Harmonic Analysis on a Spatially Distributed GaugeArray in a Tidal Marsh, with a Non-stationary Signal

Ryan S. Mieras

Center for Applied Coastal ResearchUniversity of Delaware

Newark, DE 19716

May 22, 2013

Abstract

The coast of Delaware along the Delaware Bay is comprised primarily of salt/tidal marshenvironments, serving as a protective barrier to the inland areas during storms and imminentsea level rise. It is characteristic of these marshes to have a main channel which drains andfills with the tide. In an effort to better understand the dynamics of these systems, a 14-dayfield experiment was conducted in the Brockonbridge Marsh (BM) in Kent County, DE. Usingtraditional harmonic analysis (HA) available in T_TIDE, the resulting signals of five pres-sure gauges show strong non-linear interaction between fundamental and shallow water con-stituents, causing distortion of the water level further up the channel. In particular, the M2 tidedecays considerably, further into the channel. The decay of M2 as the tide propagates throughthe channel is due to strong bottom frictional damping in the system. Further investigation intothe phase differences between M2 and its shallow water harmonics show strong asymmetrytoward longer falling tide. There is also a larger phase lag between the five gauge locationsduring the falling tide. Computing cross-correlations between the gauges show that the overallphase difference 2.3 km upstream from the mouth is about 65 minutes. This can be explainedtheoretically by representing the tide by a diffusive process as it propagates upstream, with atime variable forcing at the mouth. A roughly 24-hour surge event occured during data collec-tion, which likely contaminated the resulting constituents from T_TIDE. This is made evidentby non-zero residuals in the entire signal. The unfiltered signals show higher low-tides duringspring phase than the low-tides during the neap phase. This can be interpreted as the effectof the compound fortnightly tidal constituent, MSf . Storm surge can contaminate this fort-nightly tide by changing the phase characteristics of MSf , which can possibly explain whythe fortnightly modulation wasn’t seen in the filtered time series.

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1 Introduction

As the tide propagates from the deep ocean into the more shallow waters of estuaries and saltmarshes commonly found on the U.S. East Coast, it undergoes considerable distortion. When thetide propagates into these estuarine environments, it is influenced by the shallow water (Dronkers1964). Several factors govern the distortion of the tide, one of them being non-linear interactionsbetween principal astronomical tides and their harmonics. In particular, the modulation of the M2

tide by its first harmonic,M4, can be very pronounced in these shallow water environments (Aubreyand Speer 1985; Speer and Aubrey 1985). Consequently, M2 and its quarter-diurnal harmonic M4

can be used to demonstrate the tidal asymmetries in estuaries. Furthermore, the astronomical tidegets distorted because its trough is retarded more than its crest due to notable differences in thewater depth between low and high tide, introducing different frictional effects (Dronkers 1964).

As the tide propagates into shallow regions, there is a transfer of energy from lower frequency,astronomical components to higher frequency components, such as the higher harmonics of M2.The behavior of the astronomical tide (semi-diurnal and diurnal constituents) through an estuarycan be used to quantify dissipation due to friction as well as the level of non-linear spectral energytransfer. Friction serves to reduce the amplitude of the tide, as well as introduce phase lags. Othernon-linear transfers reduce tidal amplitude while imparting characteristic relative phase changes(Aubrey and Speer 1985).

Tidal ConstituentsSpecies Symbol Period

Fortnightly MSf 14.0 daysDiurnal O1 25.8 hrs

K1 23.93 hrsSemi-Diurnal N2 12.66 hrs

M2 12.42 hrsS2 12.00 hrs

Ter-Diurnal MK3 8.18 hrsQuarter-Diurnal MN4 6.27 hrs

M4 6.21 hrsMS4 6.10 hrs

Sixth-Diurnal M6 4.14 hrsS6 4.00 hrs

Eight-Diurnal M8 3.11 hrs

Table 1: Governing tidal constituents for Delaware Bay

Table 1 lists the species of tide that are of particular interest in this study. The basis of theseconstituents dates back to the work done by Doodson (1921) who was able to derive the tidegenerating potential and use it to determine the frequencies of many components which make up

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the tide. His work was used in the original theory on harmonic analysis for prediction of tidalmotion and still serves as a fundamental basis today.

