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Auto- and crosscorrelograms of particulate trace metals in the rhine estuary, Southern Bight and Dutch Wadden Sea
.Netherlands ,~ournal of Sea Research 10 (1) : 59-70 (1976) AUTO- AND CROSSCORREIOGRAMS OF PARTICULATE TRACE METALS IN THE RHINE ESTUARY,
SOUTHERN BIGHT WADDEN SEA
AND
DUTCH
by B. G. M. V A N D E G I N S T E , P . J . M . S A L E M I N K (Department of Analytical Chemistry, Catholic University of .Nijmegen, The .Netherlands) and J. C. D U I N K E R (.Netherlands Institute for Sea Research, Texel, The .Netherlands) CONTENTS I. Introduction . . . . . . . . . . . . . . . . . II. Definitions and Notations . . . . . . . . . . . . . . I. Autocorrelation . . . . . . . . . . . . . . . . 2. Crosscorrelation . . . . . . . . . . . . . . . . III. Results and Discussion . . . . . . . . . . . . . . 1. Autocorrelation calculations applied to data from the Southern Bight . . . . . . . . . . . . . . . . . . . 1.1. The stability of the concentration profiles . . . . . . . 1.2. The existence of concentration isopatterns . . . . . . . 2. Autocorrelation calculations applied to data from the Dutch Wadden Sea . . . . . . . . . . . . . . . . . . . 3. Crosscorrelation calculations in the Southern Bight and the Dutch Wadden Sea . . . . . . . . . . . . . . . . . IV. Conclusions . . . . . . . . . . . . . . . . . . V. Summary . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . .
59 60 60 69 63 63 64 65 66 68 69 69 70
I. I N T R O D U C T I O N As a rule, t i m e series or position series, o b t a i n e d f r o m the analysis of c h e m i c a l species in n a t u r a l waters, at different positions at the s a m e time, or at a n y position at different times, show considerable fluctuations due to o t h e r t h a n strictly h y d r o c h e m i c a l , biological or physical processes. S a m p l i n g a n d analytical errors d e t e r i o r a t e the d a t a quality. T h e r e f o r e , analytical noise is one o f the serious limitations for disc o v e r i n g h y d r o l o g i c a l processes f r o m e n v i r o n m e n t a l d a t a . F o r the detection o f signals h i d d e n in noise, a s m o o t h i n g p r o c e d u r e is n e e d e d . H o w e v e r , c o n v e n t i o n a l s m o o t h i n g p r o c e d u r e s e.g. e x p o n e n t i a l a v e r a g i n g (BRowN, 1962) i n t r o d u c e a n undesired d e f o r m a t i o n o f the signal. T h i s is i m p r a c t i c a l w h e n the signal to noise ratio is low. A u t o c o r r e l a t i o n techniques i m p r o v e the signal to noise ratio w i t h o u t d e t e r i o r a t i o n o f the signal, r e c o v e r i n g a periodicity w h i c h m i g h t be
60
B . G . M . VANDI~.GINSTE, P. J. M. S A L E M I N K & J. c. D U I N K E R
present in the data, even when hidden in noise. Moreover a parameter is obtained which is inversely proportional to the velocity of the fluctuations in the signal. The basic technique can also be used to obtain an autocorrelogram of a position series, giving a parameter which is related to the correlation of the concentrations at adjacent sampling positions. This could reveal the presence of a concentration isopattern. Position series measured at different times, can be combined in a new data set (Fig. 1). The appearance of a synchronization pulse in the autocorrelogram of such a series (Fig. 2) gives information about the stability in time of the concentration profile. Crosscorrelation calculations offer the possibility to obtain a quantitative description of correlation. This may reveal more details than the qualitative description (PHILLIPS, 1972; PRICE & CALVERT, 1973; DUINKER & NOLTING, 1974). We applied this technique that is currently used in process control, to some series of concentration measurements of particulate Mn, Fe, Zn and Cu in the Southern Bight and the Dutch Wadden Sea (DUINKER • NOLTING, 1976). Acknowledgements.--We would like to thank Prof G. Kateman, Department of Analytical Chemistry, Catholic University of Nijmegen, for his helpfull discussions. II. DEFINITIONS AND NOTATIONS 1. AUTOCORRELATION Consider a time series x(t), with equally spaced data (interval At), a finite sample size n, and a mean :~. For any time shift ~: = n' At, the autocovariance function is defined by: t - (n-- n')At 'rxx(
) =
(n -
x
-
x, +
-
xt)
(1)
t=0
It is clear that for -: = 0, ~xx(0) --~ var(x). The autocorrelation function, that is more convenient to use is defined as the normalized autocovariance function. Cxx(~) = "rx~(~) / "rxx(0)
(2)
An important parameter which can be derived from the autocorrelogram is the time constant (Tx) which is inversely proportional to the velocity of the fluctuations of the time series. For a stationary time series with Tx # O, the autocorrelogram takes the form: Oxx ~
e-x/
Tx
PARTICULATE
TRACE
METALS
61
(VAN DER GRINTEN & LENOIR,1973). Fig. 2 shows the autocorrelation function of a noisy periodic signal, demonstrating the power of autocorrelation calculations for the discovery of periodicity. In comparison to the original signal (Fig. 1), the signal to noise ratio of the autocorrelated signal increases considerably because noise superimposed on the signal is only correlated within a correlation time much smaller than the time constant of the signal itself. The improvement of the signal to noise ratio by autocorrelating a finite time series, however decreases with a decreasing number of data pairs taken in calculation (decreasing value of n -- n'). Therefore for small values of n -- n' a procedure is needed to distinguish real oscillations of the autocorrelogram from fluctuations due to noise. The variance of the autocorrelation function can be used to determine the reliability limits. Qualitatively the variance of the autocorrelogram, var(~xz(X)), tends to increase when -r increases. This is related to the fact that the autocorrelogram gives no practical information for "r = nat. The variance of an autocorrelation function can be expressed quantitatively in simple terms as a function of z (HANNAN, 1960; VAN DER GRINTEN & LENOIR, 1973) : var(¢xx('r, nat)) = 1
11r ~ 1 1
nat
n'=-n
Z
{¢~x('~) + Cxx(n'At + ~) qJ~(n'At -- x)}
(3)
For an exponential autocorrelogram, and -r = 0, equation (3) becomes very simple: var(¢xx(0)) = 2Tx / nAt (4) and for z >> T x: var(CP.x(V)) = Tx / nAt (5) Applying the Student's t-test (BROWN, 1962) a correlation level is obtained to decide whether the autocorrelogram differs from zero or not, depending on the number of degrees of freedom (n -- n' -- 1) and the desired reliability (e.g. 95 %). Analogous to time series, autocorrelograms can be calculated for data obtained from equidistant sampling positions. The time constant in that case has to be replaced by a correlation length, as a measure for the correlation between neighbouring position data. When the number of data in one position series is relatively small, the noise on the autocorrelogram becomes considerable. This can be avoided by combining several position series, sampled at different times, to make one data set (Figs 1 and 2). Now the first data point of the m-th series with n elements becomes
62
B.O.M. VANDEGINSTE,
P.J.M.
