTIDAL OSCILLATORY CURRENTS
Freq. Band
(cyc/hr)
.008 - .016
.016 - .024
.024 - .032
.05 - .07
.09 - .2
Center Freq.
rad/hr
.075
.125
.176
.377
.911
Amplitude
Strait 11
2.2
2.2
2.2
3.5
8.1
of Discrete Sinusoid (cm/sec)
Strait 12 Strait 13
2.2 2.0
2.2 2.0
2.2 2.0
3.5 3.2
8.1 7.4
fields that extend over the whole region. It is reasonable to consider these
motions omnidirectional.
Figure 5 depicts the magnitude and direction of the average current
observed at a number of locations within the central basin and Strait of Juan
de Fuca regions. These vectors are taken from Figure 38 of Cannon (1978).
With the exception of six vectors taken from the mouths of Rosario and Haro
Straits, which show a uniform out-strait flow, the remainder are widely scat-
tered and show no obvious channeling. These averages are the net flow seen
over various periods ranging from 10 to 30 days, and we reiterate that they
are not good estimates of the true average because of this relatively short
record length.
Although these vectors are not good estimates of the true, statistical,
average current, it is reasonable to presume that they are representative of
the net current one might encounter in any given drifter experiment. The
vectors themselves come from many different locations, and it would make no
sense to try to combine these values in an elaborate statistical model. We
therefore limit our analysis to the simple determination of the average mag-
nitudes of the observed net current and let this value characterize the ran-
dom, net drift we should expect in a drifter experiment. This average was
found to be .66 km/hour.
Figure 6 shows three groups of concentric circles centered on Straits 11,
12, and 13 respectively. These circles depict the hourly transport caused by
an average drift of .66 km/hour. The circular geometry reflects our uncer-
tainty in determining the direction of the net drift. These circles are to be
interpreted as a characteristic transport distance. If there were no tidal
currents or other motions, and if the streamlines were straight, then a
drifter would, on the average, be located somewhere on the circumference of
one of these circles. Figure 4 and Figure 6 are the counterpart to Figure 2.
A comparison of Figure 4 and Figure 6 shows that the very-low-frequency, den-
sity-related motions depicted in Figure 6 are the critical elements in the
case 2 problem.
A comparison of the area of the 12-hour envelope of the net drift portion
of case-2 to the case-1 envelope (which encompasses all position errors from
the release to hour 12), shows them to be of comparable size. A better com-
21
-------
parison is to plot the radius of the position error as a function of time for
the two cases. This is done in Figure 7, where we can see that the error
associated with complete ignorance of the tides (case 1) is somewhat greater
than the error caused by ignorance of the nontidal currents, for trajectory
durations less than 10 hours. After that time, the case 2 current errors
become dominant. The contribution of the high-frequency, nontidal oscilla-
tions in this figure can be seen to be of only modest importance.
CASE 3: Existing Model with Assumed Steady Current
The uncertainty in position due to our ignorance of the very-low-fre-
quency currents (Figure 6) is so great as to require the adoption of some
hypothesis regarding the behavior of these currents. The alternative, the
treatment of these currents as a simple random process, results in a region of
uncertainty that is so large as to obscure whatever insights the tidal-current
model might provide. This is not a novel observation. In a previous study of
oil spill trajectories in the region (Stewart, 1978), the results were framed
in light of the contrast between a no-steady-current hypothesis and a uniform,
ebb-directed, steady-current hypothesis. The necessity for such an assumption
that is provided by Figure 6 is simply another way of arguing the same point.
The case 3 assumptions are that we know these long-period currents. The
purpose of this analysis is to outline the region of uncertainty associated
with the use of our model in such a situation. There are two ways we can pro-
ceed. We can combine our previous analytical results, principally Figures 3
and 4, or we can integrate the differences between the observed current
velocities of the simulated current velocities to estimate the time behavior
of the error. We have adopted the latter method because it is less dependent
on the many assumptions required by the analytical method, and because it will
serve as a check on our previous calculations.
Three difference-current time series were created and then divided into
48-hour intervals. An initial time was randomly selected within each inter-
val. We then integrated the difference velocity for durations of 1, 3, or 10
hours to determine one realization of the error growth. Using distinct 48-
hour intervals, we acquired between 20 and 30 independent realizations for
each of the three durations, at all three stations. The resultant (x, y)
positions were then analyzed to determine the variance-covariance matrix, and
the coordinates of the ellipse containing 68% of the observations were then
calculated assuming the (x, y) variates were drawn from a bivariate normal
distribution. This ellipse was compared with the scatter plot to ensure
qualitative agreement with the data.