While traditional HA and tidal prediction treat the tide as a stationary process, tides in the realworld are inherently non-stationary, caused by several different mechanisms. One such mechanismis storm surge in shallow coastal regions and estuaries (Horsburgh and Wilson 2007), resulting infurther non-linear interactions. While wavelet transforms have traditionally been used to analyzenon-stationary signals (Flinchem and Jay 2000), new techniques of HA are being developed. Onesuch tool is NS_TIDE (Matte, et al. 2013), which is built to better handle non-stationarity in signalswhile doing HA. This is accomplished by directly building the forcing that causes the nonstationarytidal response into the basis functions employed in the analysis.

A fully coupled quasi-3D nearshore circulation model is currently being developed which willhopefully provide us with a better understanding about how these marsh systems evolve with time(see Shi, et al. 2003; Shi, et al. 2012; Svendson, et al 2004). While the details of the model areoutside the scope of this paper, it’s worth mentioning because a field experiment was conducted inBrockonbridge Marsh to gather data necessary for validating the model. A sufficient explanationof the methods used for data collection and analysis is provided below, followed by a thoroughdiscussion of the results of data analysis, and finally, some concluding remarks are made.

2 Field Study

2.1 Marsh Description

The Brockonbridge Marsh (BM) is located on the Delaware Bay coast of Kent County, DE, justsouth of the Murderkill River Estuary, which splits the coastal towns of Bowers and South Bowers.The marsh dynamics are primarily governed by the presence of a channel which essentially bisectsthe entire marsh (Figure 1). However, it was also noted during the experiment that wind seems toplay an important role in the propagation of the tide, especially in the more inland, low-lying areas.BM encompasses an area of roughly 5 km-sq.

The offshore tide is predominantly semi-diurnal with a range of about 2-m. The channel depthwas found to be highly variable throughout. The inlet is approximately 3-m deep during a higher-high tide; however, just offshore from the inlet is a very extensive shallow flat, composed of amud-sand mixture. Inland, the channel depth can vary anywhere from < 1-m deep to > 3-m deepduring low tide, depending on the location. It’s also worth noting that the depth certainly doesnot decrease inward from the bay as one would expect (Phase II of the field study will focus onmapping the channel bathymetry and sediment composition in more detail). In this study, thechannel is assumed to be well mixed, and freshwater input into the system is negligible.

The marsh platform elevation decreases inland, away from the bay. This creates a pondingeffect in the back end of the channel during higher-high tides and especially after storms (Figure2). The main channel, known as Brockonbridge Gut (BG), is roughly 20-m wide at its mouth, andsteadily decreasing in width inland. The channel narrows to about 5-m wide at the location of thefurthest inland pressure gauge. The BG is bordered on either side by small dykes, which have a

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Figure 1: Aerial view of Brockonbridge Gut in Kent County, DE with the channel mouth open toDelaware Bay in the upper-right (Courtesy of Google Earth)

Figure 2: Aerial photo of Brockonbridge Marsh after Hurricane Sandy passed by in early November2012 causing substantial storm surge and flooding, looking inland with Delaware Bay at the bottomof the image (Photo courtesty of Art Trembanis)

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higher elevation than the local surrounding marsh platform. As a result, the secondary and tertiarychannels in the marsh play an important role in the distribution of water, as well as the modulationof the tide as it propagates inland. The dykes are very distinct in Figure 2, as is the "pondingeffect" mentioned earlier. During higher-high tides, as well as during surge events, the water levelbecomes higher than the elevation of the berms. During these phases, the marsh platform becomesinundated with water and the role of the secondary and tertiary channels might play less of a rolein tidal distortion, while instead, friction takes over. The spilling of water over the dykes alsoserves to retard the maximum level the tide can reach, due to a sudden horizontal flow of waterover an extensive area. The effect of the secondary channel network on flow distribution can alsobe inferred from Figure 2. In order to more fully understand the role these channels might play indistorting the tide, a more in depth field experiment would need to be done. However, the resultsfrom this field study can still provide us with important insight into the harmonic interactionswith astronomical tides. The full data report is still in preparation, including measurements ofbathymetry and computations of discharge (Mieras and Kirby 2013).

