Computers & Geosciences 44 (2012) 196–202
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Simplified empirical astronomical tide model—An application for the Rı´o de la Plata estuary Enrique D’Onofrio a,b,n, Fernando Oreiro a,b, Mo´nica Fiore a,b a b
´noma de Buenos Aires, Argentina (C1127AAR). Instituto de Geodesia y Geofı´sica Aplicadas, Facultad de Ingenierı´a, Universidad de Buenos Aires. Av. Las Heras 2214, Ciudad Auto ´noma de Buenos Aires, Argentina (C1270ABV). Departamento Oceanografı´a, Servicio de Hidrografı´a Naval, Ministerio de Defensa. Montes de Oca 2124. Ciudad Auto
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 August 2011 Received in revised form 23 September 2011 Accepted 23 September 2011 Available online 25 October 2011
The Rı´o de la Plata has a complex astronomical tide due to the effect of the shallow depth of this extensive estuary, the complicated geometry and bathymetry and the huge discharge of the rivers Parana´ and Uruguay and therefore the simulation of the astronomical tide is complicated. This paper presents a Simplified Empirical Astronomical Tide model (SEAT) which overcomes the foreseen difficulties in a straightforward way. The program developed can be applied to other regions by only changing a file of the dynamic-link library. SEAT provides predictions, harmonic constants and the mean water level referred to the tidal datum used to calculate the prediction. The storage of information in the form of images reduces the size of the application. The equations considered for the prediction do not require the use of special computer processors. A personal computer with minimum hardware that supports Microsoft Framework 3.5 is suitable to run SEAT. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Tide Harmonic prediction Empirical model Rı´o de la Plata
1. Introduction The Rı´o de la Plata, formed by the confluence of the Parana´ and Uruguay rivers, is one of the world’s largest estuaries (Fig. 1). It is located on the eastern coast of South America at about 351S and it is the entrance of the Parana´-Paraguay waterway, an important transportation artery that links five countries (Bolivia, Brazil, Uruguay, Paraguay and Argentina). Communication and transportation along this rivers system are critical issues for the development of the countries involved, in an area of about 700,000 km2 with a population close to 45,000,000 inhabitants (Del Carril, 2008). The estuarine system of approximately 35,000 km2 with only 5–15 m water depth (Guerrero et al., 1997), has a width of 2 km at the inner part which increases to 220 km at the mouth. The estuary can be divided into three regions: Inner, middle and exterior (Comisio´n Administradora del Rı´o de la Plata, 1989; D’Onofrio et al., 1999; Dragani and Romero, 2004). The Inner region extends from the confluence of Parana´ and Uruguay rivers to an imaginary straight line joining Buenos Aires and Colonia cities. The middle region spans from the Buenos Aires–Colonia line to another imaginary line connecting Punta Piedras with
n Corresponding author at: Departamento Oceanografı´a, Servicio de Hidrografı´a Naval, Ministerio de Defensa. Montes de Oca 2124. Ciudad Auto´noma de Buenos Aires, Argentina (C1270ABV). Tel./fax: þ54 114301 3091. E-mail addresses:
[email protected],
[email protected] (E. D’Onofrio).
0098-3004/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2011.09.019
Montevideo and the Exterior area extends from the Punta Piedras–Montevideo line to the mouth of the river (Fig. 1). The hydrological regime of the Rı´o de la Plata is highly influenced by the tidal wave coming from the Atlantic Ocean. The circulation is sensitive to the complicated geometry and bathymetry of the estuary (Simionato et al., 2004a). The main semidiurnal constituent (M2) takes about 12 h to reach the interior limit, so a full cycle of the M2 component is present at every moment inside the Rı´o de la Plata (Simionato et al., 2004b; D’Onofrio et al., 2009). The astronomical tide in the Rı´o de la Plata is mixed mainly semidiurnal type (SHN, 2011). The range of the astronomical tide is about 1.44 m at the mouth, but in the interior areas it can reach 0.40 m (D’Onofrio et al., 2009). The knowledge of the astronomical tide at each point of the estuary becomes an important issue for navigation safety and the planning of different activities related to the level of the River. As an example, the river is the most important source of drinking water for Buenos Aires city, which population reaches almost 2,900,000 inhabitants (INDEC, 2011). An alternative way to obtain the astronomical tide at any point of the Rı´o de la Plata is using harmonic constants from global models based on satellite data. Four of the most widely used global tidal models are the Finite Element Solution Tide Model (FES2004), the Goddard Ocean Tide Model (GOT4.7), the Inverse Tide Model (TPXO7.2) and the Empirical Ocean Tide Model (EOT08a). FES2004 is a fully revised version of the global hydrodynamic tide solutions based on the solution of the tidal barotropic equations. The accuracy of these solutions was improved by assimilating tide gage and altimetry data (Lyard et al., 2006).