SALEMINK
& J. c. D U I N K E R
the (m × n ~ 1)-th d a t a point of the new d a t a set. This calculation m e t h o d gives rise to a more reliable correlation length, that is the m e a n of the correlation lengths of the different series. Moreover it enables the detection of an increased correlation at -~ = n, when the position series are stable in time. I n d e e d one or more synchronization pulses appear in the autocorrelograms of correlated position series. Fig. 2 does show a distinct pulse for M n and Fe at -~ = 82, according to the length of one series. 2. CROSSCORRELATION Autocorrelation gives information about d a t a series separately. O n the other h a n d crosscorrelation, informs about the correlation between two d a t a series. Analogous to equation (2), the crosscorrelafion function is defined as: •
/
= t = (n-
(n -- n') -1
=
n')At
Z
(ff -- xt) (~< - - y t + ~) var(x,y) -t
(6)
t=o
where for T >> Tx:
-
r x f nAt
(7)
T w o identical d a t a series show m a x i m u m correlation when v = 0. I f one series is delayed with respect to the other, m a x i m u m correlation is found w h e n ~ = Xdel,corresponding to the phase shift between the two series. Equation (6) indicates that Oxy(0) = 1. For other values of the correlation function it can reach each value between -- oo and + co. More convenient is the normalized crosscorrelation function with values between + 1 a n d --1. T h e correlation coefficient of perfect correlated d a t a series equals one. T h e crosscorrelafion coefficient is defined as :
p~(~)
=
V~(~) / ~/~xx(O) 'm~(O)
(8)
x
(9)
=
T h e significance level of the correlation coefficient is calculated using the r-test (EZEKIEL & FOX, 1959).
PARTICULATE TRACE METALS
63
III. R E S U L T S AND D I S C U S S I O N 1. AUTOCORRELATION
CALCULATIONS THE
SOUTHERN
APPLIED
TO
DATA
FROM
BIGHT
Monthly analytical data of particulate Mn, Fe, Zn and Cu concentrations (in vg metal per gram particulate matter on a basis of dry weight) were obtained from 82 sampling stations in the Southern Bight (DuINKER & NOLTINO, 1976). In this way 3 position series were obtained, measured at different times. Because the time elapsed between the measurement of the first and the last station in a cruise was small as compared with the time between two cruises, the elements of a position series were considered as been taken at the same time. In order to investigate the stability of the concentration profiles from the appearance of synchronization pulses, the 3 series were combined as described in the preceeding section. Fig. 1 shows such a combined data ,/ug.g-1 2800 T'TI
T'T2
T-T~
2400
I
I
2000
1600
1200-
800,
400.
" o
o
1 ~'o ' ,:o"
~
;o
I " ~o"
Jo
;o
I
~o
1o
~o
r:o
;o
elation number
Fig. 1. Particulate Mn concentrations in the combined position series, sampled at different times in the Southern Bight. Position of stations given in Fig. 3a.
series tbr the particulate M n concentration. Although the data in the position series were not equidistant and the series were not stationary, conclusions could be drawn from the calculated autocorrelograms, shown in Fig. 2.
64
B. 0 . M. V A N D E O I N S T E ,
P. J. M. S A L E M I N K
& J. c . D U I N K E R
~xx('E) 1.0-
0.8-
0,6 1 0.4
t
/\,
i
IV',v :',A i:',¢
\\ I '1:" -0.2
\r",
~
I
/_x
)I
I
~.4~ ~.,:~-] ,o. ""~
,A !':
"
- 0.4
'13
Fig. 2. A u t o c o r r e l o g r a m s of t h e p a r t i c u l a t e Fe ( - - - ) ,
Mn (----),
Cu ( .... ) and
Zn (-.-.) concentrations in the Southern Bight. Tx values for Mn, Fe, Cu and Zn indicate the correlation of the concentrations in consecutive stations (Table I).
