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Statistical models and distributions of current velocities with application to the prediction of extreme events
Estuarine, Coastal and Shelf Science 58 (2003) 601–609
Statistical models and distributions of current velocities with application to the prediction of extreme events J.A. Mattias Green*, Anders Stigebrandt Department of Oceanography, Go¨teborg University, P.O. Box 460, SE-40530 Gothenburg, Sweden Received 5 May 2002; accepted 6 May 2003
Abstract A multiple input system cross-spectral analysis method for ocean current velocities, involving the major meteorological and hydrographical forcing terms, is described and analyzed. The model uses joint observed data sets of forcing and current velocities to calculate initially unknown transfer functions that correlate each forcing term to the current velocity in transform space. From further forcing data sets, spectral estimates of the current velocity can be calculated with a higher level of significance than from a shorter, observed, series. The method is applied to data from the O¨resund, between Sweden and Denmark, and the Gullmar Fjord at the Swedish west coast. The results show that this method is suitable to estimate the statistical parameters of a current in a specific area, using the transfer functions, if the forcing is known for a long time and the current for a briefer period. It has advantages over purely statistical models, as it accounts for the coupling between the statistical parameters of the current and the forcing. It also has advantages over complex hydrodynamic model, as it is simple to use and interpret. The paper also contains a brief analysis of statistical distributions of current velocities sampled with various instruments in different hydrographic regimes. It is concluded that currents can be regarded as normally distributed when dominating tidal components have been removed by filtration. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: multiple input system cross-spectral analysis; currents velocities; statistical models; dispersion model; statistical distribution
1. Introduction 1.1. Statistics of ocean currents For many human activities in coastal waters and open sea, it is important to know the velocity range of currents and the frequency of extreme conditions. The current regime and range at a location are best described using local current records, but to estimate the frequency of rarely occurring, i.e. extreme, events would require a very long record. The high cost to do measurements and await the results makes it valuable to have methods to predict rarely occurring extreme conditions from short records. If ocean currents in general are distributed according to some well-known statistical distribution, * Corresponding author. E-mail address: [email protected] (J.A. Mattias Green). 0272-7714/03/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0272-7714(03)00138-0
already a relatively short record would suffice to estimate the parameters of the distribution, and thereby the properties of the current field. Many statistical methods require data to be normally distributed, and the central value theorem indicates that this is fulfilled if data are random and independent (e.g. Emery & Thomson, 1996; Larsen & Marx, 1986). Unfortunately, this is not always the case. First, observations close in time are generally dependent due to finite amplitude long-term variability. Second, tidal currents are strictly periodic and therefore not entirely random. Third, rotor current meters have thresholds of a few centimeters per second and some values in the currentrange are usually missing (e.g. Stigebrandt & Aure, 1995). Under such circumstances, there should be an overestimate of values at 0 cm s1 and an underestimate of values between the threshold value and zero and the data are not entirely random.
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A normal distribution is characterized by its mean and variance. The variance gets contributions from a spectrum of frequencies and spectra with the same variance should have the same normal distribution around their means. The particular shape of the spectrum of a data series should not influence the statistical properties of the series that may be derived from the distribution but the shape should be important for the length of the record needed to estimate the variance. If the low-frequency contribution vanishes, the variance may, obviously, be estimated from a short data series but otherwise, a longer record should be needed. 1.2. Dispersion models Local current conditions are important for several issues in aquaculture and dispersion from fixed point sources (e.g. Ervik et al., 1997; Gowen, Smyth, & Silvert, 1994; Hansen et al., 2001; Signell, Jenter, & Blumberg, 2000). One issue concerns computations of spreading and deposition of suspended particulate organic waste from fish farm net pens, with settling velocities in the range 0–0.15 m s1. Dispersion of suspended matter from point sources is caused by the variability of the current whereas the mean flow simply displaces the dispersion pattern relative to the farm. Estimates of dispersion using observed currents from rotor instruments can therefore be biased by the threshold effect. Another issue concerns the oxygen supply to and ammonium removal from the fauna in the net pens and on the bottom, respectively. It is essential for the water quality in the pens that the flushing is more or less continuous and sufficiently intense to prevent development of low oxygen and high ammonia concentrations. A short period without flushing can be accepted if the water quality is good enough at the beginning of the period. Most farms are well flushed if the current velocity is greater than about 0.02 m s1 (Stigebrandt & Aure, 1995). To make reliable estimates of current properties it is better to use models coupling the current to its forcing, e.g. weather and tides, than just the statistical distributions of the observed currents. This coupling is generally complicated but it is usually possible to find long time series of the forcing functions to be used in a statistical model. One goal should then be to establish how the mean and variance of the current are related to the forcing to possibly catch some of the dynamics of a system. For strait-flows, the velocity can be calculated from simple hydrodynamic models using the observed offshore stratification and sea-level variations and the statistical parameters are then easily calculated (e.g. Stigebrandt, 1990, 1992). For more open areas, hydrodynamic ocean circulation models can be used but at the present state of development, they only resolve the larger spatial scales of the spectrum of motions. The
variability of currents is therefore underestimated and such models cannot estimate extreme current conditions. If the goal is to predict the statistical properties of the current, it is not necessary to calculate the timedependent velocity if the properties can be estimated in another way. A statistical model that involves the forcing is evaluated in this study.