Figure 8 depicts these ellipses for Straits 11, 12, and 13 at durations
of 1, 3, and 10 hours. The steady flow used in creating these figures was
taken from the net flow in the current meter observation. Notice that Straits
11 and 13 have flows consistent with the CODAR streamlines used in our pre-
vious representations, but the net flow at Strait 12 is oriented in the oppo-
site direction. This graphically illustrates the importance of the slowly
varying currents, and it gives warning of the dangers in assuming knowledge of
this phenomenon.
22
-------
SECTION 6
WIND FIELD SIMULATION
The wind field in the region is modeled using a time series of hourly
wind velocity observations at a master station and a large-scale wind pattern
that extrapolates the wind velocity to areas that are distant from the master
station. The wind pattern is determined using a numerical model that calcu-
lates the winds that would result from the application of a steady pressure
gradient across the region (Overland, Hitchman, and Han, 1980). This submodel
accounts for the topographic channeling of the dense marine boundary layer.
The spatial resolution of the program is such that the major features of the
Olympic Mountain Range are discernable, but hills such as those that are seen
in the city of Seattle do not appear. The cost of computing these regional
wind patterns is great, and so the model uses a library of 8 patterns, each
selected to represent a characteristic pressure gradient condition. A se-
quence of pattern types is selected for the period of the master station wind
velocity observations, based on the evolution of the regional pressure gra-
dient. This is presently a tedious and somewhat subjective manual task. The
wind simulation is thus based on empirical observations, but the interpola-
tion/extrapolation junction i-s determined objectively from a physically based
numerical model. In this respect the wind model differs from the tidal-
current model in which the interpolation methods were based on manually deter-
mined coefficients of uncertain underpinning.
Some numerical algorithm is required to combine the wind patterns and
master station observations. Stewart (1978), in an oil-spill study using the
MESA-Puget Sound model, developed a technique wherein a scaled and rotated
hourly difference velocity was applied throughout the region. This technique
caused the simulated velocity at the master station to equal the observed
velocity, and at surrounding points the simulated velocity changed in ampli-
tude and direction in a fashion determined by the wind pattern. A charac-
teristic of this method of extrapolating the observed velocity is that if the
pattern velocity equals the observed velocity, then the pattern velocity is
applied unperturbed throughout the region.
The purpose of this algorithm was twofold. First it caused the simulated
wind field to vary hourly. These high-frequency variations would not be
present in the simulation if the wind field had been created by using a se-
quence of patterns. Secondly, the algorithm assured agreement between the
simulated and observed winds at the master station.
A complete evaluation of the wind field simulation should be based on an
examination of both the patterns and the algorithm used to turn these patterns
into extrapolation and interpolation functions. However, for this study, we
have judged it sufficient to examine just the characteristics of the patterns.
23
-------
We have taken this course because Stewart's algorithm was selected largely on
subjective grounds, and whatever merit the algorithm might have, it is predi-
cated on the assumed agreement between the model's wind patterns and the
observed winds. If the patterns are a poor fit to the data, then any agree-
ment between the algorithm's velocities and the observations would be purely
fortuitous. Little would be learned if the fit was less than perfect. Fur-
ther, an examination of the pattern errors should uncover any simple method
for perturbing the wind patterns so as to achieve the purposes of the extra-
polation algorithm.
Because the determination of the pattern sequence for an extended period
is a laborious task, we decided to closely examine the wind field simulation
over a short time interval. To be compatible with the analysis of the joint
experiment data, the period 22-26 August 1978 was selected. Large scale
isobaric weather maps and the hourly wind observations at Race Rocks were
obtained and given to the wind model's creator, Dr. James E. Overland of PMEL,
who determined a pattern sequence based on 6-hour pattern intervals. The
period selected for study turned out to have rather anomalous weather for the
summer months, but we judged it to be as fair a test as any other that might
be devised around a five day period. The pattern sequence for this period was
determined to be:
6,5,5,5,5,2,2,3,3,3,3,4,4,5,5,3,3,4,4,4
where the integer serves as an index to the pattern type. It can be seen that
the bulk of the interval was accounted for by patterns 3, 4, and 5. These
patterns correspond to pressure gradients such that the large scale, extra-
regional flow is from the south, southwest, and west respectively. Figures 9,
10, and 11 show the regional wind fields for these cases.