1.1 T H E S T A B I L I T Y
OF THE
CONCENTRATION
PROFILES
W h e n the particulate metal concentration profile shows a certain stability in time, a significant increase of correlation must be found at z = 82 in the autocorrelogram of the combined position series. This appears as a synchronization pulse. Table I gives the values of the autocorrelograms at • = 82 and the correlation values needed for a 95°/o and 99% level of reliability that ~xx(82) ~ 0. As shown, the significance tests, applied to the values of the autocorrelation show that the synchronization pulses for particulate M n a n d Fe differ from zero with a reliability up to 99%. F r o m these calculations we conclude that the concentration profile of particulate M n and Fe is stable over a period of at least one month. T h e detection of a stability over a period, longer t h a n one m o n t h will require more position series, in order to obtain reliable correlation values at z ---- 164.
PARTICULATE TRACE METALS
65
Assuming that the height of the synchronization pulse above the significance level is a measure for the stability of these concentration profiles, one may conclude that this stability in the Southern Bight decreases as follows: Mn > F e > C u ~ Z n From our observations not any important stability in time could be detected for the particulate Cu and Zn concentrations.
1.2.
THE
EXISTENCE
OF CONCENTRATION
ISOPATTERNS
As pointed out the correlation length obtained from the correlogram gives an estimation of the correlation of the concentration between consecutive sampling stations. Since these stations are not equally spaced over the area, this correlation length has no absolute value, but offers a good reladve measure to compare the behaviour of different trace metals (Fig. 2 and Table I). Because the data series are small, the TABLE I
Value and reliability of the autocorrelation at v ~ 82 for data series from the Southern Bight, and the correlation length of the correlation function, Tx in units of distance between sampling stations. Element
Mn Zn Cu Fe
Tx
4 2 1 5
q~x. (82)
0.79 0.17 0.17 0.52
Statistical significance level at probability o~.
a = 0.99
a = 0.95
0.28 0.28 0.20 0.33
0.20 0.19 0.14 0.23
accuracy of the calculated correlation length is low. Nevertheless we suppose that a relatively large correlation length (say 4 to 5 stations) suggests the existence of an isopattern. Autocorrelation of the concentrations in sequence of increasing position number, shows the greatest correlation length for particulate Mn and Fe. This suggests that an isopattern exists for both. Wether the isopatterns are congruent or not cannot be decided on the basis of their autocorrelograms. According to the pronounced synchronization pulse in the autocorrelograms, these isopatterns of both elements seem to be stable in time. On the other hand the particulate Cu has a very short correlation length, revealing the absence of any isopattern. Isopatterns for Mn and Fe
66
B.G.M. VANDEOINSTE,
)
P. J. M. S A L E M I N K
:'
..
..
.J
I/
/./FI./
o
i
.@¢
•
& J. c. D U I N K E R
rr
..
...
//./< lJ
~.././-
o
Fig,
~
_
t
_
..................................................._ b
3. I s o p a t t e r n of particulate M n (a) a n d Fe (b) c o n c e n t r a t i o n in the S o u t h e r n Bight, averaged over 3 cruises in p p m (a) a n d % (b).
have been constructed by averaging the concentration values of the 3 data series (Fig. 3). It is a remarkable result that the presence or absence of a stable concentration profile coincides with the presence or absence of an isopattern. 9. A U T O C O R R E L A T I O N CALCULATIONS APPLIED T H E D U T C H W A D D E N SEA
TO
DATA FROM
Analytical data of the concentration of particulate trace metals (Mn, Fe, Zn and Cu) were obtained from 76 sampling stations situated in the Dutch Wadden Sea. Unfortunately only 2 complete data series were available for this type of calculations. These 2 position series were combined to one series; the same remarks, as pointed out in the previous section, are valid. Therefore, higher correlation values are needed to conclude for sigrdficancc of the autocorrclograms shown in Fig. 4. As the sampling stations are very unequally spaced over the area, an interpretation of the concentration lengths found in the correlograms is not allowed. Table II shows again that particulate Mn and Fe possess significant synchronization pulses. However, these pulses are lcss pronounced than those found for the measurements in the Southern Bight. Assuming that the height of the synchronization pulse above the significance level is a measure for the stability of the concentration profile, one can conclude that the stability decreases as follows: Mn ~ F e ~ Z n
~;>Cu
PARTICULATE
67
TRACE METALS
This sequence agrees with the sequence for the trace metals in the Southern Bight. AdditionaUy, Zn in the Wadden Sea belongs to the elements having a significant synchronization pulse.
1.0 -[
0.8,
0.6 ,.tB
Jt~i o.4.
.
•
\j
1
1
,J
i!
",. o
'
it
-0!4-
'~
go
~o
io
do
'IT
,~o
Fig, 4. hutocorrelograms of the particulate Fe ( - - - ) , M n ( ), Cu ( . . . . ) and Zn ( - . - . ) concentrations in the Dutch Wadden Sea. TABLE
II
Value and reliability of the autocorrelation at x = 76 for data series from the Dutch W a d d e n Sea.
Element
M.n Zn Cu Fe
~xx (76)
0.51 0.52 0.08 0.57
Statistical significance level at probability ot a = 0.99
a = 0.95
0.36 0.45 0.50 0.45
0.26 0.32 0.36 0.32
68
B.G.M.
VANDEGINSTE,
3. CROSSCORRELATION BIGHT
P.
j.
M. S A L E M I N K
CALCULATIONS AND
THE
DUTCH
& IN
WADDEN
J.