1.3. Multiple input system cross-spectral analysis The statistical parameters of a current field can be estimated from a multiple input systems cross-spectral analysis (referred to as MISCSA; e.g. Bendat & Piersol, 1966; Emery & Thomson, 1996). The output, in this case current velocity, is written as a sum of N forcing inputs, e.g. wind, sea-level and stratification. The output yðtÞ is correlated to each of the forcing functions xk ðtÞ, k ¼ 1; 2; . . . ; N through initially unknown transfer functions hk ðtÞ and a residual function eðtÞ containing random noise and contributions from processes not correlated to the forcing, like small-scale internal waves. The formulation assumes that there is a negligible lag between the forcing and the response of the output. In mathematical terms yðtÞ ¼ eðtÞ þ
N X
hk ðtÞxk ðtÞ
ð1Þ
k¼1
The residue is, if the correct forcing is provided, small compared to the other terms. In most practical cases, it may therefore be ignored unless the forcing is weak. It may then be important to the statistics, and the problem is more complex. For simplicity, eðtÞ is ignored through the rest of the analyses without any large loss of correlation. Eq. (1) can be Fourier transformed and the functions written in terms of their spectral densities. The result is a system for the unknown transfer functions in the spectral estimates of the inputs and outputs at each frequency f. On matrix form, the spectral estimates of Eq. (1) are Sky ð f Þ ¼ Sky ð f ÞHð f Þ
j; k ¼ 1; 2; . . . ; N
ð2Þ
Here Sky is a vector with the cross-spectral densities among the different forcing inputs xk ðtÞ and the output yðtÞ, Skj the matrix of cross-spectral densities between inputs xk(t) and xj(t) and H is the Fourier transformed response function matrix. If one has observations of the current velocity and the different forcing functions for a period, the transfer functions can be calculated from those data. Further records of the forcing functions can then be used in a prognosis model where Skj and H are known and Sky is unknown. The different statistical parameters of the velocity can then be estimated directly from Skj. The mean, l, and variance, ry2, of the data series yðtÞ are given in terms of the power spectrum matrix and transfer functions as
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ly ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi R0 Sð f ÞHð f Þ df 0
ð3Þ Öresund
r2y ¼
Z
1
Sð f ÞHð f Þ df
ð4Þ
o
0
o Sweden o
56 N
1.4. Correlation between probability density functions and histograms The correlation between the histogram and the normal probability density functions (PDF) for the data sets can be calculated as the correlation between the data points of the histogram and its corresponding points in the (discrete) normal PDF for the data. This gives a simple estimate of how normally distributed the data are, but some caution is needed. A high correlation means that the histogram closely resembles the PDF and thus is approximately normally distributed. A low correlation can mean either that data are not normally distributed, or that the normal PDF and the histogram do not match, but the data can still be normally distributed but askew from their theoretical normal PDF. 1.5. Purpose and outline The purpose of this study is to evaluate the MISCSA method to see if it is useful to predict oceanic current velocities. It is applied to two coastal areas forced by different mechanisms and its ability to predict the power spectrum of the current velocity is evaluated. Further, data sets from six different regimes are evaluated to show that ocean current velocities are normally distributed if tides are filtered out. The outline of this paper is as follows. In Section 2, the data sets and their applications are presented. The results are given in Section 3 and the paper ends with a discussion of the results and the conclusions in Section 4.
2. Data and data analysis 2.1. Multiple input system cross-spectral analysis data The first MISCSA was applied to data from the O¨resund, the second largest connection between the Baltic Sea and the Kattegat (Fig. 1). Its wide (15 km) and shallow (5 m) sill is some 10 km long. The typical flow in either direction is 30,000 m3 s1, although flows greater than 100,000 m3 s1 occur occasionally. The sealevel difference between the Baltic and the Kattegat provides the dominating forcing of the flow through the strait. The tidal range in southeastern Kattegat is only a few centimeters and even less in the Baltic. The
Denmark
x
x
o
Southwestern
o
Baltic Sea o
55 N o 12 E
o
13 E
o
14 E
Fig. 1. Map of the O¨resund with the locations of the sampling sites marked. The circles mark the sea-level stations and the crosses mark the locations of the ADCP.
meteorologically forced sea-level variations are about 10 times greater than this (Svansson, 1975). Current data were sampled in October–November 1994 by two bottom-mounted ADCP, one in each of the two channels at the sill. Sea-level observations come from two stations outside each end of the strait. All data were averaged to hourly values. The sea-level difference between the Baltic and the Kattegat and the mean flow through the strait are shown in Fig. 2. The analysis was made with the square root of the sea-level difference between the northern and southern ends as forcing, because the coupling between the sea-level difference and volume flow is quadratic (e.g. Stigebrandt, 1992). The averaged velocities from the two ADCP-rigs were used as current velocity, with outflows from the Baltic taken as positive. Data from the first 15 days were used to compute the transfer functions. These were then used with the remaining forcing data to estimate the spectrum of the current for the entire period of time. The modeled spectrum was then compared with the spectrum of the observed current. The second MISCSA was applied to the current at the mouth of the Gullmar Fjord at the Swedish west coast (Arneborg & Liljebladh, 2001; Fig. 3). The fjord is 28 km long and 1–2 km wide with a sill (maximum) depth of 43 m (116 m). The three-layer stratification in the fjord is maintained by the offshore stratification and a constant local freshwater supply. The 5–20 m deep
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[m]
1
(a)
0 −1
(b) [m s−1]
0.5 0 −0.5 0
7
14
21
28
35
42
49
days Fig. 2. The data sets used in the MISCSA from O¨resund. (a) The sea-level difference between the southern and northern ends of the strait. (b) The observed current velocity. Positive flows are outflows from the Baltic Sea.
surface layer with free contact with the coastal water has a freshwater influenced layer about 1 m thick at the sea surface. Between the surface layer and the sill depth is the intermediary layer, which also has free contact with the coastal water. The stagnant bottom water below the sill depth is usually renewed once a year. Data were taken from mid-January to mid-March 1997. The current came from ADCP observations at the sill taken with 2-m vertical resolution and stratification from CTloggers from seven depths at a rig outside the fjord, which sampled every 10 min. Sea-level observations are hourly values, whereas wind speed and wind direction were observed every 3 h. The current, temperature and salinity observations were depth averaged into a surface layer (5–15 m) and an intermediary layer (15–40 m), and the following analyses were made separately on each. The freshwater supply to the fjord was neglected in the present analysis, as it contributes little to the variability of the exchange (Arneborg & Liljebladh, 2001). All data were averaged or interpolated to 2-h values and are shown in Fig. 4. Owing to a 10-day data gap at the surface measurements of the ADCP, only data from day 11 to 20 were used to estimate the surface transfer functions, whereas data from the first 20 days were used for the intermediary water. Both sets of transfer functions were then used with the remaining forcing data to estimate the spectra of the currents in each layer for the remaining period. The modeled spectra were compared with the spectra of the observed currents.
bution function if the probability for extreme events is to be estimated. It is therefore important to make a brief analysis of current velocity data from different hydrographic regimes to try to determine their distributions. Six different data sets were analyzed. Low-velocity ocean data came from the Arctic Ocean and high velocity ocean data came from Torungen in the Skagerrak. A strait with large tidal flows was found in Bamburi lagoon, Kenya, a strait with some tidal flow was found in No¨tesund in Sweden and a strait with low tidal influence was found in O¨resund. The sets are summarized in Table 1. The dominating velocity components 30’
Gullmar fjord
Skagerrak
ct adcp
o
58 N 15.00’
sea−lvl
2.2. Distributions o
11 E
As discussed previously in the paper, normally distributed data are essential for the statistical methods that are used. The statistical parameters that can be calculated from the spectrum must be used in a distri-
30’
Fig. 3. Map of the Gullmar Fjord. The labels ct, adcp and sea-lvl show the sampling sites for stratification, currents speed and sea-level, respectively. Meteorological data came from a station just to the north of the map edge.