To see whether there is a basis for separating the time series into
segments, we performed a principal component analysis of the wind observations
from Smith Island, New Dungeness, Race Rocks, and Tatoosh Island. These
locations are shown in Figure 12. This analysis revealed that 42% of the
variance observed at the four stations could be accounted for by perturbations
to the mean wind speed that would simultaneously lie to the east-northeast at
Tatoosh, south at Race Rocks, west-northwest at New Dungeness, and north-
northwest at Smith Island. The mean wind velocity and these perturbations ere
shown in Figure 13a. This result may be interpreted as follows. Under condi-
tions that would cause the wind at Tatoosh to become more southerly (in the
meteorological sense), the wind at Smith Island would swing a little to the
east, the wind at Race Rocks would swing to the north from its mean north-
westerly direction, and the wind at New Dungeness would swing to the southeast
from its mean southerly value.
The second principal component is depicted in Figure 13b, and it accounts
for 18% of the observed variance. Notice that positive perturbations to this
component will partially cancel positive perturbations to the first principal
component at Tatoosh and Smith Islands, while enhancing the northerly and
easterly swings at Race Rocks and New Dungeness respectively. Alternatively,
if the second component is given a positive perturbation while the first
component is negatively perturbed, then the Race Rocks and New Dungeness
24
-------
velocities would tend to remain constant while the wind at Smith Island would
drop and swing south, and the wind at Tatoosh would drop and swing to the
east. The third principal component accounted for 14% of the variance, and it
is the last significant descriptor of the data, the remaining, smaller compo-
nents accounting for 9%, 7%, 6%, 3%, and 1% of the variance respectively.
This component is shown in Figure 13c. Whereas the first two principal compo-
nents caused variations of approximately equal size at all four locations,
this third component is effective mostly at the two eastern-most stations,
Smith Island and New Dungeness. It can be seen that negative amplitudes
applied to this component will result in a westerly rotation and increase in
wind speed at Smith Island, and a calming at New Dungeness. The wind at Race
Rocks simultaneously increases and swings to the north. This third component
thus represents some kind of eddy in the lee of the Olympic Mountains.
An obvious question is whether these components are related to the pat-
tern sequences. Figure 14 shows the amplitudes of these three components at
three hourly intervals. The three plots, W2 vs Wj, W3 vs Wj, and W3 vs W2,
are two-dimensional plan views of the three dimensional (Wj, W2, W3) space.
That is, any given wind velocity observation can be approximated by a summa-
tion of the three principal components with weights Wlf W2 and W3. The (Wx,
W2> ws) space thus provides a means of representing the observations. The
bulk of the observations form a cluster in the vicinity of the origin, with a
slightly negative average value for the W2 component. However, the large
amplitude observations tend to fall into two groups. Observations that were
made during Pattern 3 intervals have positive weights on the first component
and negative weights on the second and third. This results in an increasing
southerly wind at Tatoosh and Smith Islands, a southerly wind at New Dungeness
that drops and swings to the east, and a north-northwest wind at Race Rocks
that increases slightly. Behavior of this type would seem to fall somewhere
in between Patterns 3 and 4 and is not well represented by either (see
Figures 9 and 10).
Observations made during Pattern 4 and Pattern 5 intervals, that had
large negative weights on the first component, had large positive weights on
the second component and negative weights on the third. This corresponds to
an easterly swing at Tatoosh, a westerly swing at Race Rocks, a slight drop at
New Dungeness, and a slight westerly swing at Smith Island. The Tatoosh
behavior does not correspond to either Pattern 4 or 5. The New Dungeness
behavior seems more characteristic of Pattern 4 than Pattern 5, and yet Pat-
tern 5 intervals had the extreme weights in this cluster.
Another way of looking at the data is to draw a wind velocity scatter
diagram for each pattern type at several stations. Focusing now on the cen-
tral basin, Figure 15 shows the wind velocity clusters at Smith Island, Point
Wilson, New Dungeness, and Race Rocks. Notice that the Pattern 3 data clus-
ters nicely at Smith Island and Point Wilson. It is rather scattered at Race
Rocks and strongly bimodal at New Dungeness, with the predicted wind rather
too easterly for the large amplitude cluster. Wind Pattern 4 correctly places
the wind velocity at New Dungeness at calm. Other stations tend to be rather
scattered, although the model predictions are indicative of the predominant
quadrant. Wind Pattern 5 does a poor job of sorting or predicting the ten-
dency of the Smith Island, New Dungeness, and Point Wilson observations, but
25
-------
it picks up one mode of what appears to be a bimodal distribution at Race
Rocks.