2.2 Field Methods

Phase I of the data collection involved deploying a total of 7 instruments throughout the mainchannel (Figure 3). Among the instruments we depoloyed were five pressure gauges (PG), which

Figure 3: Layout of instrument locations in Brockonbridge Marsh

were recording at 3-minute intervals for a 14-day period. They are depicted in Figure 3 as red dots.The gauges were deployed for a 14-day period in order to encompass a full neap-spring cycle. A

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concerted effort was made to equally distribute them throughout the channel, as well as to coverthe largest extent of the channel as possible. As a result, the distance between gauge B (locatedapproximately at the mouth) and gauge F (located as far inland as was navigably possible with theoutboard motor during low-tide) are spaced at about 2.3-km from each other, along the channel.The gauges were attached to steel mesh plates and lowered to sit on the bottom of the channel. EachPG was surveyed for its (Easting,Northing) coordinates using a GPS device. Saltwater correctionswere made during post-processing. Finally, it’s worth pointing out that after the 14-day recordingperiod, the time discrepancy between computer time and instrument time was on the order of acouple seconds for each gauge, which is expected. However, two seconds over a period of 14-daysis negligible. The data analysis from these 5 PGs, sites B-F in Figure 3, will be the focus of the restof this paper.

3 Data Analysis

3.1 Initial Data Analysis

Figure 4 (lower panel) shows the unprocessed signal from each gauge. The surge event with strongwinds blowing primarily out of the northeast, as was observed during the stay at the field site,occurred between the dates of Mar 25-26. However, the gagues further up the channel seem tohave a delayed return back to normal tide levels. As was mentioned earlier, this is caused by aninability for the water to drain properly. This is due to "ponding" of water in the lower lying areasinland from the bay, caused by the negative slope of the marsh platform away from the bay. Acloser look at the top panel in Figure 4 reveals that tide levels are affected by the surge event muchless at the inlet to the Murderkill River (black dot in Figure 3). It appears to return to normal tidaloscillation much faster than at site F in the BG.

Furthermore, the site F signal in the bottom panel of Figure 4 shows when the water, which wasbeing pushed up into the channel, perhaps spills over the levees onto the flats on either side. Duringthe coincidence of the storm with high tide, the water level at site F is nearly 0.5-m higher thanat site B. This non-stationarity is due to strong friction caused by a large mass of water flowingpast vegetation at site F, while at the mouth, which has dykes of higher elevation keeping waterfrom spilling over, water is able to flow back out with the tide much more easily. An exaggeratedaccount of this phenomenon can be seen in the aerial photo taken post-Hurricane Sandy in Figure2.

3.2 Harmonic Analysis

Harmonic analysis dates back to work done by Doodson (1921), which provided an understandingof the tide generating potential. Knowing the frequencies of each component making up the tideallowed for much more accurate prediction of tide levels. Godin (1972) provided a slight reformu-lation of Doodson’s derivation, serving as a basis for traditional tidal anaysis methods, includingthe modern formulations used in the Matlab code of T_TIDE (Pawlowicz, et al. 2002). While

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Figure 4: Top Panel - predicted tidal extremes (red) and measured tidal elevation (black), withmean subtracted out, at the USGS gauge station (black dot in figure 3; Middle Panel - wind speedand direction plotted over time from a DEOS weather station about 5 miles north of BM (the stationwas not functional during storm, Mar 25-26); Lower Panel - gauge signals at sites B-F, blue closestto mouth and magenta furthest inland.

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T_TIDE doesn’t explicitly include non-stationary forcing in its basis functions, robust estimatorswere added to T_TIDE via iteravely reweighted least squares (Leffler and Jay 2009). This servesto reduce non-tidal variation on the overall fit by downweighting outliers.

HA is useful because of its representation of the tide by a line spectrum consisting of a finitenumber of infinitely narrow peaks at fixed, predetermined frequencies, each with a componentamplitude and phase (Flinchem and Jay 2000). The tidal signal can be reconstructed by summingeach constituent together

f(t) =

N∑n=1

cncos(ωnt− εn) (1)

where ωn is the predetermined frequency of the nth constituent, while cn and εn are the amplitudeand phase of the nth component, respectively. The cn’s and εn’s are determined in T_TIDE byusing the well known least-square error analysis,

ε2 =1

T

T∫0

(f − fn)2dt (2)

There are several benefits of using the least-squares method in HA: (1) the method can be ap-plied to any length time series; (2) unlike Fourier Analysis, it is not dependent on the samplinginterval and there are no restrictions on resolution; (3) due to the fact that each constituent fre-quency is predetermined, rather than being an integer multiple of a fundamental frequency, there isno aliasing (Aubrey and Speer 1985).

Because of these useful features, the Matlab code of T_TIDE was employed in the process-ing of the data from all five pressure gagues. However, attention should be paid to the possiblecontamination of the data by the surge event. This will be discussed in the next section.