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(a) −3
[kg m ]
28 26 24 22
(b)
[m]
1 0 −1
(c) −1
[m s ]
15 0 −15
(d) −1
[m s ]
0.25 0 −0.25 0
7
14
21
28 days
35
42
49
56
Fig. 4. The data sets used in the MISCSA from the Gullmar Fjord. Stratification and current data are taken for the intermediary layer, i.e. 15–40 m. (a) The density outside the fjord, shown as r-1000. (b) The sea-level variation at the sill. (c) The along-fjord wind speed. (d) The current velocity at the mouth.
from each station were plotted as histograms with fitted normal PDFs. The correlation between the PDF and histogram was calculated for each station.
3. Results 3.1. Multiple input system cross-spectral analysis results The MISCSA of O¨resund gave a very good fit between the observed and modeled spectrum, as could be expected from a series with just one dominating forcing (Fig. 5). Other studies have shown that about 90% of the current variability in the strait is given by the sea-level differences (Mattsson, 1995). The study of the Gullmar Fjord gave good results for large parts of the spectrum, but the prognosis is not reliable for the lower frequencies (Fig. 6). The explanation is that the series used to calculate the transfer functions are too short, especially for the surface layer. The low-frequency fluctuations are likely dominated by the intermediary circulation driven by fluctuations in the offshore stratification. About half of the variability in
the latter should come from periods longer than one month as shown for the northern Skagerrak, where the conditions should be similar to those outside the Gullmar Fjord (Aure, Molvaer, & Stigebrandt, 1997). As an example of an application of the MISCSA, the transfer function for O¨resund was used on a 10-year record of daily sea-level observations from the area. The analysis was made on monthly ensembles of the data, i.e. one spectrum for the January data, one for the February data, etc. The spectra of the currents were used Table 1 Summary of the data sets used in the analyses of the statistical distributions Name
Instrument and sampling period
Bamburi lagoon, Kenya Lomonosov Ridge, Arctic Ocean Nordre Ro¨se, Denmark Torungen, Norway
SD6000 RCM, 10 min RCM7 and ACM-2, 60 min ADCP, 30 min RCM7, 10 min
No¨tesund, Sweden
ADCP, 15 min
Observation depth(s) 1 m from bottom 1109 and 1698 m 4–9.5 m 13, 20 30, 50, 75 and 100 m 2–8 m
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Fig. 5. The observed (solid) and modeled (dots) power spectral densities (PSD) for the velocity in the O¨resund using MISCSA. The transfer function was calculated from 15 days of joint data and the prediction on 48 days of forcing data.
to estimate the mean and variance for each month and were then used to estimate the probabilities to encounter a current greater than 0.5 m s1 or a current less than 0.02 m s1 during each month. The resulting histograms
are shown in Fig. 7. This analysis demonstrates that one may obtain estimates of current statistics for the whole period for which forcing is available, using only a short record (15 days) to calibrate a physically relevant model.
6
10
(a)
4
PSD [m2 s−1]
10
2
10
0
10
−2
10
6
observed modelled
10
(b)
4
PSD [m2 s−1]
10
2
10
0
10
−2
10
observed modelled −1
0
10
10 frequency [cpd]
Fig. 6. The observed (solid) and modeled (dots) PSD for the velocity across the sill in the Gullmar Fjord using MISCSA. The transfer functions for the surface (intermediary) layer were estimated from 10 (20) days of joint data and the prediction on 38 days of forcing data. (a) The surface layer model (data from 0 to 15 m depth). (b) The intermediary layer model (data from 15 to 40 m depth).
J.A. Mattias Green, A. Stigebrandt / Estuarine, Coastal and Shelf Science 58 (2003) 601–609
607
P(u>0.5)
(a) 0.02 0.01
P(u<0.02)
0
(b)
0.02 0.01 0
1
2
3
4
5
6 7 month
8
9
10
11
12
Fig. 7. Histogram over the probability for a given month to encounter a current velocity greater than 0.5 m s1 (a) or less than 0.02 m s1 (b) in O¨resund. Estimates are from a MISCSA on 10 years of sea-level data, with transfer functions calculated from 15 days of combined forcing and current data.
The result may be used, for example, to choose the best time of year for operations in the strait that cannot be done during conditions with strong currents, which in this case should be January and November (Fig. 7).
3.2. Statistical distributions and parameters Observations taken with an RCM in low-current environments have a large overrepresentation of zerovelocity observations that could not be found in the ACM data from about the same depth (Fig. 8a and d). Rotor data from a sea with high velocities (Torungen) were normally distributed with a very good correlation between the histogram and the PDF (Fig. 8c). This supports the theory of overrepresentation at low currents due to the threshold effect. The tidally dominated currents at Bamburi (Fig. 8b) have an M shape, typical of flows with rather constant current speed in either direction. ADCP data from Nordre Ro¨se and No¨tesund (Fig. 8e and f ) show more normal distributions than the RCM data. The correlation coefficients between the PDF and the histograms are summarized in Table 2. The difference in the correlation for the RCM and ACM data at the Lomonosov Ridge accentuates the inability of rotor gauges to sample low currents, although the correlation is not low for either of the sets. The large share of observations at or close to 0 m s1 gives a sharp histogram, whereas the normal PDF is more flattened, probably because of a rather large standard deviation of the data. The histogram and normal PDF for the data from Bamburi lagoon have a very low correlation coefficient. If the data are filtered and the dominating harmonic frequencies removed, the correlation improves (not shown). The ADCP data from Nordre Ro¨se and No¨tesund have good correlations between the normal
PDF and the histograms. There are no known instrument-induced artifacts that could cause a distorted distribution for the ADCP. The lower correlation for Nordre Ro¨se is caused by some scattering of the data around the peak of the PDF, but the fit is still good. The RCM data from Torungen are sampled in a highcurrent environment and have a very good fit between the normal PDF and histogram.