These observations are too limited in number to warrant further statis-
tical analysis. However, they do suggest some general conclusions. First,
there appears to be a strong eddylike motion between Race Rocks and New
Dungeness. This motion apparently results from extraregional flows that fall
somewhere between the Pattern 4 and Pattern 5 boundary condition. Further,
the bimodal behavior of the wind at Race Rocks and New Dungeness suggests that
there is some threshold value of pressure gradient below which light and
variable winds prevail. Once this threshold is exceeded, the model's predic-
tions are somewhat more reliable, but still subject to relatively large errors
in direction. The behavior of the wind at Tatoosh under Pattern 4 and 5 con-
ditions is paradoxical, but perhaps this simply reflects the importance of
sub-grid-scale topography on local winds.
The fundamental precept of the wind model is that a substantial fraction
of the regional wind patterns can be represented using a library containing a
fairly small number of patterns based on time-steady pressure gradients. This
assumption is critical, and it requires a great deal more study. In the
five-day period we examined, the principal component analysis of the wind
observations at four stations determined that 74% of the variance could be
accounted for by using three wind patterns. Unfortunately the three patterns
determined from the model did not show much agreement with the principal
component patterns.
Another important point that can be made regarding the wind field simu-
lation is the importance of matching the master station selection with the
wind field behavior. None of the stations that we examined had uniformly good
agreement with the three pattern types that occurred during the observation
interval. For example, only one Race Rocks observation in Pattern 3 intervals
fell within ten degrees of the predicted wind. By using Race Rocks as a
master station for Pattern 3 winds, we essentially negated the good fit seen
at the other three stations (Figure 15), since a large error velocity would be
determined from the difference between the Race Rocks observations and the
model predictions, and this error would then be applied to the other stations.
More generally, the selection of an algorithm for combining observed
winds and the regional patterns appears to be a difficult problem. If the
wind field actually has a threshold for its response to the large-scale pres-
sure gradient, as suggested by the histograms at Race Rocks and New Dungeness,
then the algorithm will have to create very small-scale, weak winds for condi-
tions below the threshold, coupled with patternlike winds once the threshold
is exceeded. The statistical properties of the sub-threshold local winds will
have to be studied to determine spatial coherence scales and the characteris-
tics of the time variability.
SIGNIFICANCE OF WIND MODEL ERRORS
Because of the limited amount of data and the subsequently qualitative
nature of the analysis, there is no reasonable method for estimating in a
statistical sense the magnitude of the simulated wind velocity errors. Thus,
26
-------
even if we postulated a formula relating wind and drifter velocities, it would
not be possible to make quantitative estimates of the dispersion caused by the
model errors. However, it is possible to draw some conclusions regarding the
adequacy of the simulation method.
To use the model in its present state, with its limited pattern reper-
toire, is to chance misestimating the wind by 45° to 180° in certain key
regions. The area off Port Angeles is one such region when the extraregional
wind is from the south to southwest. The model predictions could be in error
by 180°; and with a 3% wind drift factor, the resultant drift could easily be
in error by 15 cm/sec to 30 cm/sec, or an amount equivalent to the steady
baroclinic currents.
27
-------
SECTION 7
DRIFTER RESPONSE STUDIES
When we first proposed this work, we envisioned an analysis of the Cox,
Ebbesmeyer and Helseth (1979) drifter data. This study was to reveal the
response of drifters to surface winds, and it was to be the subject of a
separate report. Because of the relatively large errors in the tidal-current
and wind models, the proposed analysis was not successful. Although this work
may seem somewhat tangential to the present report, it does provide further
insight into the importance of the model errors. Certainly, any model that
cannot make good hindcasts must be of questionable validity in the forecast
mode.
The results of this failed investigation are important also because they
bear on the concept of the oil-spill-of-opportunity research program, one task
of which was to infer the transport equations for oil slicks from data
acquired in actual oil spills. In contrast to most oil spill investigations,
we had the benefit of many current meter and anemometer records, we had good
data on the positions of easily recognized drifters, and we had our computer
model with which to perform the various calculations and interpolations. The
fact that our analysis was inconclusive despite all these advantages ought to
be considered in the design of any future spill-of-opportunity experiments.
The drifter data were taken from the Plates in Cox, Ebbesmeyer, and
Helseth (1979). This data consisted of the position of 97 drift sheets
deployed over the period 22-26 August 1978. Positions were determined by air-
craft overflying the sheets, and the error in position was estimated by Cox
et al., at 31 meters. The data were acquired only during daylight hours and
the time between fixes was nominally about 30 minutes. Using the known posi-
tion, we determined the average velocity by dividing the distance traveled by
the time between fixes. This velocity was applied at the midpoint between the
known positions. If the time interval exceeded two hours, the point was cast
out.