E. D’Onofrio et al. / Computers & Geosciences 44 (2012) 196–202
Fig. 1. Geographical location of Rio de la Plata estuary.
The model provides fifteen tidal constituents distributed on 1/81 grids. GOT4.7 is the last update of the GOT99.2 (Ray, 1999); this model is based on six years of sea-surface height measurements performed by the TOPEX/POSEIDON satellite altimeter. The model provides ten tidal constituents distributed on 1/21 grids. EOT08a is an empirical ocean tide model from multi-mission satellite altimetry. The model provides ten tidal constituents are distributed on 1/81 grids. EOT08a uses empirical co-tidal and co-range mapping to incorporate tidal constants (Savcenko and Bosch, 2008). TPXO7.2, the update of TPXO7.1 (Egbert and Erofeeva, 2002), assimilates different cycles of satellite missions to improve its accuracy. The model provides eleven tidal constituents distributed on 1/41 grids. All these models have achieved an accuracy of 70.2–0.3 m in deep oceans (Shum et al., 1997, 2001), but they are much less accurate in coastal seas (Shum et al., 2001, Padman et al., 2008) with errors of up to 1 m (http://amcg.ese.ic.ac.uk/ index.php?title=Local:Global_Tidal_Models). Relative errors (%), for the M2 constituent, between observed and modeled amplitudes, in the Rı´o de la Plata, were calculated by Dragani et al. (2010) finding values up to 61.4%, 53.4% and 11.5% for FES2004, TPXO7.2 and GOT4.7, respectively. For O1 constituent the values found were up to 60.6%, 50.2% and 15.6% for GOT4.7, TPXO7.2 and FES2004 respectively. Greenwich epoch differences (in hours), for the M2 constituent, between observed and modeled were calculated by Dragani et al. (2010), in the Rı´o de la Plata, obtaining values up to 3:53 h, 3:44 h and 1:70 h, for FES2004, TPXO7.2 and GOT4.7, respectively. For O1 constituent the values found were up to 5:47 h, 1:30 h and 0:75 h for FES2004, TPXO7.2 and GOT4.7, respectively. Due to these results it is not advisable to use harmonic constants from global models to predict tides in the Rı´o de la Plata. In academic circles, regional models supported by hydrodynamic equations have been applied to the Rı´o de la Plata. They do not only provide the astronomical tide, but also calculate the storm surge and tidal currents. Simionato et al. (2004a,2004b) implemented the three-dimensional primitive equation regional model HamSOM (Hamburg Shelf Ocean Model, Backhaus, 1983, 1985) with a resolution of 3 km. Palma et al. (2004) implemented Princeton Ocean Model (Blumberg and Mellor, 1987), with a curvilinear grid that spans from 551S to 181S and from 701W to 401W, and with a variable horizontal resolution that goes from 5 km near the coast to 20 km near the open boundary. Patagonian Shelf 2010-OSU tidal data inversion regional model developed for the southwest Atlantic Ocean also gives harmonic constants for the Rı´o de la Plata estuary (http://volkov.oce.orst.edu/tides/PatS. html). It has a resolution of 1/301 and assimilates data from satellite altimeters and tide gauges. These models have not been
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developed in user friendly software and can’t be used without previous knowledge, but the model outputs could be used for tidal prediction. The Patagonian Shelf 2010 regional model output can be used by the Tide Model Driver (TMD) to get harmonic constant and calculate tidal prediction (http://polaris.esr.org/ ptm_index.html). All these outputs could also be used by SEAT program to get harmonic constant and tidal prediction. Most users consult the Tide Tables to obtain the astronomical tides. Nowadays the Tide Tables for the Rı´o de la Plata (SHN, 2011) provide eight predictions for coastal locations and two predictions in the middle of the estuary. This information is not enough to satisfy the needs of some users. The objective of this work is to present the development of a high resolution Simplified Empirical Astronomical Tide Model (SEAT) applied to the Rı´o de la Plata, with free access to all users of tide tables. The SEAT is a program that can be run in any computer with Microsoft Framework 3.5, and the size of the executable file is of only 126 kB. To obtain the tide predictions, a dynamic-link library, 5.94 MB large, must be loaded. The SEAT model uses the amplitudes and epochs of the main tide constituents which have been calculated using observed water levels obtained with conventional tide gages. Results from SEAT are more accurate in height and time than those provided by the above mentioned global models. SEAT also provides an unlimited choice of locations. This is a great advantage compared to the ten predictions provided by the Tide Tables (SHN, 2011).