C. D U I N K E R
THE
SOUTHERN
SEA
A p p l y i n g e q u a t i o n ( 6 ) , crosscorrelation functions were calculated f r o m the s a m e d a t a as used for the calculation of the a u t o c o r r e l a t i o n functions in the S o u t h e r n Bight a n d the D u t c h W a d d e n Sea. N o n e of the crosscorrelation functions shows a n y phase shift. T h e r e fore, no processes t h a t involve a position shifted correlation could be detected, as we could expect. An interesting p a r a m e t e r is f u r t h e r the value of the crosscorrelation coefficient at z = 0, which is shown in T a b l e I I I . W e observe TABLE III
Cross correlation coefficients at v = 0 of particulate elements in the Southern Bight and Dutch Wadden Sea. For the Southern Bight, a correlation coefficient of 0.15 is different from zero with a reliability of 99%, a coefficient of 0.10 is different from zero with a reliability of 95% ; for the Dutch Wadden Sea a correlation coefficient of 0.18 is different from zero with a reliability of 99%, a coefficient of 0.13 is different from zero with a reliability of 950/o. Element
Southern Bight Mn
Mn Fe Cu Zn
Fe
1 0.4 1 --0.13 --0.04 --0.01 0.15
Cu
1 0.53
Dutch Wadden Sea Zn
1
"
Mn
Fe
Cu
Zn
1 0.39 0.25 0.51
1 0.16 0.25
t 0.55
1
high correlation values for F e / M n a n d Z n / C u in b o t h the S o u t h e r n Bight a n d the D u t c h W a d d e n Sea. This m e a n s t h a t in b o t h areas the patterns of the M n a n d Fe c o n c e n t r a t i o n in suspended m a t t e r are significantly correlated. I n c o m b i n a t i o n with the f o r m e r conclusion t h a t b o t h elements show a stable isopattern, the conclusion is justified that the isopatterns of the M n a n d Fe concentrations in the suspended m a t t e r h a v e identical gradients in the S o u t h e r n Bight. This suggests t h a t the concentrations of b o t h elements in the p a r t i c u l a t e m a t t e r are influenced b y similar processes. A l t h o u g h we did not detect a n y i s o p a t t e r n or stable profile for the p a r t i c u l a t e Z n a n d Cu c o n c e n t r a tions, the concentrations of b o t h elements t h a t look r a n d o m l y distributed at first sight, a p p e a r to be strongly correlated. A low crosscorrelation coefficient should be found for the concentration of two metals, one h a v i n g a n d one missing a p r o n o u n c e d synchronization pulse in the a u t o c o r r e l o g r a m . This is true for the
PARTICULATE
TRACE METALS
69
correlation coefficients found for Mn/Cu, Mn/Zn, Fe/Cu and Fe/Zn in the Southern Bight. Therefore, these results confirm the former conclusions from the autocorrelograms, namely that the Fe and Mn concentrations in the suspended matter show a dynamic behaviour that is different from the corresponding Cu and Zn concentrations. O n the other hand some apparently significant values of crosscorrelation coefficients of Cu and other metals in the Wadden Sea do not agree with the absence of the synchronization pulse for Cu. DUINKER & NOLTING (1976) try to analyse the processes within the marine environment and to interpret the present findings. IV. CONCLUSIONS The present investigation shows that auto- and crosscorrelation studies applied to data from surface waters offers possibilities for the detection ofisopatterns and for the quantitative evaluation of their stability. The quantitative description of correlation, even when time delayed or position shifted, between various chemical species is possible. The presence of correlation is important for the detection of hydrochemical, physical or biological processes. O n the other hand, a knowledge of the dynamic properties of concentrations of any chemical component in surface water is important from an analytical point of view. In fact these data may form a basis from which the analysis conditions (analysis accuracy, -time and -frequency) for an optimal observation of the pollution in the Southern Bight and the Dutch Wadden Sea can be derived. V. SUMMARY Auto- and crosscorrelation calculations have been applied to leacheable particulate trace metals concentration data (Mn, Fe, Zn and Cu) from the Southern Bight and the Dutch Wadden Sea. Although the amount of data is limited, some features on the dynamic properties of the four metals could be established. The stability of the concentration profile in the two areas decreases in the order Mn, Fe, Zn, Cu. Manganese and iron show a concentration profile that is stable over at least one month in the Southern Bight; moreover as the correlation length is rather large, the existence of an isopattern for M n and Fe in the Southern Bight is justified. The patterns of particulate M n and Fe (the same conclusion applies Cu and Zn) are significantly correlated. This indicates that the isopatterns for M n and Fe have identical gradients in the Southern Bight, suggesting that the particulate (leacheable) Fe and M n concentrations are influenced
70
B . G . M . VANDEGINSTE, P. J. M. SALEMINK & J. c. D U I N K E R
by similar processes. A l t h o u g h in the results a n y i s o p a t t e r n or stable profile for c o p p e r a n d zinc was absent, t h e o b s e r v e d s t r o n g c o r r e l a t i o n suggests t h a t well d e f i n e d processes m u s t be responsible for the a p p a r e n t r a n d o m c o n c e n t r a t i o n values o f b o t h elements. VI. R E F E R E N C E S BROWN, R. G., 1962. Smoothing, forecasting and prediction of discrete time series. Prentice-Hall, Englewood Cliffs, N.J. : 1-468. DUINKeR, J. C. & R. F. NOLTINO, 1974. Trace metals in water and suspended matter in the Southern Bight (copper, zinc, manganese and iron). ICES C.M. 1974/E:26 (mimeo). , 1976. Distribution model for particulate trace metals in the Rhine estuary, Southern Bight and Dutch Wadden Sea.--Neth. J. Sea Res. 10 ( 1) : 71-102. EZEKmL, M. & K. A. Fox, 1959. Methods of correlation and regression analysis. Wiley & Sons, Edinburgh: 1-548. GmN'rEN, P. M. E. M. VAN DER & L. LENOm, 1973. Statistische procesbeheersing. Spectrum, Antwerpen : 1-598. HANNa-~, E.J., 1960. Time series analysis. Wiley & Sons, New York: 1-152. PmLLIPS,J., 1972. Chemical processes in estuaries. In : R. S. K. BARNES& J. GREEN. The estuarine environment: 33-50. PRICE, N. B. & S. E. CALVERT, 1973. A study of the geochemistry of suspended particulate matter in coastal waters.--Mar. Chem. 1: 169-189.