We then sorted the data by day and by location. The latter sorting was
based on a nearly square grid in which the north-south cell dimension was 2
minutes of latitude and the east-west cell dimension was 3 minutes of longi-
tude. This corresponds to a square about 3.6 km on a side. This data set was
then examined, and grid squares that over the five-day experiment had fewer
than ten distinct drifters were combined with adjacent squares to form a cell
with sufficient data to allow statistical treatment.
Because the wind model had proven itself unreliable in this area, we
decided to use the observed winds at Race Rocks and at New Dungeness for the
estimate of the surface wind. The wind data from Race Rocks and New Dungeness
28
-------
were stored in our data arrays at hourly intervals. The hourly value at New
Dungeness was the average of three readings taken at twenty-minute
intervals centered on the hour. The drifter velocities, however, were deter-
mined at irregular times as dictated by the times of the position fixes. To
accommodate this problem, the wind was linearly interpolated from its hourly
value to the time of the drifter velocity determination. In this interpola-
tion, we might have done better at New Dungeness had we used the original,
twenty-minute wind records.
Several data files were then created containing the interpolated wind
velocities and a residual velocity equal to the difference of the drifter
velocity and the tidal-current velocity as determined by the model. These
data files were then analyzed to see if there was any statistical basis to
support a drift equation of this general form:
(13)
cose
n lu.i + Kl * fu 1 * F£ll
j (v*J + [v*J + [vjj * UJ
where (uj»vj) is the drifter velocity, (un>v/>) *s tne l°cal wind velocity,
(u ,v ) is the tidal current velocity, (u ,v ) is the slowly varying baro-
w. t S S
clinic current velocity, 0' is an angle of rotation between the local
surface wind and the motion it induces at the surface, and e. and e» are
random measurement errors. Since we used a shore-based station to estimate
the local winds, we must also include a possible rotation and amplification
of the shore station wind. That is, in any given grid cell,
cose" -
" -sine"! fu
" cos«"J Lv
(14)
Thus the complete model is:
= or [~cose
a [sine
"Sin6
cose
l Fuwl + K
j [v*J LV^
(15)
where a = a'*a", 6=6' +6".
Let (uf,v') be the tidal currents predicted by the model, so
The (Up, Vp, u , v ) covariance matrix is then given by:
J\ *» W W
= a cosB a2, - 2 sinO cose a + sin e a + a
Vw wvw w w
(16a)
VR
2 22
* ' 8in9Vw
+2 sin6 costr
Vw
22
cosO a
Vw
°ee <16b>
29
-------
2 2
a cos6 sine (a
u u
w w
- a
V V
w w
222
) + a (cos 6 - sin 6)
U V
w w
(16c)
^ »-y
u
K W
COS0 a
u u
W W
Sin6 a
u v
W W
(16d)
n v
K w
= a cos6 a
u v
w w
- a sin6 a
v v
w w
(16e)
v J
v U
W
cos9 a
u v
W W
Sin6 CT
u u
W W
(l6f)
vv
w
= a cos6 a
vv
w w
a sine a
uv
w w
(I6g)
<• &'
-• -
provided u~ - u_ is neglible. Here (u ,v ) is assumed the independent vari-
able, and we assume
= a
=a
In addition to the seven conditions of equation (16), we may also show
22 222
that a >0, a > a and a > a are implied conditions.
ee VRVR 8£
The covariances on the left hand sides of equations (16b) through (I6g) may
22 2
be estimated from the data, as well as the a , o , and a terms.
' u u u v ' v v
w w w w w w
Thus we have seven equations, three constraints, but only three unknowns, Of,
6, a2 . There are as many as 50 different ways of formulating a solution in
Co
this over-determined system, and each should yield reasonably similar esti-
mates of a, 6, and a2 , if the model is correct.
OO
We performed this analysis using four different methods for evalua-
ting a, 6, and a2 for the grid cells lying just off New Dungeness. In
o£
no case did we find a data set that yielded four estimates of a, 6, and
a2 that simply satisfied the constraints, let alone exhibit any con-
Co
sistency. In short, the model, equation (15), did not predict the proper
covariance structure.
30
-------
The reason for this failure may be directly attributed to the errors in
the tidal-current model. The variances of the residual velocity components
were typically in the range of 10 cm/sec 2 to 300 cm/sec 2. The amplitude
errors in the tidal-current simulation have been shown to be in the range 10
cm/sec 2 to 30 cm/sec 2, which correspond to variances that are equal in
size to the residual velocity variances. Thus, VL, - ul is far from negli-
gible, and, in fact, we could argue that most, if not all, of the variability
seen in (u_, v ) was due to model error and not the wind.