2. Model description Tides can be represented as the sum of waves, which amplitude and epoch are known as the tidal harmonic constituents. These constituents are calculated by applying harmonic analysis to observations of water level. Tidal height at any time can be written as a function of harmonic constituents as (Schureman, 1988): hðtÞ ¼ Z 0 þ
n X
Hi f i cosðsi t þðV 0 þ uÞi K 0i Þ
ð1Þ
i¼1
where hðtÞ is the height of the tide at any time t; Z 0 is the mean height of the water level above datum used for prediction; Hi is the amplitude of its constituent; f i is the nodal factor; si is the speed of the constituent; ðV0 þ uÞi is the value of the equilibrium argument of the constituent when t¼0; K 0i is the epoch of the constituent; n is the number of constituents The empirical model developed herein, employs Eq. (1) to calculate harmonic tidal prediction at the nodes of a grid defined in the domain of the model. For each node the mean height of the water level and the amplitude and epoch of each constituent must be known. SEAT calculates the equilibrium arguments and the nodal factors for the selected date. The model assigns the predictions of tide corresponding to the nearest node of the location chosen by the user. This fact is taken into account for the calculation of the grid resolution. Then the maximum error of prediction will be half the difference of the predicted height between adjacent nodes. In order to apply the model to another region, the code is designed to use any combination of the 123 constituents shown in Table 1. To store lightweight files, the harmonic constituents and mean water level data are saved in Portable Network Graphics (PNG) format, which uses a lossless data compression algorithm. The color space Red, Green, Blue (RGB) of each one of the pixels is used to store the information corresponding to every node. Another image, in the same format, is used to identify those points where tide predictions can be computed. In this case the value of the Red component of each pixel is 0 and the value of the Green component stores the number of constituents involved in
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Table 1 Angular Speed of the 123 tidal constituents that can be used by SEAT model. Constituent
Speed (1/h)
Constituent
Speed (1/h)
Constituent
Speed (1/h)
SA SSA MM MSF MF 2Q1 SIGMA1 Q1 RHO1 O1 MP1 M1 CHI1 PI1 P1 S1 K1 PSI1 PHI1 THETA1 J1 2PO1 SO1 OO1 2NS2 2NK2S2 OQ2 MNS2 MNK2S2 2MS2K2 2N2 MU2 N2 NU2 2KN2S2 OP2 M2 MKS2 M2(KS)2 2SN(MK)2 LAMBDA2
0.0410666776 0.0821372786 0.5443746958 1.0158957627 1.0980330413 12.8542862065 12.9271398353 13.3986609022 13.4715145311 13.9430355980 14.0251728766 14.4920521000 14.5695476000 14.9178646831 14.9589313607 15.0000000000 15.0410686393 15.0821353000 15.1232059180 15.5125897000 15.5854433351 15.9748271234 16.0569644020 16.1391016806 26.8794590830 26.9615963616 27.3416964000 27.4238337789 27.5059710574 27.8039339174 27.8953548458 27.9682084746 28.4397295415 28.5125831704 28.6040040987 28.9019669587 28.9841042373 29.0662415160 29.1483787945 29.3734880256 29.4556253042
L2 T2 S2 R2 K2 MSN2 KJ2 2KM(SN)2 2SM2 SKM2 NO3 MO3 M3 SO3 MK3 2MQ3 SK3 N4 3MS4 MN4 MNKS4 M4 SN4 KN4 MS4 MK4 SL4 S4 SK4 MNO5 2MO5 3MP5 MNK5 2MP5 2MK5 MSK5 3NKS6 2NM6 2NMKS6 2MN6 2MNKS6
29.5284789331 29.9589333224 30.0000000000 30.0410667000 30.0821372786 30.5443747000 30.6265119744 30.7086492530 31.0158957627 31.0980330413 42.3827651395 42.9271398353 43.4761563560 43.9430355980 44.0251728766 44.5695475724 45.0410686393 56.8794590830 56.9523127119 57.4238337788 57.5059710574 57.9682084746 58.4397295415 58.5218668201 58.9841042373 59.0662415160 59.5284789331 60.0000000000 60.0821372786 71.3668693768 71.9112440726 71.9933813512 72.4649024181 72.9271398353 73.0092771139 74.0251728766 85.4013259031 85.8635633203 85.9457005989 86.4079380162 86.4900752947