SOUTHERN BIGHT WADDEN SEA
AND
DUTCH
by B. G. M. V A N D E G I N S T E , P . J . M . S A L E M I N K (Department of Analytical Chemistry, Catholic University of .Nijmegen, The .Netherlands) and J. C. D U I N K E R (.Netherlands Institute for Sea Research, Texel, The .Netherlands) CONTENTS I. Introduction . . . . . . . . . . . . . . . . . II. Definitions and Notations . . . . . . . . . . . . . . I. Autocorrelation . . . . . . . . . . . . . . . . 2. Crosscorrelation . . . . . . . . . . . . . . . . III. Results and Discussion . . . . . . . . . . . . . . 1. Autocorrelation calculations applied to data from the Southern Bight . . . . . . . . . . . . . . . . . . . 1.1. The stability of the concentration profiles . . . . . . . 1.2. The existence of concentration isopatterns . . . . . . . 2. Autocorrelation calculations applied to data from the Dutch Wadden Sea . . . . . . . . . . . . . . . . . . . 3. Crosscorrelation calculations in the Southern Bight and the Dutch Wadden Sea . . . . . . . . . . . . . . . . . IV. Conclusions . . . . . . . . . . . . . . . . . . V. Summary . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . .
59 60 60 69 63 63 64 65 66 68 69 69 70
I. I N T R O D U C T I O N As a rule, t i m e series or position series, o b t a i n e d f r o m the analysis of c h e m i c a l species in n a t u r a l waters, at different positions at the s a m e time, or at a n y position at different times, show considerable fluctuations due to o t h e r t h a n strictly h y d r o c h e m i c a l , biological or physical processes. S a m p l i n g a n d analytical errors d e t e r i o r a t e the d a t a quality. T h e r e f o r e , analytical noise is one o f the serious limitations for disc o v e r i n g h y d r o l o g i c a l processes f r o m e n v i r o n m e n t a l d a t a . F o r the detection o f signals h i d d e n in noise, a s m o o t h i n g p r o c e d u r e is n e e d e d . H o w e v e r , c o n v e n t i o n a l s m o o t h i n g p r o c e d u r e s e.g. e x p o n e n t i a l a v e r a g i n g (BRowN, 1962) i n t r o d u c e a n undesired d e f o r m a t i o n o f the signal. T h i s is i m p r a c t i c a l w h e n the signal to noise ratio is low. A u t o c o r r e l a t i o n techniques i m p r o v e the signal to noise ratio w i t h o u t d e t e r i o r a t i o n o f the signal, r e c o v e r i n g a periodicity w h i c h m i g h t be
60
B . G . M . VANDI~.GINSTE, P. J. M. S A L E M I N K & J. c. D U I N K E R
present in the data, even when hidden in noise. Moreover a parameter is obtained which is inversely proportional to the velocity of the fluctuations in the signal. The basic technique can also be used to obtain an autocorrelogram of a position series, giving a parameter which is related to the correlation of the concentrations at adjacent sampling positions. This could reveal the presence of a concentration isopattern. Position series measured at different times, can be combined in a new data set (Fig. 1). The appearance of a synchronization pulse in the autocorrelogram of such a series (Fig. 2) gives information about the stability in time of the concentration profile. Crosscorrelation calculations offer the possibility to obtain a quantitative description of correlation. This may reveal more details than the qualitative description (PHILLIPS, 1972; PRICE & CALVERT, 1973; DUINKER & NOLTING, 1974). We applied this technique that is currently used in process control, to some series of concentration measurements of particulate Mn, Fe, Zn and Cu in the Southern Bight and the Dutch Wadden Sea (DUINKER • NOLTING, 1976). Acknowledgements.--We would like to thank Prof G. Kateman, Department of Analytical Chemistry, Catholic University of Nijmegen, for his helpfull discussions. II. DEFINITIONS AND NOTATIONS 1. AUTOCORRELATION Consider a time series x(t), with equally spaced data (interval At), a finite sample size n, and a mean :~. For any time shift ~: = n' At, the autocovariance function is defined by: t - (n-- n')At 'rxx(
) =
(n -
x
-
x, +
-
xt)
(1)
t=0
It is clear that for -: = 0, ~xx(0) --~ var(x). The autocorrelation function, that is more convenient to use is defined as the normalized autocovariance function. Cxx(~) = "rx~(~) / "rxx(0)
(2)
An important parameter which can be derived from the autocorrelogram is the time constant (Tx) which is inversely proportional to the velocity of the fluctuations of the time series. For a stationary time series with Tx # O, the autocorrelogram takes the form: Oxx ~
e-x/
Tx
PARTICULATE
TRACE
METALS
61
(VAN DER GRINTEN & LENOIR,1973). Fig. 2 shows the autocorrelation function of a noisy periodic signal, demonstrating the power of autocorrelation calculations for the discovery of periodicity. In comparison to the original signal (Fig. 1), the signal to noise ratio of the autocorrelated signal increases considerably because noise superimposed on the signal is only correlated within a correlation time much smaller than the time constant of the signal itself. The improvement of the signal to noise ratio by autocorrelating a finite time series, however decreases with a decreasing number of data pairs taken in calculation (decreasing value of n -- n'). Therefore for small values of n -- n' a procedure is needed to distinguish real oscillations of the autocorrelogram from fluctuations due to noise. The variance of the autocorrelation function can be used to determine the reliability limits. Qualitatively the variance of the autocorrelogram, var(~xz(X)), tends to increase when -r increases. This is related to the fact that the autocorrelogram gives no practical information for "r = nat. The variance of an autocorrelation function can be expressed quantitatively in simple terms as a function of z (HANNAN, 1960; VAN DER GRINTEN & LENOIR, 1973) : var(¢xx('r, nat)) = 1
11r ~ 1 1
nat
n'=-n
Z
{¢~x('~) + Cxx(n'At + ~) qJ~(n'At -- x)}
(3)
For an exponential autocorrelogram, and -r = 0, equation (3) becomes very simple: var(¢xx(0)) = 2Tx / nAt (4) and for z >> T x: var(CP.x(V)) = Tx / nAt (5) Applying the Student's t-test (BROWN, 1962) a correlation level is obtained to decide whether the autocorrelogram differs from zero or not, depending on the number of degrees of freedom (n -- n' -- 1) and the desired reliability (e.g. 95 %). Analogous to time series, autocorrelograms can be calculated for data obtained from equidistant sampling positions. The time constant in that case has to be replaced by a correlation length, as a measure for the correlation between neighbouring position data. When the number of data in one position series is relatively small, the noise on the autocorrelogram becomes considerable. This can be avoided by combining several position series, sampled at different times, to make one data set (Figs 1 and 2). Now the first data point of the m-th series with n elements becomes
62
B.O.M. VANDEGINSTE,
P.J.M.