We also performed a number of tests using the Race Rocks wind records,
and we tried lagging the wind one hour. None of these tests resulted in a
significant relationship.
Figures I6a and I6b show the average residual currents calculated for the
days of 25 and 26 August 1978, respectively. These figures show the same
intrusion reported in Frisch et al. (1980), and which is also seen in Plates
3d and 3e of Cox et al. (1979). The tidal-current model error is smaller than
assuming ul is zero, and so averages formed using the model should be slightly
superior to those made without the model.
31
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SECTION 8
REFERENCES
1. Cannon, G. A. (editor) 1978: Circulation in the Strait of Juan de Fuca:
Some Recent Oceanographic Observations, NOAA Technical Report ERL-
399-PMEL-29, Environmental Research Laboratories, Boulder, Colorado,
49 pp.
2. Cox, J. M., C. C. Ebbesmeyer, J. M. Helseth 1979: "Surface Drift Sheet
Movements Observed in the Inner Strait of Juan de Fuca" Evans-
Hamilton Inc. 6306 21st Ave. NE, Seattle, Washington 98115.
Completion Report submitted to MESA Puget Sound Project Office,
NOAA.
3. Friedman, J. H. and L. C. Rafsky 1979: "Multivariate Generalizations
of the Wald-Wolfowitz and Smirnov Two-Sample Tests" The Annals
of Statistics Vol. 7, New York, pp. 697-717.
4. Frisch, A. S., J. E. Holbrook, and A. B. Ages 1980: "Observations of a
Summertime Reversal in the Circulation in the Strait of Juan de
Fuca" submitted to J G R.
5. Godin, G. 1972: The Analysis of Tides, Univ. of Toronto Press, Toronto
and Buffalo, 264 pp.
6. Gradshteyn, I. S. and I. M. Ryzhik 1965: Table of Integrals, Series and
Products, Academic Press, New York and London, 1086 pp.
7. National Ocean Survey 1973a: Tidal Current Charts, Puget Sound North
Part, NOAA, National Ocean Survey, Rockville, Maryland, 12 pp. +
endpapers.
8. National Ocean Survey 1973b: Tidal Current Charts, Puget Sound Southern
Part, NOAA, National Ocean Survey, Rockville, Maryland, 12 pp +
endpapers.
9. National Ocean Survey 1979: Tidal Current Statistics for Puget Sound
and Adjacent Waters (unpublished data sets), NOAA, National Ocean
Survey, Rockville, Maryland, 38 pp.
10. Overland, J. E., M. H. Hitchman, and Y. J. Han 1979: "A Regional Surface
Wind Model for Mountainous Coastal Areas" NOAA Technical Report
ERL-407-PMEL-32 Environmental Research Laboratories, Boulder,
Colorado, 34 pp.
32
-------
11. Parker, B. B. 1977: Tidal Hydrodynamics in the Strait of Juan de Fuca -
Strait of Georgia, NOAA Technical Report, NOS-69, National Ocean
Survey, Rockville, Maryland, 56 pp.
12. Pease, C. H. 1980: An Empirical Model for Tidal Currents in Puget Sound,
Strait of Juan de Fuca, and Southern Strait of Georgia, DOC/EPA
Interagency Energy/Environment R & D Program Report, in press.
13. Schureman, P. 1958- Manual of Harmonic Analysis and Prediction of Tides,
Coast and Geodetic Survey Special Publication No. 98, U.S. Govern-
ment Printing Office, Washington, B.C., 317 pp.
14. Stewart, R. J. 1978: "Oil Spill Trajectory Predictions for the Strait
of Juan de Fuca and San Juan Islands for the Bureau of Land
Managements Review of the Northern Tier Pipeline Company's
Proposal," Pacific Marine Environmental Laboratory, NOAA Environ-
mental Research Laboratories, Seattle, Washington.
33
-------
APPENDIX A
CALCULATION OF TIDAL PHASE AND AMPLITUDE
FROM MODEL COEFFICIENTS
The tidal model is based on the 5 constituents found in Table A of
Parker (1977), M2, S2, N2, Klt and 0 . Parker's Epochs (or phase lag) are
measured with respect to local time (LT). Since the model also uses LT at
the subprogram level, the phase for one of the constituents is given by:
Qla = (v° + u) ' Mls,
where p denotes Parker's station number, £ denotes the tidal constituent
(e.g., M2,N2, etc.)i t denotes local time, and the "equilibrium argument," (V0
+ u), is taken from Table 15 of Schureman (1958). The time in the model is
measured relative to 0000 January 1, 1978, and, for example, the equilibrium
argument for the M2 constituent is 201.8°.