M6 MSN6 MKN6 2MS6 2MK6 2SN6 NSK6 2SM6 MSK6 S6 3MO7 2NMK7 2MNK7 2MSO7 2(MN)8 3MN8 3MNKS8 M8 2MSN8 2MNK8 3MS8 3MK8 MSNK8 2(MS)8 2MSK8 2M2NK9 3MNK9 4MK9 3MSK9 4MN10 M10 3MNS10 4MS10 2MNSK10 3M2S10 4MSK11 M12 4MNS12 5MS12 3MNKS12 4M2S12
86.9523127120 87.4238337789 87.5059710574 87.9682084746 88.0503457533 88.4397295415 88.5218668201 88.9841042373 89.0662415160 90.0000000000 100.8953483100 100.9046319596 101.4490066554 101.9112440726 114.8476675576 115.3920422534 115.4741795320 115.9364169493 116.4079380161 116.4900752947 116.9523127119 117.0344499905 117.5059710574 117.9682084746 118.0503457532 129.8887361969 130.4331108927 130.9774855885 131.9933813512 144.3761464907 144.9205211000 145.3920422534 145.9364169492 146.4900752947 146.9523127119 160.9774855885 173.9046254000 174.3761464907 174.9205211865 175.4741795320 175.9364169492
the prediction. When the value of the Red component is other than 0, no data are available for the calculation. Then, according to the stored value, the software can identify islands, continent or points which are located outside the computational domain. In this image the value of the Blue component is not used. The harmonic constituent amplitude and the mean water level have the same storage system on each node. Given that in this case values (in millimeters) can exceed the limit of 255 (maximum value of each RGB component), then they are previously converted to base 256 number. The values of the amplitude and the mean water level can then be reconstructed as follows: 2
Amplitudeðor mean water levelÞ ¼ Red256 þ Green256þ Blue ð2Þ The values of the epochs, with an accuracy of 10 4 degree, can be computed using Red Red Red 100 100 100 þ Blue ð3Þ þ Epoch ¼ Green2þ 10000 100 In this algorithm, the Green component stores the integer epoch value divided by two in each node. Then, the image obtained using only this component is an approximation of the constituent cotidal chart, allowing the detection of errors in the stored epochs. In Eq. (3), the hundreds of the Red component are 0 or 1 when the integer part of the epoch (in degrees) is even or odd,
Table 2 Three examples of storage of epochs using the color space Red, Green, Blue. Epoch
32410.6877
231
12510.8926
Green Red Blue
162 068 077
11 100 000
62 189 026
respectively. The tens and units of the Red component represent the first 2 decimals of the epoch. The tens and units of the Blue component store the third and fourth decimals of the epoch. Table 2 shows 3 examples of storage of epochs. The SEAT, developed with Microsoft Visual Basic.NET, solves the prediction equation using the stored information of the images from a dynamic-link library (dll) file. This file also contains the computational domain limits, the number of the constituents, the size of the grid, the time range available for the calculation and a picture of the area to help the user on the identification of the location (source: Google maps, http://maps. google.com.ar/?ie=UTF8&ll=-35.451721,-56.590576&spn=3.5125 1.8.453979&t=k&z=8). The reason for choosing this kind of file is that it also permits the inclusion of coordinate transformation subroutines. These subroutines are needed because whereas the user enters the selected location in geographic coordinates, the
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information contained in the images may be associated with planar coordinates. To run the SEAT, as a first step, the dynamic-link library must be loaded. Then, the user must provide the date and geographical coordinates of the chosen location. The user can also provide an offset value to modify the tidal datum and/or to add a height to consider a storm surge. SEAT provides the graphical output of the predicted tides for the chosen day, the high and low water values, heights every 15 min with and without offset and the harmonic constants used for the calculation. This information can be saved in a text file and/or can be print.