4. Discussion MISCSA is more statistically reliable than regression analysis and other purely statistical methods. An understanding of the processes in the studied area is required though, because all major forcing processes should be included to get a good result over the entire spectrum. A typical application of the MISCSA is to evaluate probabilities for extreme events, as is exemplified here with the probability study from the O¨resund. To construct probability diagrams for extreme events during certain periods is of importance for, for example, construction works and the location of fish farms. The present study only covers two geographical areas, although they are of different hydrographic character and have very different forcing. The good results from both areas and the general and solid mathematical and physical theory behind the MISCSA method indicate that it is a useful method for this type of study. The difference in results between the two areas is a consequence of the different dynamics of the areas. The MISCSA on O¨resund gave high correlation for low frequencies and lower correlation for high frequencies whereas the opposite holds for the Gullmar Fjord. This is not surprising as the flow through O¨resund is forced by essentially one function—the sea-level difference. In
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Fig. 8. The histogram (dots) and normal PDF (solid) for the different data sets used in the distribution analysis. Note the different scales on the axes. (a) East–west component from an RCM7 at 1109 m depth at the Lomonosov Ridge with positive flows to the east. Note the very high share of samples at 0 m s1. The threshold value for the instrument was 0.015 m s1. (b) The along-strait component from the southern channel at Bamburi lagoon, Kenya. Inflows to the lagoon are positive. (c) The along-coast component in the surface layer at Torungen, Skagerrak. Outflows from the Skagerrak are positive. (d) East–west component from a bottom-mounted ACM2 at 1698 m depth at the Lomonosov Ridge. Positive flows are to the east. (e) The along-strait component from 2 m depth in No¨tesund, Sweden. Outflows from the fjord are positive. (f) The along-strait component from 4.5 m depth at Nordre Ro¨se, O¨resund. Outflows from the Baltic are positive.
the Gullmar Fjord, currents are driven by a larger number of forcing functions that increase the complexity of the study. The needed record length to perform a successful MISCSA depends on the frequency of the dominating forcing. In the Gullmar Fjord, one of the dominating forcing functions has much variability on time-scales much longer than the calibration period. To get a statistically reliable spectrum, some cycles of the forcing should be included. This can be difficult to know in advance, but meteorological forcing can often be analyzed before any measurements are made. The needed record length also increases with the number of forcing functions, as the dynamics of the area are more complex. The needed correlation between the modeled and observed spectra is dependent on the purpose of the study. If low-frequency fluctuations are of interest, the correlation in the lower part must of course be high, which may require larger data sets from which to calculate the forcing functions. But if all the forcing functions are included and the data sets are large enough, the correlation should be good enough for the modeled data to be useful. The generally good result of the application to two different areas with varying dynamics shows that the method is applicable if the
physical processes of the studied area are known. Provided the physics of the area is understood, the mathematical reliability makes the MISCSA a safe model to use. Of course, further applications are of interest, and it has yet to be implemented in dispersion models. Current velocities in the ocean generally follow a normal distribution, except when dominated by tidal flows. Low-current environments in combination with RCM can cause peaks at certain parts of the sampling histogram. Resampling zero-current observations by normally distributed random numbers could be a way to
Table 2 The correlation coefficients between the normal PDF and the histograms for the different data sets Location
Correlation
Lomonosov Ridge RCM Lomonosov Ridge ACM Bamburi Lagoon Nordre Ro¨se Torungen No¨tesund
0.72 0.80 0.47 0.80 0.92 0.90
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remove peaks in the sampling histogram as done by Stigebrandt and Aure (1995). The final conclusion of the study is that the MISCSA is useful to estimate the statistical parameters of current velocities. These can then be used in, for example, a normal PDF to obtain probabilities of extreme events to occur.
Acknowledgements Go¨ran Bjo¨rk, Bengt Liljebladh and Lars Rydberg at the Department of Oceanography, Go¨teborg University, provided the data from No¨tesund, Gullmar Fjord, Arctic Ocean and Bamburi Lagoon. The data from O¨resund were sampled by O¨resundskonsortiet and made available by Klaus Dynesen, and Jan Aure, at the Institute of Marine Research, Norway, provided the data from Torungen. The participants of the summer 2000 project-course made the distribution analyses and raw-data treatment. The manuscript was improved by the comments of two anonymous reviewers. Financial support came from the Faculty of Science at Go¨teborg University, and the Swedish Foundation for Strategic Research through MARE—Marine Research on Eutrophication. References Arneborg, L., & Liljebladh, B. (2001). The internal seiches in Gullmar fjord. Part I: Dynamics. Journal of Physical Oceanography 31:9, 2549–2566. Aure, J., Molvaer, J., & Stigebrandt, A. (1997). Observations of inshore water exchange forced by a fluctuating offshore density field. Marine Pollution Bulletin 33, 112–119.