4 Results & Discussion

There is a pronounced decay in the amplitude of the high tide as the wave propagates into thechannel; however, the low tide at each gauge location drops to about the same level (Figure 5). Thisis a consequence of the negative slope of the marsh platform away from the bay. The water canonly rise to the level of the platform in the channel before it spills out horizontally, at which pointits vertical rise is severely retarded. This is also a consequence of the decay of the semi-diurnaltide through the estuary from a non-linear response of tidal forcing. The lag in tidal propagation isconsiderably larger during falling tide, again due to frictional effects being more pronounced. As aresult, the falling tide is of longer duration than the rising tide.

The rest of this section is separated into three parts: first we will discuss the possible contam-ination of harmonic analysis due to a surge event present in the record, introducing a highly non-stationary response. Cross-correlations between certain gauge pairs will then be used to determinethe lag in the sea surface response with increasing distance upstream. Finally we will investigate

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03/25 03/26 03/26 03/26 03/26 03/26−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

η (

m)

SITE B

SITE C

SITE D

SITE E

SITE F

Figure 5: Sample of two tidal periods of filtered data after harmonic analysis using T_TIDE

the ratio of M2 and its first harmonic, M4, as well as the generation of compound tides and forcedlow frequency constituents (MSf for example), to demonstrate non-linearities which arise as thetide propagates upstream. In addition, the role of friction will be discussed by examining both thedecay of the astronomical tide and sea surface phase adjustment, M2 - M4.

4.1 Contamination due to presence of non-stationary signal

As mentioned before, T_TIDE isn’t built to handle highly non-stationary signals. The residualsshow T_TIDE’s inability to handle the surge event (Figure 6). The lower panel shows the tidalresiduals for each gauge. One would expect the residuals to be nearly zero if the harmonic anal-ysis was accurate in reproducing the original signal. But this expectation is really only valid forstationary signals due to the fundamental stationarity assumption in traditional HA.

It was expected to see several peaks in the residuals during the surge, due to the surge eventexisting in the original record, but as the lower panel of Figure 6 shows, the residuals on either sideof the storm oscillate about a mean value. This suggests that there is contamination of the outputof HA due to the surge event. To circumvent this problem, more advanced HA techniques that canhandle non-stationary signals, like that of Matte, et al. (2013), should be used instead. However,the assumption from here on is made that T_TIDE’s results can be reasonably used to demonstratethe non-linear and frictional nature of the Brockonbridge Marsh. We will readdress this assumption

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03/21 03/23 03/25 03/27 03/29 03/31 04/02−2

0

2

4

6

8

Original tidal signals

B

C

D

E

F

03/21 03/23 03/25 03/27 03/29 03/31 04/02−2

0

2

4

6

8

su

rfa

ce

dis

pla

ce

me

nt

(m) Tidal signals AFTER T__TIDE harmonic analysis

03/21 03/23 03/25 03/27 03/29 03/31 04/02−1

0

1

2

3

4

5

Tidal residuals

Figure 6: Top Panel - Original pressure gauge signals, unfiltered; Middle Panel - Signals afterharmonic analysis in T_TIDE; Bottom Panel - Tidal residuals (original signals less the harmoniccomponents). *Signals offset vertically for clarity

at the end of the results and discussion section.The field experiment conducted by Aubrey and Speer (1985) also had a surge event occur

during data collection, which induced a 0.75-m super-elevation within the estuary, with a decaytime of approximately one week. They point out that the energetic event swamped the non-linearforced component of MSf , such that the storm surge affected estimates of MSf . This fortnightlytide is created mostly by interactions between the M2 and S2 semi-diurnal components (Dronkers1964). The same phenomenon is seen in the lower panel of Figure 6, as well as in the top panel inFigure 4. In both cases, the mean of the entire signal seems to shift to a higher value after the surgeevent, decreasing slowly over the following weeek.

Furthermore, the amplitude of the MSf component eventually surpasses the amplitudes of theastronomical components, leading to higher low-tides during spring than during the neap phase(LeBlond 1979). During the 14-day deployment, spring phase occured in the second week ofdeployment, which is when lower low-tides are expected; however, both the top and lower panelsin Figure 6 show a higher low-tide during this period.

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4.2 Cross-correlations, determining lag times

During the time spent in the field, we immediately noticed an obvious lag between the predictedtime for high tide at the mouth and the arrival of high tide 1.5-km upstream (also, see Figure 5).To quantify this observation, cross-correlations between select gauge pairs were computed. Thelag times that gave the highest correlation for each pair are representative of the amount of time ittakes the tide to propagate between the two gauges (Figure 7).