3. An application to the Rı´o de la Plata estuary The tidal constituents considered by the SEAT application to the Rı´o de la Plata are Q1, K1, O1, P1, S1, M2, S2, N2, K2, L2, M4 and SA. These components were chosen from D’Onofrio et al. (1999). The root-mean-square deviation (RMSD) calculated as the differences between predicted height using the SEAT constituent and those predicted using all constituents obtained by D’Onofrio et al. (1999), was less than 0.03 m in average. To obtain differences less than 0.01 m between two neighbor nodes for the calculation of tide prediction, a grid of 500 m 500 m resolution and 126,584 nodes was chosen. The boundaries of this application are the coastlines and the exterior limit of the Rı´o de la Plata defined by the line between Punta Rasa (Argentina) and Punta del Este (Uruguay). The coastline was obtained by digitalization of the nautical charts H-113 and H-116 of the Naval Hydrographic Service of Argentina (SHN, 1999a,1999b). In order to simplify the calculations, the results are transformed from geographic
199
coordinates to planar Gauss Kruger. The model uses the digitalized coastline (including islands) to identify the locations where predictions can be made. The amplitudes and phases of the twelve tidal constituents were calculated for each node of the grid as described in 3.2. To simplify the use of the model onboard, a mean level referred to the chart datum, established by the nautical charts H-113 and H-116 (SHN, 1999a,1999b) is assigned to each node. This information (3,164,600 data) and the geographical limits of the domain are stored in a dynamic-link library file. The prediction period is limited between 1950 and 2050, to avoid possible variations in harmonic constants that can cause errors in the calculation. In this dynamic estuary a potential increase in mean level and/or changes in the bathymetry might produce a variation in the harmonic constituents (D’Onofrio et al., 2010).
3.1. Data Table 3 shows the latitude, longitude, type of device and length of the time series where water level has been measured. Geographical position of locations in the table can be found in Fig. 2 by means of their associated indexes. The locations shaded in gray in Table 3 correspond to permanent tidal stations which belong to Naval Hydrographic Service or Hidrovia S.A. Tidal records were measured at standard tidal stations in nine locations, where water levels were gathered by a basic tide gage with a floater and counterweight inside a stilling well (UNESCO, 1985, 2002). Six pressure sensors are placed on piles in the river to measure hydrostatic pressure of the water column and convert the pressure into a level. Nine self-contained mooring pressure
Table 3 Location of water level gage, duration (hours) of the series analyzed to obtain harmonic constants and type of device. Indexes in brackets refer locations to Fig. 2. In type of device the following abbreviations are used: PS (Pressure Sensor), SCPS (Self-contained Pressure sensor), TP (Tide Pole) and F (Floater). Location (index) Nueva Palmira (1) Guazu Desembocadura (2) Carmelo (3) Isla Martin Garcı´a (4) Conchillas (5) Canal Emilio Mitre—Braga (6) Baliza Diamante (7) San Fernando (8) Muelle Platero (9) Colonia del Sacramento (10) Olivos (11) Buenos Aires (12) Torre Norden (13) La Paloma (14) Banco Ortiz (15) Bernal (16) Banco Jesu´s Marı´a (17) La Plata (18) Montevideo (19) Magdalena (20) Isla de Flores (21) Punta del Este (22) Canal Punta Indio (239.1 km) (23) Torre Oyarvide (24) Canal Punta Indio (201.6 km) (25) Spar Brasileira (26) Punta Indio (Costa) (27) Rı´o de la Plata Exterior I (28) Punta Piedras (29) Samborombom I (30) Rio de la Plata Exterior II (31) Samborombom II (32) San Clemente del Tuyu (33) General Lavalle (34)
Latitude 331 341 341 341 341 341 341 341 341 341 341 341 341 341 341 341 341 341 341 341 321 341 351 351 351 351 351 351 351 351 351 361 361 361
0