The major axis velocity of one of the model's reference stations thus
assumes the following form:
V|(T) = A| cos (a^T + 6^) (A2)
where the superscript again denotes the reference station number, and the
subscript denotes the tidal constituent. Note that the model calculation is
based on arrays of 0^ . and not on the epoch, K, and equilibrium argument.
t ,*
The simulated current for tidal constituent "£" at a grid location (i,j)
is determined from the summation of as many as three, weighted, reference
station velocities. Thus:
= I W(i,j,m)A^1'J»nuco.(oDT 1- eri^'^), (A3)
.. Jv- Jt X • J&-
m=l '
where the reference station index, P(i,j,m), and weighting factor W(i,j,m)
are determined from arrays in which the east-west and north-south positions
determine the principal indices i and j. The index 'm' is just the sumnation
index and is of no further consequence. Our present discussion is devoted
to the demonstration that equation (A3) is properly implemented in the model.
34
-------
Dispensing with the cumbersome notation of (A3) we can see that our
simulated tidal current for constituent, K, is created numerically using
an equation of the general form:
3 .
A, COS(CTT + , in terms of the b and 6 parameters.
iiu-s is readily done by expanding tfie cosine function with its (at + 8 )
argument into sine and cosine constituents of argument.CTt, each multiplied
by sines and cosines of argument 8 , (or 8,) and the b (or a.) amplitudes.
by sines and cosines of argument . ,
This results in two equations for ij>lw and
= tan
m
mI1bmcosem
(A4a)
a, =
(A4b)
These are the phase and amplitude of the simulated tidal current assuming
the model is functioning properly. They are to be compared with the
R2SPEC analysis of a time series generated by the model.
The model creates its simulated time series through the summation of
the five tidal constituents mentioned above. Thus,
. .
Vlt3(t) = Z Vj'j(t)
*
(A5)
This velocity lies along an axis determined by the flood-ebb direction. It
is converted to u (east-west), v (north-south) components with a simple
rotation transformation. It is this time series of u, v components that
comprise the data analyzed by R2SPEC.
The only difficulty in this comparison is the conversion of the model
phase lag parameters into forms suitable for comparison with R2SPEC's output.
R2SPEC lists the constituent Epochs in a form consistent with Schureman
(1958), which is to say that it is referenced to GMT time, not local time.
The conversion is made as follows:
and
u)* ~ 8tl(see Eq-
;j =: 00^j + of (8 hrs.)
(A6)
where T denotes GMT time and a. is constituent Jfc's frequency in degrees per
hour.
35
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APPENDIX B
MOMENTS FOR RANDOMLY ORIENTED CONSTITUENT DISPLACEMENTS
If the major axis of the 2th constituent is randomly oriented with
respect to a reference axis, then the amplitude of the motion along the
reference axis will be given by the product
x = 6cos<|>0 cos£_; (Bl)
where 6 is the amplitude of the motion,
4>. is the angle between the reference axis and the axis of motion, and
4- is the phase of motion.
We assume <|>, and £p are independent, uniformly distributed random variates
lying in the interval (0,27r).
We now define y = 6 cos<)>. and z = coslj., and note that the distributions
on y and z are given by:
fz(z) = 1 i ; fy(y) = 1 ; (B2a,b)
7t(l-z2)% 7i6(l-/y\2)%
where -l^z^+1, and -6^y^
For XQ greater than zero, we can readily derive the probability that
x is less than or equal to XQ:
P[xgx0] = i-^fW* dz fy z(y>z> = l-2/*dy fy(y)[l-Fz(^)];6>x>0 (|3)
y
where F is the cumulative distribution on z.
z
The marginal distribution is obtained by taking the derivative of the
cumulative distribution, P[X^XQ], whence:
Using similar arguments, a corresponding form can be found for -
Upon inspection, it will be seen that the equation for -6£xo-0 *s Just a
transformation of (B4) in which x0 = -x0. Thus, (B4) holds for the whole
range of x, -
36
-------
The integral (B.4) can be integrated in term's of elliptic integrals
of the first kind, with the result:
£ (x0) = 2 1 F( n, l-(^)2 ) (B5)
x nz 6 2 °
(See Gradshteyn and Ryzhik, 1965, §3.152(10) pg 246.)