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Bendat, J. S., & Piersol, A. G. (1966). Measurement and analysis of random data (390 pp.). New York: Wiley. Emery, W. J., & Thomson, R. E. (1996). Data analysis methods in physical oceanography (2nd ed., 650 pp.). Oxford: Pergamon press. Ervik, A., Hansen, P. K., Aure, J., Stigebrandt, A., Johannessen, P., & Jahnsen, T. (1997). Regulating the local environmental impact of intensive marine fish farming. I. The concept of the MOM-system (Modelling-Ongrowing fish farms-Monitoring). Aquaculture 158(1– 2), 85–94. Gowen, R. J., Smyth, D., & Silvert, W. (1994). Modelling the spatial distribution and loading of organic fish farm waste to the seabed. In B. T. Hargrave (Ed.), Modelling benthic impact of organic enrichment from marine aquaculture. Canadian Technical Report of Fisheries and Aquatic Sciences 1949, 19–30. Hansen, P. K., Ervik, A., Schaanning, M., Johannessen, P., Aure, J., Jahnsen, T., & Stigebrandt, A. (2001). Regulating the local environmental impact of intensive, marine fish farming. II. The monitoring programme of the MOM-system (Modelling-Ongrowing fish farms-Monitoring). Aquaculture 194(1–2), 75–92. Larsen, R. J., & Marx, M. L. (1986). An introduction to mathematical statistics and its applications (630 pp.). Englewood cliffs: Prentice-Hall. Mattsson, J. (1995). Observed linear flow resistance in the O¨resund due to rotation. Journal of Geophysical Research 100, 20779–20791. Signell, R. P., Jenter, H. L., & Blumberg, A. F. (2000). Predicting the physical effects of relocating Boston’s sewage outfall. Estuarine, Coastal and Shelf Science 50, 59–72 (doi: 10.1006/ecss. 1990.0532). Stigebrandt, A. (1990). On the response of the horizontal mean vertical distribution in a fjord to low-frequency density fluctuations in the coastal water. Tellus 42A, 605–614. Stigebrandt, A. (1992). Bridge-induced flow reduction in sea straits with reference to effects of a planned bridge across O¨resund. Estuarine, Coastal and Shelf Science 212, 130–134. Stigebrandt, A., & Aure, J. (1995). A model for the dispersion of organic waste from a fish farm (7 pp.). Appendix II in Fisken og Havet. Technical Report No. 26, Institute of Marine Research, Bergen, Norway. Svansson, A. (1975). Physical and chemical oceanography of the Skagerrak and the Kattegat (88 pp.). Technical Report 1, Fishery Board of Sweden, Institute of Marine Research.
Statistical models and distributions of current velocities with application to the prediction of extreme events J.A. Mattias Green*, Anders Stigebrandt Department of Oceanography, Go¨teborg University, P.O. Box 460, SE-40530 Gothenburg, Sweden Received 5 May 2002; accepted 6 May 2003
Abstract A multiple input system cross-spectral analysis method for ocean current velocities, involving the major meteorological and hydrographical forcing terms, is described and analyzed. The model uses joint observed data sets of forcing and current velocities to calculate initially unknown transfer functions that correlate each forcing term to the current velocity in transform space. From further forcing data sets, spectral estimates of the current velocity can be calculated with a higher level of significance than from a shorter, observed, series. The method is applied to data from the O¨resund, between Sweden and Denmark, and the Gullmar Fjord at the Swedish west coast. The results show that this method is suitable to estimate the statistical parameters of a current in a specific area, using the transfer functions, if the forcing is known for a long time and the current for a briefer period. It has advantages over purely statistical models, as it accounts for the coupling between the statistical parameters of the current and the forcing. It also has advantages over complex hydrodynamic model, as it is simple to use and interpret. The paper also contains a brief analysis of statistical distributions of current velocities sampled with various instruments in different hydrographic regimes. It is concluded that currents can be regarded as normally distributed when dominating tidal components have been removed by filtration. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: multiple input system cross-spectral analysis; currents velocities; statistical models; dispersion model; statistical distribution
1. Introduction 1.1. Statistics of ocean currents For many human activities in coastal waters and open sea, it is important to know the velocity range of currents and the frequency of extreme conditions. The current regime and range at a location are best described using local current records, but to estimate the frequency of rarely occurring, i.e. extreme, events would require a very long record. The high cost to do measurements and await the results makes it valuable to have methods to predict rarely occurring extreme conditions from short records. If ocean currents in general are distributed according to some well-known statistical distribution, * Corresponding author. E-mail address: [email protected] (J.A. Mattias Green). 0272-7714/03/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0272-7714(03)00138-0
already a relatively short record would suffice to estimate the parameters of the distribution, and thereby the properties of the current field. Many statistical methods require data to be normally distributed, and the central value theorem indicates that this is fulfilled if data are random and independent (e.g. Emery & Thomson, 1996; Larsen & Marx, 1986). Unfortunately, this is not always the case. First, observations close in time are generally dependent due to finite amplitude long-term variability. Second, tidal currents are strictly periodic and therefore not entirely random. Third, rotor current meters have thresholds of a few centimeters per second and some values in the currentrange are usually missing (e.g. Stigebrandt & Aure, 1995). Under such circumstances, there should be an overestimate of values at 0 cm s1 and an underestimate of values between the threshold value and zero and the data are not entirely random.
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A normal distribution is characterized by its mean and variance. The variance gets contributions from a spectrum of frequencies and spectra with the same variance should have the same normal distribution around their means. The particular shape of the spectrum of a data series should not influence the statistical properties of the series that may be derived from the distribution but the shape should be important for the length of the record needed to estimate the variance. If the low-frequency contribution vanishes, the variance may, obviously, be estimated from a short data series but otherwise, a longer record should be needed. 1.2. Dispersion models Local current conditions are important for several issues in aquaculture and dispersion from fixed point sources (e.g. Ervik et al., 1997; Gowen, Smyth, & Silvert, 1994; Hansen et al., 2001; Signell, Jenter, & Blumberg, 2000). One issue concerns computations of spreading and deposition of suspended particulate organic waste from fish farm net pens, with settling velocities in the range 0–0.15 m s1. Dispersion of suspended matter from point sources is caused by the variability of the current whereas the mean flow simply displaces the dispersion pattern relative to the farm. Estimates of dispersion using observed currents from rotor instruments can therefore be biased by the threshold effect. Another issue concerns the oxygen supply to and ammonium removal from the fauna in the net pens and on the bottom, respectively. It is essential for the water quality in the pens that the flushing is more or less continuous and sufficiently intense to prevent development of low oxygen and high ammonia concentrations. A short period without flushing can be accepted if the water quality is good enough at the beginning of the period. Most farms are well flushed if the current velocity is greater than about 0.02 m s1 (Stigebrandt & Aure, 1995). To make reliable estimates of current properties it is better to use models coupling the current to its forcing, e.g. weather and tides, than just the statistical distributions of the observed currents. This coupling is generally complicated but it is usually possible to find long time series of the forcing functions to be used in a statistical model. One goal should then be to establish how the mean and variance of the current are related to the forcing to possibly catch some of the dynamics of a system. For strait-flows, the velocity can be calculated from simple hydrodynamic models using the observed offshore stratification and sea-level variations and the statistical parameters are then easily calculated (e.g. Stigebrandt, 1990, 1992). For more open areas, hydrodynamic ocean circulation models can be used but at the present state of development, they only resolve the larger spatial scales of the spectrum of motions. The
variability of currents is therefore underestimated and such models cannot estimate extreme current conditions. If the goal is to predict the statistical properties of the current, it is not necessary to calculate the timedependent velocity if the properties can be estimated in another way. A statistical model that involves the forcing is evaluated in this study.