SALEMINK
& J. c. D U I N K E R
the (m × n ~ 1)-th d a t a point of the new d a t a set. This calculation m e t h o d gives rise to a more reliable correlation length, that is the m e a n of the correlation lengths of the different series. Moreover it enables the detection of an increased correlation at -~ = n, when the position series are stable in time. I n d e e d one or more synchronization pulses appear in the autocorrelograms of correlated position series. Fig. 2 does show a distinct pulse for M n and Fe at -~ = 82, according to the length of one series. 2. CROSSCORRELATION Autocorrelation gives information about d a t a series separately. O n the other h a n d crosscorrelation, informs about the correlation between two d a t a series. Analogous to equation (2), the crosscorrelafion function is defined as: •
/
= t = (n-
(n -- n') -1
=
n')At
Z
(ff -- xt) (~< - - y t + ~) var(x,y) -t
(6)
t=o
where for T >> Tx:
-
r x f nAt
(7)
T w o identical d a t a series show m a x i m u m correlation when v = 0. I f one series is delayed with respect to the other, m a x i m u m correlation is found w h e n ~ = Xdel,corresponding to the phase shift between the two series. Equation (6) indicates that Oxy(0) = 1. For other values of the correlation function it can reach each value between -- oo and + co. More convenient is the normalized crosscorrelation function with values between + 1 a n d --1. T h e correlation coefficient of perfect correlated d a t a series equals one. T h e crosscorrelafion coefficient is defined as :
p~(~)
=
V~(~) / ~/~xx(O) 'm~(O)
(8)
x
(9)
=
T h e significance level of the correlation coefficient is calculated using the r-test (EZEKIEL & FOX, 1959).
PARTICULATE TRACE METALS
63
III. R E S U L T S AND D I S C U S S I O N 1. AUTOCORRELATION
CALCULATIONS THE
SOUTHERN
APPLIED
TO
DATA
FROM
BIGHT
Monthly analytical data of particulate Mn, Fe, Zn and Cu concentrations (in vg metal per gram particulate matter on a basis of dry weight) were obtained from 82 sampling stations in the Southern Bight (DuINKER & NOLTINO, 1976). In this way 3 position series were obtained, measured at different times. Because the time elapsed between the measurement of the first and the last station in a cruise was small as compared with the time between two cruises, the elements of a position series were considered as been taken at the same time. In order to investigate the stability of the concentration profiles from the appearance of synchronization pulses, the 3 series were combined as described in the preceeding section. Fig. 1 shows such a combined data ,/ug.g-1 2800 T'TI
T'T2
T-T~
2400
I
I
2000
1600
1200-
800,
400.
" o
o
1 ~'o ' ,:o"
~
;o
I " ~o"
Jo
;o
I
~o
1o
~o
r:o
;o
elation number
Fig. 1. Particulate Mn concentrations in the combined position series, sampled at different times in the Southern Bight. Position of stations given in Fig. 3a.
series tbr the particulate M n concentration. Although the data in the position series were not equidistant and the series were not stationary, conclusions could be drawn from the calculated autocorrelograms, shown in Fig. 2.
64
B. 0 . M. V A N D E O I N S T E ,
P. J. M. S A L E M I N K
& J. c . D U I N K E R
~xx('E) 1.0-
0.8-
0,6 1 0.4
t
/\,
i
IV',v :',A i:',¢
\\ I '1:" -0.2
\r",
~
I
/_x
)I
I
~.4~ ~.,:~-] ,o. ""~
,A !':
"
- 0.4
'13
Fig. 2. A u t o c o r r e l o g r a m s of t h e p a r t i c u l a t e Fe ( - - - ) ,
Mn (----),
Cu ( .... ) and
Zn (-.-.) concentrations in the Southern Bight. Tx values for Mn, Fe, Cu and Zn indicate the correlation of the concentrations in consecutive stations (Table I).
1.1 T H E S T A B I L I T Y
OF THE
CONCENTRATION
PROFILES
W h e n the particulate metal concentration profile shows a certain stability in time, a significant increase of correlation must be found at z = 82 in the autocorrelogram of the combined position series. This appears as a synchronization pulse. Table I gives the values of the autocorrelograms at • = 82 and the correlation values needed for a 95°/o and 99% level of reliability that ~xx(82) ~ 0. As shown, the significance tests, applied to the values of the autocorrelation show that the synchronization pulses for particulate M n a n d Fe differ from zero with a reliability up to 99%. F r o m these calculations we conclude that the concentration profile of particulate M n and Fe is stable over a period of at least one month. T h e detection of a stability over a period, longer t h a n one m o n t h will require more position series, in order to obtain reliable correlation values at z ---- 164.