Longitude 00
52 30 000 0000 000 0000 110 0000 120 3000 190 0000 250 3600 260 0000 260 3000 280 3000 300 0000 330 4500 380 0000 390 2000 400 0000 400 5500 410 0000 500 0000 540 3000 560 0000 150 3000 570 4200 050 0000 060 0000 060 0000 100 3000 160 0000 180 1600 270 0000 430 3400 540 2300 090 1300 210 3000 240 0000
581 581 581 581 581 581 571 581 571 571 581 581 571 541 571 581 561 571 561 571 581 541 551 571 571 561 571 551 571 561 561 571 561 561
0
00
25 00 250 0000 180 0000 150 0000 040 3000 300 0000 570 2400 300 0000 380 3000 510 0000 290 0000 240 0000 550 0000 080 5000 100 0000 130 3300 540 0000 530 0000 130 3000 270 0000 060 0000 570 0600 590 0000 080 0000 080 0000 370 0000 130 1500 560 3200 080 2000 470 4600 170 4000 010 2700 420 3000 570 0000
Hours
Type of device
785 8760 785 8784 791 40,504 18,130 8109 5831 11,639 4415 166,559 17,519 17,543 793 2343 785 35,063 8760 17,134 1378 8760 21,281 105,936 8784 8760 4718 760 617 1948 760 781 19,007 8760
TP TP TP F TP PS PS TP TP F TP F PS F SCPS SCPS SCPS F F PS SCPS F PS F SCPS PS TP SCPS TP SCPS SCPS SCPS F TP
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sensors were deployed on the Rı´o de la Plata and ten data series were obtained from tide pole. 3.2. Harmonic constants and mean water level in the model domain Harmonic constituents were computed following the leastsquares method (Foreman, 1977) for every tidal series (Table 3). According to the Rayleigh criterion (Pugh, 1987, Schureman 1988), 17 from the 34 tidal series available were too short to separate the required 12 components. For these 17 series, therefore the methodology developed by Godin (1972) was followed to decontaminate the components included in the analysis and to infer contaminant components at close frequencies. The harmonic constants at the nodes of the grid are obtained by interpolation of the harmonic constants at the 34 tide stations. In order to interpolate, the geographic coordinates of each station are transformed to planar Gauss Kruger coordinates, and weights are assigned, taking into account the length of the tide series. Different interpolation methods were tested: radial basis functions, Kriging, minimum curvature and Shepard modified. Kriging was the finally chosen because it had the least mean square error
Fig. 2. Geographical locations of tidal stations.
from the residues (difference between harmonic constants calculated from measurements and those computed from interpolation between nodes). Even though they are outside the model domain, the stations La Paloma (14) and General Lavalle (34) were used for
Table 4 Slope, correlation coefficient (R) and NRMSD of filtered series vs. modeled predicted heights. Indexes in brackets refer locations to Fig. 2. Location (index)
R
Slope
NRMSD
Nueva Palmira (1) Guazu Desembocadura (2) Carmelo (3) Isla Martin Garcı´a (4) Conchillas (5) Canal Emilio Mitre—Braga (6) Baliza Diamante (7) San Fernando (8) Muelle Platero (9) Colonia del Sacramento (10) Olivos (11) Buenos Aires (12) Torre Norden (13) Banco Ortiz (15) Bernal (16) Banco Jesu´s Marı´a (17) La Plata (18) Montevideo (19) Magdalena (20) Isla de Flores (21) Punta del Este (22) Canal Punta Indio (Km 239.1) (23) Torre Oyarvide (24) Canal Punta Indio (Km. 201.6) (25) Spar Brasileira (26) Punta Indio (Costa) (27) Rı´o de la Plata Exterior (28) Punta Piedras (29) Samborombom I (30) Rio de la Plata Exterior II (31) Samborombom II (32) San Clemente del Tuyu (33)
0.82 0.82 0.81 0.84 0.88 0.85 0.83 0.85 0.81 0.84 0.85 0.86 0.86 0.92 0.87 0.92 0.87 0.85 0.88 0.88 0.92 0.88 0.92 0.91 0.92 0.94 0.96 0.84 0.94 0.92 0.93 0.95
0.75 0.79 0.85 0.96 1.10 0.94 1.04 0.99 0.93 1.01 1.03 0.97 1.03 1.07 1.04 0.98 1.00 0.91 0.89 1.01 1.06 1.06 0.97 1.01 1.04 1.10 0.95 1.10 1.09 0.98 0.94 1.01
11% 10% 9% 9% 8% 7% 8% 8% 7% 6% 7% 7% 6% 7% 8% 8% 7% 5% 6% 7% 6% 6% 6% 8% 5% 8% 6% 7% 4% 5% 4% 4%
Fig. 3. Cotidals and corange charts of the M2 and O1 components.