This form can then be used to find the higher moments of the dis-
tribution. The function is symmetrical with -6^x^6, and so all odd moments
are zero. The second moment is found as follows:
_/2\2 62 r1
' W J o
M2 62 1 jl +
"W 2 2
E(k)dk . (G. and R. , §6.147, pg 637)
0 2
G = .2865 62. (G. and R. , §6.l48,pg 637)
where K(k) = F( ^,k), is the complete elliptic integral; and
G = .915 965 594.. .,
is Catalan's constant. Higher moments can be readily deduced from §6.147.
37
-------
co
00
©STRAIT 12
p-fc,
P"31 ADMIRALTY
INLET
factors
-------
to
I2HRS
IOHRS
5HRS
IHR
12 MRS
IOHRS
5HRS
HR
STRAITS 12
SMITH
G>
I2HRS
IOHRS
5HRS
HR
STRAITS 13
PROTECTION
ADMIRALTY
INLET
PT. WILSON
Figure 2. Dispersion envelopes assuming no knowledge of tidal currents
(Case 1).
-------
12 MRS
10 MRS
5 MRS
IHR
I2HRS
IOHRS
5HRS
HR
I2HRS
10 MRS
5 MRS
IHR
STRAITS 12
SMITH
e>
STRAITS 13
PROTECTION I
ADMIRALTY
INLET
PT. WILSON
Figure 3. Tide model contribution to reducing position error relative
to Case 1.
-------
Figure 4. Dispersion envelopes associated with ignorance of high
frequency non-tidal current oscillations (Case 2).
-------
20 cm/sec
.72km/hr
Figure 5. Observed net currents at various locations in the Strait of
Juan de Fuca and the Central Basin.
42
-------
CO
Figure 6. Dispersion caused by randomly oriented net current of
.66 km/hr (case 2).
-------
I
§
CC
I
2
9-
8-
7-
6-
5-
4-
3-
2-
I-
0.
O STRAIT II
D STRAIT 12
• STRAIT 13
CASE I ERRORS
CASE 2 ERRORS
01 23456789 10
TRAJECTORY DURATION (HOURS)
ERROR CAUSED BY NON-TIDAL
"HIGH" FREQUENCY OSCILLATIONS
D
Figure 7. A comparison of Case 1 and Case 2 errors.
-------
Figure 8. Difference - current dispersion ellipses at 1, 3, and 10 hours.
-------
Figure 9. Wind Pattern No. 3
46
-------
Figure 10. Wind Pattern No. 4
47
-------
PATTERN 5
Figure 11. Wind Pattern No. 5
-------
49C
48.5£
VO
48C
TATOOSH
47C
125C
124.5° 124° 123.5°
Figure 12. Wind station locations.
123C
122.5C
-------
1st COMPONENT
NEW DUNGENESS SMITH I.
2nd COMPONENT
RACE ROCKS
NEW DUNGENESS SMITH I
3rd COMPONENT
RACE ROCKS
TATOOSH ONEW DUNGENESS & SMITH I
Figure 13. Principal components of surface wind observations.
50
-------
a
5- o
41
j. 00
21 °
11
0
-I- 0 <
.2. o °
-3-
4 - c
-cj a
-5 -4 -3 -Z -i
*
a . .0
* *
°B W, W3 * o °
o
•
o
a • a ' •
*o
a
T 2 3 •> 5 -5 -4 -3 -2 -i
4 •
V °
o
2- «
I • •
: .„ ..
a °
o r5
4
0 0 .3
o .2
1
o
0 o
0 O "1
o •
• "3
-4
-5
*
" 1 2 3 4 ~5
^3 PATTERN
• 3
o 4
a
•
•
D
oL W,
-I' a' ° %
-2. o o • f
-5-4-3-2-1 12345
Figure 14. Scatter plot of principal component weights.
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5 m/sec
SMITH 1.
POINT WILSON
RACE ROCKS NEW DUNGENESS
WIND
PATTERN
3
ro
WIND
PATTERN
4
WIND
PATTERN
5
Figure 15. Scatter plot of wind observation, sorted by pattern type
at four central basin stations.
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en
\
VELOCITY SCALE
50 cm/sec
DECEPTION
PASS
SMITH I.
O
ADMIRALTY
INLET
PT. WILSON
PROTECTION
Figure 16a. Average residual currents on August 25, 1978
(residual = drifter - model).
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tn
CATTLE PT.
Cl
0
-J
VELOCITY SCALE
50 cm/sec
ADMIRALTY
INLET
PT. WILSON
PROTECTION I
Figure 16b. Average residual currents on August 26, 1978
(residual = drifter - model).
ft
to ^,
rv t—
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