PARTICULATE TRACE METALS
65
Assuming that the height of the synchronization pulse above the significance level is a measure for the stability of these concentration profiles, one may conclude that this stability in the Southern Bight decreases as follows: Mn > F e > C u ~ Z n From our observations not any important stability in time could be detected for the particulate Cu and Zn concentrations.
1.2.
THE
EXISTENCE
OF CONCENTRATION
ISOPATTERNS
As pointed out the correlation length obtained from the correlogram gives an estimation of the correlation of the concentration between consecutive sampling stations. Since these stations are not equally spaced over the area, this correlation length has no absolute value, but offers a good reladve measure to compare the behaviour of different trace metals (Fig. 2 and Table I). Because the data series are small, the TABLE I
Value and reliability of the autocorrelation at v ~ 82 for data series from the Southern Bight, and the correlation length of the correlation function, Tx in units of distance between sampling stations. Element
Mn Zn Cu Fe
Tx
4 2 1 5
q~x. (82)
0.79 0.17 0.17 0.52
Statistical significance level at probability o~.
a = 0.99
a = 0.95
0.28 0.28 0.20 0.33
0.20 0.19 0.14 0.23
accuracy of the calculated correlation length is low. Nevertheless we suppose that a relatively large correlation length (say 4 to 5 stations) suggests the existence of an isopattern. Autocorrelation of the concentrations in sequence of increasing position number, shows the greatest correlation length for particulate Mn and Fe. This suggests that an isopattern exists for both. Wether the isopatterns are congruent or not cannot be decided on the basis of their autocorrelograms. According to the pronounced synchronization pulse in the autocorrelograms, these isopatterns of both elements seem to be stable in time. On the other hand the particulate Cu has a very short correlation length, revealing the absence of any isopattern. Isopatterns for Mn and Fe
66
B.G.M. VANDEOINSTE,
)
P. J. M. S A L E M I N K
:'
..
..
.J
I/
/./FI./
o
i
.@¢
•
& J. c. D U I N K E R
rr
..
...
//./< lJ
~.././-
o
Fig,
~
_
t
_
..................................................._ b
3. I s o p a t t e r n of particulate M n (a) a n d Fe (b) c o n c e n t r a t i o n in the S o u t h e r n Bight, averaged over 3 cruises in p p m (a) a n d % (b).
have been constructed by averaging the concentration values of the 3 data series (Fig. 3). It is a remarkable result that the presence or absence of a stable concentration profile coincides with the presence or absence of an isopattern. 9. A U T O C O R R E L A T I O N CALCULATIONS APPLIED T H E D U T C H W A D D E N SEA
TO
DATA FROM
Analytical data of the concentration of particulate trace metals (Mn, Fe, Zn and Cu) were obtained from 76 sampling stations situated in the Dutch Wadden Sea. Unfortunately only 2 complete data series were available for this type of calculations. These 2 position series were combined to one series; the same remarks, as pointed out in the previous section, are valid. Therefore, higher correlation values are needed to conclude for sigrdficancc of the autocorrclograms shown in Fig. 4. As the sampling stations are very unequally spaced over the area, an interpretation of the concentration lengths found in the correlograms is not allowed. Table II shows again that particulate Mn and Fe possess significant synchronization pulses. However, these pulses are lcss pronounced than those found for the measurements in the Southern Bight. Assuming that the height of the synchronization pulse above the significance level is a measure for the stability of the concentration profile, one can conclude that the stability decreases as follows: Mn ~ F e ~ Z n
~;>Cu
PARTICULATE
67
TRACE METALS
This sequence agrees with the sequence for the trace metals in the Southern Bight. AdditionaUy, Zn in the Wadden Sea belongs to the elements having a significant synchronization pulse.
1.0 -[
0.8,
0.6 ,.tB
Jt~i o.4.
.
•
\j
1
1
,J
i!
",. o
'
it
-0!4-
'~
go
~o
io
do
'IT
,~o
Fig, 4. hutocorrelograms of the particulate Fe ( - - - ) , M n ( ), Cu ( . . . . ) and Zn ( - . - . ) concentrations in the Dutch Wadden Sea. TABLE
II
Value and reliability of the autocorrelation at x = 76 for data series from the Dutch W a d d e n Sea.
Element
M.n Zn Cu Fe
~xx (76)
0.51 0.52 0.08 0.57
Statistical significance level at probability ot a = 0.99
a = 0.95
0.36 0.45 0.50 0.45
0.26 0.32 0.36 0.32
68
B.G.M.
VANDEGINSTE,
3. CROSSCORRELATION BIGHT
P.
j.
M. S A L E M I N K
CALCULATIONS AND
THE
DUTCH
& IN
WADDEN
J.