Figure 7: Cross-correlations between select gauge pairs, plotted against lag times; figure in lowerright zooms in on highest correlation values, marked by stars. cBC in legend signifies the cross-correlation of gauge C with gauge B, for example.

Despite the highly variable depth of BG, as well as its many meanders, there appears to bea linear relationship between lag time and distance along the channel (Figure 8). Using the wellknown expression for phase speed of a linear propressive shallow water wave (Dean and Dalrymple1991),

C =√gh (3)

and inserting a characteristic depth of the BG, we can compare this value to that of the ratio betweenthe distance along channel to the lag time to propagate between two points.

In doing so, it was determined that these lags cannot be fully explained by a progressive natureof the wave. The following section describes in detail what causes this lag in phase further inland.

4.3 Decay of semi-diurnal constituents, associated with large phase lags

Tidal motions in channels in shallow estuarine environments can be described by a simple first-order differential equation with one wave, which diffuses inland from time variable forcing atthe mouth (LeBlond 1978). The dynamic balance is between surface slope and friction. The

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0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

distance from SITE B, along channel (km)

lag

tim

e,

τ (

min

ute

s)

Figure 8: Lag time between gauge stations based on highest correlation between signals

frictional nature of Brockonbridge Marsh is demonstrated by a substantial decay of the semi-diurnalcomponents, as well as large accompanying phase lags (Dronkers 1964; Speer and Aubrey 1985)(Figures 9 and 10).

Speer and Aubrey (1985) use a 1-D numerical model to show that more time-variable geometryand strong friction produce largerM4/M2 ratios. This corresponds with the results of the field study(lower panel in Figure 10). The quarter-diurnal harmonic has significant influence on the tide inthe Brockonbridge Marsh.

The phase relationship between M2 and M4 (top panel in Figure 10) demonstrates a fallingtide whose duration exceeds that of the rising tide, with the asymmetry increasing further inland.This is apparent by a phase lag between 90 and 180 degrees. By revisiting Figure 5, we definitelysee the existence of a longer falling tide than rising. It is also apparent, as mentioned earlier, thatthe assymetry is much more pronounced during the falling tide. This result is consistent with thefindings in the field study by Aubrey and Speer (1985), whose estuary is characterized by similargeometrical parameters to Brockonbridge Marsh. Therefore, it’s not surprising to see a similarresult to their study.

The ratio M4/M2 in the lower panel of Figure 10 doesn’t show any sort of increase into thechannel until almost 2-km inland. This tells us that the quarter-diurnal harmonic, M4, is actuallydecaying faster than the semi-diurnal constituent, M2, in the bayward regime. We can clearly seefrom Figure 9 that the M2 tide decays at each gauge location in the channel. So we can at leastsay there is a decay of the astronomical, semi-diurnal tide through the channel. However, basedon the lower panel of Figure 10, we cannot definitively say there is an accompanying growth of itsharmonics.

It’s worth pointing out a couple of factors that might cause the above result to be slightly

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

1

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Distance

from S

ITE B

(km)

M8 3MK7M6

2SK5

2MK5

M4

Frequency (cph)

M3

M2

K1 O1

Am

plit

ud

e (

m)

Figure 9: Amplitudes of each constituent as they vary with distance from site B, determined fromHA in T_TIDE

0 0.5 1 1.5 2 2.5

100

120

140

160

180

M2 −

M4 p

has

e (d

egre

es)

0 0.5 1 1.5 2 2.50.13

0.14

0.15

0.16

0.17

Distance from SITE B (km)

M4/M

2

Figure 10: Summary of the tidal response in Brockonbridge Marsh. Top Panel - sea surface phaseadjustment, M2 - M4; Lower Panel - ratio of sea surface amplitudes M4/M2.

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skewed. First, the spacing between the gauges is very large. We cannot accurately resolve anythingin between the five data points. In reality, the ratio is likely more variable with distance. Second,and more importantly, the assumption was made in the beginning of the analysis that it is reasonableto neglect the non-stationarity of the signal and accept the output from T_TIDE, knowing this isn’tnecessarily correct. As a consequence, spectral energy bias can be incorrectly introduced to favorconstituents with lower frequency, reducing the ability to see the growth of higher order harmonics.

5 Conclusions

Traditional harmonic analysis serves as a good predictor for tidal levels. Although, we know thattidal records in typical fluvial/estuarine environments are inherently non-stationary, in which cases,traditional harmonic analysis doesn’t seem to work as well.