1.3. Multiple input system cross-spectral analysis The statistical parameters of a current field can be estimated from a multiple input systems cross-spectral analysis (referred to as MISCSA; e.g. Bendat & Piersol, 1966; Emery & Thomson, 1996). The output, in this case current velocity, is written as a sum of N forcing inputs, e.g. wind, sea-level and stratification. The output yðtÞ is correlated to each of the forcing functions xk ðtÞ, k ¼ 1; 2; . . . ; N through initially unknown transfer functions hk ðtÞ and a residual function eðtÞ containing random noise and contributions from processes not correlated to the forcing, like small-scale internal waves. The formulation assumes that there is a negligible lag between the forcing and the response of the output. In mathematical terms yðtÞ ¼ eðtÞ þ
N X
hk ðtÞxk ðtÞ
ð1Þ
k¼1
The residue is, if the correct forcing is provided, small compared to the other terms. In most practical cases, it may therefore be ignored unless the forcing is weak. It may then be important to the statistics, and the problem is more complex. For simplicity, eðtÞ is ignored through the rest of the analyses without any large loss of correlation. Eq. (1) can be Fourier transformed and the functions written in terms of their spectral densities. The result is a system for the unknown transfer functions in the spectral estimates of the inputs and outputs at each frequency f. On matrix form, the spectral estimates of Eq. (1) are Sky ð f Þ ¼ Sky ð f ÞHð f Þ
j; k ¼ 1; 2; . . . ; N
ð2Þ
Here Sky is a vector with the cross-spectral densities among the different forcing inputs xk ðtÞ and the output yðtÞ, Skj the matrix of cross-spectral densities between inputs xk(t) and xj(t) and H is the Fourier transformed response function matrix. If one has observations of the current velocity and the different forcing functions for a period, the transfer functions can be calculated from those data. Further records of the forcing functions can then be used in a prognosis model where Skj and H are known and Sky is unknown. The different statistical parameters of the velocity can then be estimated directly from Skj. The mean, l, and variance, ry2, of the data series yðtÞ are given in terms of the power spectrum matrix and transfer functions as
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ly ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi R0 Sð f ÞHð f Þ df 0
ð3Þ Öresund
r2y ¼
Z
1
Sð f ÞHð f Þ df
ð4Þ
o
0
o Sweden o
56 N
1.4. Correlation between probability density functions and histograms The correlation between the histogram and the normal probability density functions (PDF) for the data sets can be calculated as the correlation between the data points of the histogram and its corresponding points in the (discrete) normal PDF for the data. This gives a simple estimate of how normally distributed the data are, but some caution is needed. A high correlation means that the histogram closely resembles the PDF and thus is approximately normally distributed. A low correlation can mean either that data are not normally distributed, or that the normal PDF and the histogram do not match, but the data can still be normally distributed but askew from their theoretical normal PDF. 1.5. Purpose and outline The purpose of this study is to evaluate the MISCSA method to see if it is useful to predict oceanic current velocities. It is applied to two coastal areas forced by different mechanisms and its ability to predict the power spectrum of the current velocity is evaluated. Further, data sets from six different regimes are evaluated to show that ocean current velocities are normally distributed if tides are filtered out. The outline of this paper is as follows. In Section 2, the data sets and their applications are presented. The results are given in Section 3 and the paper ends with a discussion of the results and the conclusions in Section 4.
2. Data and data analysis 2.1. Multiple input system cross-spectral analysis data The first MISCSA was applied to data from the O¨resund, the second largest connection between the Baltic Sea and the Kattegat (Fig. 1). Its wide (15 km) and shallow (5 m) sill is some 10 km long. The typical flow in either direction is 30,000 m3 s1, although flows greater than 100,000 m3 s1 occur occasionally. The sealevel difference between the Baltic and the Kattegat provides the dominating forcing of the flow through the strait. The tidal range in southeastern Kattegat is only a few centimeters and even less in the Baltic. The
Denmark
x
x
o
Southwestern
o
Baltic Sea o
55 N o 12 E
o
13 E
o
14 E
Fig. 1. Map of the O¨resund with the locations of the sampling sites marked. The circles mark the sea-level stations and the crosses mark the locations of the ADCP.
meteorologically forced sea-level variations are about 10 times greater than this (Svansson, 1975). Current data were sampled in October–November 1994 by two bottom-mounted ADCP, one in each of the two channels at the sill. Sea-level observations come from two stations outside each end of the strait. All data were averaged to hourly values. The sea-level difference between the Baltic and the Kattegat and the mean flow through the strait are shown in Fig. 2. The analysis was made with the square root of the sea-level difference between the northern and southern ends as forcing, because the coupling between the sea-level difference and volume flow is quadratic (e.g. Stigebrandt, 1992). The averaged velocities from the two ADCP-rigs were used as current velocity, with outflows from the Baltic taken as positive. Data from the first 15 days were used to compute the transfer functions. These were then used with the remaining forcing data to estimate the spectrum of the current for the entire period of time. The modeled spectrum was then compared with the spectrum of the observed current. The second MISCSA was applied to the current at the mouth of the Gullmar Fjord at the Swedish west coast (Arneborg & Liljebladh, 2001; Fig. 3). The fjord is 28 km long and 1–2 km wide with a sill (maximum) depth of 43 m (116 m). The three-layer stratification in the fjord is maintained by the offshore stratification and a constant local freshwater supply. The 5–20 m deep
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[m]
1
(a)
0 −1
(b) [m s−1]
0.5 0 −0.5 0
7
14
21
28
35
42
49
days Fig. 2. The data sets used in the MISCSA from O¨resund. (a) The sea-level difference between the southern and northern ends of the strait. (b) The observed current velocity. Positive flows are outflows from the Baltic Sea.