C. D U I N K E R
THE
SOUTHERN
SEA
A p p l y i n g e q u a t i o n ( 6 ) , crosscorrelation functions were calculated f r o m the s a m e d a t a as used for the calculation of the a u t o c o r r e l a t i o n functions in the S o u t h e r n Bight a n d the D u t c h W a d d e n Sea. N o n e of the crosscorrelation functions shows a n y phase shift. T h e r e fore, no processes t h a t involve a position shifted correlation could be detected, as we could expect. An interesting p a r a m e t e r is f u r t h e r the value of the crosscorrelation coefficient at z = 0, which is shown in T a b l e I I I . W e observe TABLE III
Cross correlation coefficients at v = 0 of particulate elements in the Southern Bight and Dutch Wadden Sea. For the Southern Bight, a correlation coefficient of 0.15 is different from zero with a reliability of 99%, a coefficient of 0.10 is different from zero with a reliability of 95% ; for the Dutch Wadden Sea a correlation coefficient of 0.18 is different from zero with a reliability of 99%, a coefficient of 0.13 is different from zero with a reliability of 950/o. Element
Southern Bight Mn
Mn Fe Cu Zn
Fe
1 0.4 1 --0.13 --0.04 --0.01 0.15
Cu
1 0.53
Dutch Wadden Sea Zn
1
"
Mn
Fe
Cu
Zn
1 0.39 0.25 0.51
1 0.16 0.25
t 0.55
1
high correlation values for F e / M n a n d Z n / C u in b o t h the S o u t h e r n Bight a n d the D u t c h W a d d e n Sea. This m e a n s t h a t in b o t h areas the patterns of the M n a n d Fe c o n c e n t r a t i o n in suspended m a t t e r are significantly correlated. I n c o m b i n a t i o n with the f o r m e r conclusion t h a t b o t h elements show a stable isopattern, the conclusion is justified that the isopatterns of the M n a n d Fe concentrations in the suspended m a t t e r h a v e identical gradients in the S o u t h e r n Bight. This suggests t h a t the concentrations of b o t h elements in the p a r t i c u l a t e m a t t e r are influenced b y similar processes. A l t h o u g h we did not detect a n y i s o p a t t e r n or stable profile for the p a r t i c u l a t e Z n a n d Cu c o n c e n t r a tions, the concentrations of b o t h elements t h a t look r a n d o m l y distributed at first sight, a p p e a r to be strongly correlated. A low crosscorrelation coefficient should be found for the concentration of two metals, one h a v i n g a n d one missing a p r o n o u n c e d synchronization pulse in the a u t o c o r r e l o g r a m . This is true for the
PARTICULATE
TRACE METALS
69
correlation coefficients found for Mn/Cu, Mn/Zn, Fe/Cu and Fe/Zn in the Southern Bight. Therefore, these results confirm the former conclusions from the autocorrelograms, namely that the Fe and Mn concentrations in the suspended matter show a dynamic behaviour that is different from the corresponding Cu and Zn concentrations. O n the other hand some apparently significant values of crosscorrelation coefficients of Cu and other metals in the Wadden Sea do not agree with the absence of the synchronization pulse for Cu. DUINKER & NOLTING (1976) try to analyse the processes within the marine environment and to interpret the present findings. IV. CONCLUSIONS The present investigation shows that auto- and crosscorrelation studies applied to data from surface waters offers possibilities for the detection ofisopatterns and for the quantitative evaluation of their stability. The quantitative description of correlation, even when time delayed or position shifted, between various chemical species is possible. The presence of correlation is important for the detection of hydrochemical, physical or biological processes. O n the other hand, a knowledge of the dynamic properties of concentrations of any chemical component in surface water is important from an analytical point of view. In fact these data may form a basis from which the analysis conditions (analysis accuracy, -time and -frequency) for an optimal observation of the pollution in the Southern Bight and the Dutch Wadden Sea can be derived. V. SUMMARY Auto- and crosscorrelation calculations have been applied to leacheable particulate trace metals concentration data (Mn, Fe, Zn and Cu) from the Southern Bight and the Dutch Wadden Sea. Although the amount of data is limited, some features on the dynamic properties of the four metals could be established. The stability of the concentration profile in the two areas decreases in the order Mn, Fe, Zn, Cu. Manganese and iron show a concentration profile that is stable over at least one month in the Southern Bight; moreover as the correlation length is rather large, the existence of an isopattern for M n and Fe in the Southern Bight is justified. The patterns of particulate M n and Fe (the same conclusion applies Cu and Zn) are significantly correlated. This indicates that the isopatterns for M n and Fe have identical gradients in the Southern Bight, suggesting that the particulate (leacheable) Fe and M n concentrations are influenced
70
B . G . M . VANDEGINSTE, P. J. M. SALEMINK & J. c. D U I N K E R
by similar processes. A l t h o u g h in the results a n y i s o p a t t e r n or stable profile for c o p p e r a n d zinc was absent, t h e o b s e r v e d s t r o n g c o r r e l a t i o n suggests t h a t well d e f i n e d processes m u s t be responsible for the a p p a r e n t r a n d o m c o n c e n t r a t i o n values o f b o t h elements. VI. R E F E R E N C E S BROWN, R. G., 1962. Smoothing, forecasting and prediction of discrete time series. Prentice-Hall, Englewood Cliffs, N.J. : 1-468. DUINKeR, J. C. & R. F. NOLTINO, 1974. Trace metals in water and suspended matter in the Southern Bight (copper, zinc, manganese and iron). ICES C.M. 1974/E:26 (mimeo). , 1976. Distribution model for particulate trace metals in the Rhine estuary, Southern Bight and Dutch Wadden Sea.--Neth. J. Sea Res. 10 ( 1) : 71-102. EZEKmL, M. & K. A. Fox, 1959. Methods of correlation and regression analysis. Wiley & Sons, Edinburgh: 1-548. GmN'rEN, P. M. E. M. VAN DER & L. LENOm, 1973. Statistische procesbeheersing. Spectrum, Antwerpen : 1-598. HANNa-~, E.J., 1960. Time series analysis. Wiley & Sons, New York: 1-152. PmLLIPS,J., 1972. Chemical processes in estuaries. In : R. S. K. BARNES& J. GREEN. The estuarine environment: 33-50. PRICE, N. B. & S. E. CALVERT, 1973. A study of the geochemistry of suspended particulate matter in coastal waters.--Mar. Chem. 1: 169-189.