The amplitude of the astronomical semi-diurnal tide M2 decays with distance upstream in thechannel. This is due to both friction and non-linear transfer of energy from the semi-diurnal tide toits higher harmonics. Future bathymetry measurements could possibly provide more insight intothe relationship between tidal energy dissipation and channel depth.

The associated large increase in phase lag, particularly for the M2 constituent, further showsthe frictional nature of the marsh. Consequently, tidal flow in Brockonbridge Marsh can be char-acterized as dominated by bed friction, as evidenced by the phase difference between M4 and M2.Additionally, the magnitude of this phase lag is roughly between 100 and 150 degrees. This leadsto a longer falling tide, with the asymmetry increasing upstream. Such pronounced asymmetrymay have important implications on sediment transport throughout the marsh, which governs itsstability over time.

Estimates of the forced fortnightly tide MSf can be severely affected by storm surge events.However, as described by LeBlond (1979), this fortnightly component can sometimes cause higherlow-tides during spring phase, compared with neap. While the cause of the observed increase inmean water level accompanied by a steady decay over the week following the storm can’t be knownexactly, it can be reasonably justified by either explanation, if not by both.

6 References

Aubrey, D. G. & Speer, P. E. (1985), A study of non-linear tidal propagation in shallow in-let/estuarine systems Part I: Observations. Estuarine, Coastal and Shelf Science, 21, 185-205.

Dean, R. G. & Dalrymple, R. A. (1991), Water Wave Mechanics for Engineers and Scientists.World Scientific Publishing Company Incorporated, 353 pp.

Doodson, A. T., (1921), The Harmonic Development of the Tide Generating Potential. Proc. Roy.Soc. London, 100A, 305-329.

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Dronkers, J. (1964), Tidal Computations in Rivers and Coastal Waters. North-Holland PublishingCompany, Amsterdam, 516 pp.

Flinchem, E. P., & Jay, D. A. (2000), An Introduction to Wavelet Transform Tidal Analysis Meth-ods. Estuarine, Coastal and Shelf Science, 51, 177-200.

Godin, G. (1972). The Analysis of Tides. University of Toronto Press, 264 pp.

Horsburgh, K. J. & Wilson, C. (2007), Tide-surge Interaction and its Role in the Distribution ofSurge Residuals in the North Sea. J. Geophys. Res., 112, C08003, doi:10.1029/2006JC004033.

LeBlond, P. H., (1978), On Tidal Propagation in Shallow Rivers. J. Geophys. Res., 83, (C9),4717-4721.

LeBlond, P. H., (1979), Forced Fortnightly Tides in Shallow Rivers. Atmos. Ocean, 17, 253-264.

Leffler, K. E., and Jay, D. A. (2009). Enhancing Tidal Harmonic Analysis: Robust (hybrid L1/L2)Solutions. Cont. Shelf Res., 29, 78-88.

Matte, P., Jay, D. A., & Zaron, E. D. (2013) Adaptation of Classical Tidal Harmonic Analysis toNonstationary Tides, with Application to River Tides American Meteorological Society, 30,569-589.

Mieras, R. and Kirby, J. T., 2013, "Discharge, pressure and bathymetry measurements in Brock-onbridge Gut, Kent County, Delaware, March-August 2013: Data report", Research ReportCACR-13-XX, Center for Applied Coastal Research, in preparation.

Shi, F., Kirby, J. T., Hsu, T. J., Chen, J. and Mieras, R., (2012), A Hybrid TVD Sover for NearshoreCommunity Model Documentation and User’s Manual, Center for Applied Coastal ResearchReport, CACR 2012-03, University of Delaware, Newark, Delaware.

Shi, F., Svendsen, I. A., Kirby, J. T. and Smith, J. M., (2003), A curvilinear version of a quasi-3Dnearshore circulation model, Coastal Engineering, 49, 99-124.

Svendsen, I. A., Haas, K. A. and Zhao, Q., (2004), Quasi-3D Nearshore Circulation modelSHORE-CIRC: Version 2.0, Research Report, Center for Applied Coastal Research, Uni-versity of Delaware.

Pawlowicz, R., Beardsley, B., & Lentz, S. (2002), Classical Tidal Harmonic Analysis IncludingError Estimates in MATLAB using T_TIDE. Comput. Geosci., 28, 929-937.

Speer, P. E. & Aubrey, D. G. (1985), A study of non-linear tidal propagation in shallow in-let/estuarine systems Part II: Theory. Estuarine, Coastal and Shelf Science, 21, 207-224.

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