surface layer with free contact with the coastal water has a freshwater influenced layer about 1 m thick at the sea surface. Between the surface layer and the sill depth is the intermediary layer, which also has free contact with the coastal water. The stagnant bottom water below the sill depth is usually renewed once a year. Data were taken from mid-January to mid-March 1997. The current came from ADCP observations at the sill taken with 2-m vertical resolution and stratification from CTloggers from seven depths at a rig outside the fjord, which sampled every 10 min. Sea-level observations are hourly values, whereas wind speed and wind direction were observed every 3 h. The current, temperature and salinity observations were depth averaged into a surface layer (5–15 m) and an intermediary layer (15–40 m), and the following analyses were made separately on each. The freshwater supply to the fjord was neglected in the present analysis, as it contributes little to the variability of the exchange (Arneborg & Liljebladh, 2001). All data were averaged or interpolated to 2-h values and are shown in Fig. 4. Owing to a 10-day data gap at the surface measurements of the ADCP, only data from day 11 to 20 were used to estimate the surface transfer functions, whereas data from the first 20 days were used for the intermediary water. Both sets of transfer functions were then used with the remaining forcing data to estimate the spectra of the currents in each layer for the remaining period. The modeled spectra were compared with the spectra of the observed currents.
bution function if the probability for extreme events is to be estimated. It is therefore important to make a brief analysis of current velocity data from different hydrographic regimes to try to determine their distributions. Six different data sets were analyzed. Low-velocity ocean data came from the Arctic Ocean and high velocity ocean data came from Torungen in the Skagerrak. A strait with large tidal flows was found in Bamburi lagoon, Kenya, a strait with some tidal flow was found in No¨tesund in Sweden and a strait with low tidal influence was found in O¨resund. The sets are summarized in Table 1. The dominating velocity components 30’
Gullmar fjord
Skagerrak
ct adcp
o
58 N 15.00’
sea−lvl
2.2. Distributions o
11 E
As discussed previously in the paper, normally distributed data are essential for the statistical methods that are used. The statistical parameters that can be calculated from the spectrum must be used in a distri-
30’
Fig. 3. Map of the Gullmar Fjord. The labels ct, adcp and sea-lvl show the sampling sites for stratification, currents speed and sea-level, respectively. Meteorological data came from a station just to the north of the map edge.
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(a) −3
[kg m ]
28 26 24 22
(b)
[m]
1 0 −1
(c) −1
[m s ]
15 0 −15
(d) −1
[m s ]
0.25 0 −0.25 0
7
14
21
28 days
35
42
49
56
Fig. 4. The data sets used in the MISCSA from the Gullmar Fjord. Stratification and current data are taken for the intermediary layer, i.e. 15–40 m. (a) The density outside the fjord, shown as r-1000. (b) The sea-level variation at the sill. (c) The along-fjord wind speed. (d) The current velocity at the mouth.
from each station were plotted as histograms with fitted normal PDFs. The correlation between the PDF and histogram was calculated for each station.
3. Results 3.1. Multiple input system cross-spectral analysis results The MISCSA of O¨resund gave a very good fit between the observed and modeled spectrum, as could be expected from a series with just one dominating forcing (Fig. 5). Other studies have shown that about 90% of the current variability in the strait is given by the sea-level differences (Mattsson, 1995). The study of the Gullmar Fjord gave good results for large parts of the spectrum, but the prognosis is not reliable for the lower frequencies (Fig. 6). The explanation is that the series used to calculate the transfer functions are too short, especially for the surface layer. The low-frequency fluctuations are likely dominated by the intermediary circulation driven by fluctuations in the offshore stratification. About half of the variability in
the latter should come from periods longer than one month as shown for the northern Skagerrak, where the conditions should be similar to those outside the Gullmar Fjord (Aure, Molvaer, & Stigebrandt, 1997). As an example of an application of the MISCSA, the transfer function for O¨resund was used on a 10-year record of daily sea-level observations from the area. The analysis was made on monthly ensembles of the data, i.e. one spectrum for the January data, one for the February data, etc. The spectra of the currents were used Table 1 Summary of the data sets used in the analyses of the statistical distributions Name
Instrument and sampling period
Bamburi lagoon, Kenya Lomonosov Ridge, Arctic Ocean Nordre Ro¨se, Denmark Torungen, Norway
SD6000 RCM, 10 min RCM7 and ACM-2, 60 min ADCP, 30 min RCM7, 10 min
No¨tesund, Sweden
ADCP, 15 min
Observation depth(s) 1 m from bottom 1109 and 1698 m 4–9.5 m 13, 20 30, 50, 75 and 100 m 2–8 m
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Fig. 5. The observed (solid) and modeled (dots) power spectral densities (PSD) for the velocity in the O¨resund using MISCSA. The transfer function was calculated from 15 days of joint data and the prediction on 48 days of forcing data.
to estimate the mean and variance for each month and were then used to estimate the probabilities to encounter a current greater than 0.5 m s1 or a current less than 0.02 m s1 during each month. The resulting histograms
are shown in Fig. 7. This analysis demonstrates that one may obtain estimates of current statistics for the whole period for which forcing is available, using only a short record (15 days) to calibrate a physically relevant model.
6
10
(a)
4
PSD [m2 s−1]
10
2
10
0
10
−2
10
6
observed modelled
10
(b)
4
PSD [m2 s−1]
10
2
10
0
10
−2
10
observed modelled −1
0
10
10 frequency [cpd]
Fig. 6. The observed (solid) and modeled (dots) PSD for the velocity across the sill in the Gullmar Fjord using MISCSA. The transfer functions for the surface (intermediary) layer were estimated from 10 (20) days of joint data and the prediction on 38 days of forcing data. (a) The surface layer model (data from 0 to 15 m depth). (b) The intermediary layer model (data from 15 to 40 m depth).
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607
P(u>0.5)
(a) 0.02 0.01
P(u<0.02)
0
(b)
0.02 0.01 0
1
2
3
4
5
6 7 month
8
9
10
11
12
Fig. 7. Histogram over the probability for a given month to encounter a current velocity greater than 0.5 m s1 (a) or less than 0.02 m s1 (b) in O¨resund. Estimates are from a MISCSA on 10 years of sea-level data, with transfer functions calculated from 15 days of combined forcing and current data.
The result may be used, for example, to choose the best time of year for operations in the strait that cannot be done during conditions with strong currents, which in this case should be January and November (Fig. 7).