E. D’Onofrio et al. / Computers & Geosciences 44 (2012) 196–202
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Fig. 4. Scatter plots of filtered vs. modeled hourly height; the full line indicates perfect fit.
the interpolation. Twenty six images were generated, one of them contains the information of pixels corresponding to waters of the Rı´o de la Plata, another image stores the values of mean water levels and 24 were used to store amplitudes and phases of harmonic constituents. The images cover the area between the Gauss Kruger coordinates 6,359,000 m and 6,688,500 m (west– east direction) and 5,975,000 m and 6,256,000 m (north–south direction). This zone preserves the information inside the area between latitudes 361 250 S y 331 450 S and longitudes 581 350 W y 541 540 W. Cotidal and corange charts of the 12 selected components were calculated. Due to length reasons, only those corresponding to M2 and O1, the main semidiurnal and diurnal constituents respectively, are shown in Fig. 3. 3.3. Model verification To verify the empirical model, the results were compared to observed tidal records. Previously, it was necessary to remove the meteorological effect that causes differences between the astronomical and the observed tide (storm surge). For that, observed records were high-pass filtered with a 401 elements filter, devised from the Hamming window (Hamming, 1977) with a cutoff frequency of 0.03 c/h, to remove the storm surge. The convolution between the high pass filter and every of the 32 series (Table 1) were performed in the time domain. La Paloma (14) and General Lavalle (34) stations were not filtered because they are outside the domain. Then, linear regressions of filtered series vs. modeled predicted height were calculated. The slopes and the correlation coefficients obtained are shown in Table 4. To quantify the error between observed (filtered) vs. computed values the normalized root-mean-square deviation (NRMSD) is calculated as the RMSD divided by the range of the observed (filtered) values. Three scatter plots of observed (filtered) vs. computed values with a perfect fit line are shown as an example in Fig. 4. This scatter plots correspond to the three main regions of the Rı´o de la Plata. The model displays a satisfactory fit in all cases. Table 4 shows that in general, the Inner Rı´o de la Plata presents lowest regression coefficients, slopes slightly different from the ideal fit and the highest values of NRMSD. A possible explanation is that the inner estuary is sensitive to the contributions of the Parana and Uruguay rivers (25,000 m3/s jointly, Nagy et al. 1997) which modify the level of the estuary and consequently change the tide propagation. D’Onofrio et al. (2010) have shown that the flow of the river variations produce changes in harmonic constants 30 km away from the mouth of the Rı´o Negro River (Carmen de Patagones City, Argentina). 3.4. Instructions for use, example The first step to run the program is loading RiodelaPlata_19502050.dll file. It contains the harmonic constants, the mean water levels and part of complementary information stored in images.
Fig. 5. Start window of SEAT with loaded data.
Fig. 6. Result window of SEAT.
Then, the coordinates of the chosen point, the date and the offset are entered. Fig. 5 shows the window where coordinates (in this example, latitude 341 350 0300 S, longitude: 581 020 0000 W), the date (06/06/2011) and the offset (0.35 m) were entered. Fig. 6 shows the results window. The program, as well as the dynamic link library of the Rı´o de la Plata can be requested by email to
[email protected], Oceanography Department of the Naval Hydrographic Service, Argentina.
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4. Conclusions The use of SEAT model is simple and easy and only requires minimal computational capabilities. A personal computer with hardware that supports the Microsoft Framework 3.5 could be used to run the SEAT. It also provides to the user the harmonic constants and the mean water level referred to the tidal datum used to calculate the prediction. The storage of information in images reduces the size of the application. The grid used for the application to the Rı´o de la Plata Estuary has a resolution of 500 m 500 m (approximately 1/1600 1/2000 ), greater than the 1/301 used in the Patagonian Shelf 2010. The software developed presents some differences with the TMD with regard to the prediction of tides. The SEAT is an application and does not require the MATLAB software to run. On the other hand the developed software is not restricted to use the output of ESR and OSU model families, but it also allows the correction of tidal heights for meteorological effect or differences with the tidal datum. The library file can be changed to calculate tide prediction in other regions with different sizes of grids. Applications like SEAT will be a replacement for Tide Tables in a near future.
Acknowledgments This research is part of the PIDDEF 05/08 and PIDDEF 42/10 Projects (Ministry of Defense, Naval Hydrographic Service), and is a contribution to I014 UBACyT 2008-2010 and 20020100100840 UBACyT 2011-2014 Science and Technological Division, University of Buenos Aires (UBACYT) Project.
Appendix A. Supporting materials Supplementary data associated with this article can be found in the online version at doi:10.1016/j.cageo.2011.09.019.
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