3.2. Statistical distributions and parameters Observations taken with an RCM in low-current environments have a large overrepresentation of zerovelocity observations that could not be found in the ACM data from about the same depth (Fig. 8a and d). Rotor data from a sea with high velocities (Torungen) were normally distributed with a very good correlation between the histogram and the PDF (Fig. 8c). This supports the theory of overrepresentation at low currents due to the threshold effect. The tidally dominated currents at Bamburi (Fig. 8b) have an M shape, typical of flows with rather constant current speed in either direction. ADCP data from Nordre Ro¨se and No¨tesund (Fig. 8e and f ) show more normal distributions than the RCM data. The correlation coefficients between the PDF and the histograms are summarized in Table 2. The difference in the correlation for the RCM and ACM data at the Lomonosov Ridge accentuates the inability of rotor gauges to sample low currents, although the correlation is not low for either of the sets. The large share of observations at or close to 0 m s1 gives a sharp histogram, whereas the normal PDF is more flattened, probably because of a rather large standard deviation of the data. The histogram and normal PDF for the data from Bamburi lagoon have a very low correlation coefficient. If the data are filtered and the dominating harmonic frequencies removed, the correlation improves (not shown). The ADCP data from Nordre Ro¨se and No¨tesund have good correlations between the normal
PDF and the histograms. There are no known instrument-induced artifacts that could cause a distorted distribution for the ADCP. The lower correlation for Nordre Ro¨se is caused by some scattering of the data around the peak of the PDF, but the fit is still good. The RCM data from Torungen are sampled in a highcurrent environment and have a very good fit between the normal PDF and histogram.
4. Discussion MISCSA is more statistically reliable than regression analysis and other purely statistical methods. An understanding of the processes in the studied area is required though, because all major forcing processes should be included to get a good result over the entire spectrum. A typical application of the MISCSA is to evaluate probabilities for extreme events, as is exemplified here with the probability study from the O¨resund. To construct probability diagrams for extreme events during certain periods is of importance for, for example, construction works and the location of fish farms. The present study only covers two geographical areas, although they are of different hydrographic character and have very different forcing. The good results from both areas and the general and solid mathematical and physical theory behind the MISCSA method indicate that it is a useful method for this type of study. The difference in results between the two areas is a consequence of the different dynamics of the areas. The MISCSA on O¨resund gave high correlation for low frequencies and lower correlation for high frequencies whereas the opposite holds for the Gullmar Fjord. This is not surprising as the flow through O¨resund is forced by essentially one function—the sea-level difference. In
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Fig. 8. The histogram (dots) and normal PDF (solid) for the different data sets used in the distribution analysis. Note the different scales on the axes. (a) East–west component from an RCM7 at 1109 m depth at the Lomonosov Ridge with positive flows to the east. Note the very high share of samples at 0 m s1. The threshold value for the instrument was 0.015 m s1. (b) The along-strait component from the southern channel at Bamburi lagoon, Kenya. Inflows to the lagoon are positive. (c) The along-coast component in the surface layer at Torungen, Skagerrak. Outflows from the Skagerrak are positive. (d) East–west component from a bottom-mounted ACM2 at 1698 m depth at the Lomonosov Ridge. Positive flows are to the east. (e) The along-strait component from 2 m depth in No¨tesund, Sweden. Outflows from the fjord are positive. (f) The along-strait component from 4.5 m depth at Nordre Ro¨se, O¨resund. Outflows from the Baltic are positive.
the Gullmar Fjord, currents are driven by a larger number of forcing functions that increase the complexity of the study. The needed record length to perform a successful MISCSA depends on the frequency of the dominating forcing. In the Gullmar Fjord, one of the dominating forcing functions has much variability on time-scales much longer than the calibration period. To get a statistically reliable spectrum, some cycles of the forcing should be included. This can be difficult to know in advance, but meteorological forcing can often be analyzed before any measurements are made. The needed record length also increases with the number of forcing functions, as the dynamics of the area are more complex. The needed correlation between the modeled and observed spectra is dependent on the purpose of the study. If low-frequency fluctuations are of interest, the correlation in the lower part must of course be high, which may require larger data sets from which to calculate the forcing functions. But if all the forcing functions are included and the data sets are large enough, the correlation should be good enough for the modeled data to be useful. The generally good result of the application to two different areas with varying dynamics shows that the method is applicable if the
physical processes of the studied area are known. Provided the physics of the area is understood, the mathematical reliability makes the MISCSA a safe model to use. Of course, further applications are of interest, and it has yet to be implemented in dispersion models. Current velocities in the ocean generally follow a normal distribution, except when dominated by tidal flows. Low-current environments in combination with RCM can cause peaks at certain parts of the sampling histogram. Resampling zero-current observations by normally distributed random numbers could be a way to
Table 2 The correlation coefficients between the normal PDF and the histograms for the different data sets Location
Correlation
Lomonosov Ridge RCM Lomonosov Ridge ACM Bamburi Lagoon Nordre Ro¨se Torungen No¨tesund
0.72 0.80 0.47 0.80 0.92 0.90
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remove peaks in the sampling histogram as done by Stigebrandt and Aure (1995). The final conclusion of the study is that the MISCSA is useful to estimate the statistical parameters of current velocities. These can then be used in, for example, a normal PDF to obtain probabilities of extreme events to occur.
Acknowledgements Go¨ran Bjo¨rk, Bengt Liljebladh and Lars Rydberg at the Department of Oceanography, Go¨teborg University, provided the data from No¨tesund, Gullmar Fjord, Arctic Ocean and Bamburi Lagoon. The data from O¨resund were sampled by O¨resundskonsortiet and made available by Klaus Dynesen, and Jan Aure, at the Institute of Marine Research, Norway, provided the data from Torungen. The participants of the summer 2000 project-course made the distribution analyses and raw-data treatment. The manuscript was improved by the comments of two anonymous reviewers. Financial support came from the Faculty of Science at Go¨teborg University, and the Swedish Foundation for Strategic Research through MARE—Marine Research on Eutrophication. References Arneborg, L., & Liljebladh, B. (2001). The internal seiches in Gullmar fjord. Part I: Dynamics. Journal of Physical Oceanography 31:9, 2549–2566. Aure, J., Molvaer, J., & Stigebrandt, A. (1997). Observations of inshore water exchange forced by a fluctuating offshore density field. Marine Pollution Bulletin 33, 112–119.
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