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This study endeavors to deal with the least square spectral analysis on the time series, to find present significant frequencies, to analyze 40 tide components using harmonic methods and to show relationship between discovered frequencies and 40 components of tide. For the purpose of collecting data of altimetry satellites of Topex/Poseidon (T/P), Jason 1, Jason 2 and coastal tide gauges of Bandar Anzali, Noshahr, and Nekah were utilized. In this time series formed by cross over points of altimetry satellite and then using least square spectral analysis on time series derived from altimetry satellite and coastal tide gauges the significant components were found and annual, biannual, and monthly components were discovered. Then, analysis of 40 tide components was conducted using harmonic method to find the amplitude and phase. It represented that solar annual (Sa) plays the most significant role on Caspian Sea corresponded to the least square spectral analysis of the time series. The results shows that the annual (Sa) and semi-annual Solar (Ssa) constituents on all of the ports listed have the highest amplitude in comparison with the other constituents which are respectively 16 cm, 18 cm and 15 cm for annual constituent and 2.8 cm, 5.4 cm and 3.7 cm for semi-annual constituent.

Knowing the reason behind changes in seawater level is a great challenge in the scientific fields and causes of seawater level were interested issues that have been studied by scientists. Five influential factors that are effective in changing sea level are meteorological effects, oceanic effects, tide, climate change, and vertical motion of the crust [

Caspian Sea is surrounded by five countries including Islamic Republic of Iran, Russia, Republic of Azerbaijan, Turkmenistan, and Kazakhstan. The sea is the largest remaining part that emanated from dissolving the old Tethys Sea and extended from North Pole to Indian Ocean in the first to third period of geology. And in the third period it was disconnected because of folding, and mountain ranges such as Caucasus and Minor Asia and following upcoming the European continent and Iran Plain some seas were constructed like Caspian Sea. Caspian Sea is located in the North and East areas in 47˚57', and 36˚33', respectively. Also, from west and East it is restricted to meridians of 46˚43', and 54˚53', respectively. The length of Caspian Sea is about 1030 kilometers, and its width varies from 435 to 196 Kilometers [

Data of T/P, Jason 1, and Jason 2 were prepared using binary form in the different cycles through NASA spatial site and AVISO organization. Data were gathered from 1992 to 2014 and updated cycles by NASA spatial organization site and AVISO site will be available in the following addresses: http://Podacc.jpl.nassa.gov and http://www.aviso.altimetry.fr/en/home.html. As periodicity of T/P, Jason 1, Jason 2 satellites is approximately 10 days (precisely 9.915 days), the least observation interval of data is 10 days.

The quantity of observation in satellites altimetry is the length of satellite to the surface of water, namely Satellite Range. When the length of satellite in the circuit in the Reference Ellipsoid can be specified using some methods, it is possible to achieve Sea Surface Height in the observation points to reference Ellipsoid as follows [

In the stated equation

After calculating the corrected height, having altitude from Reference Ellipsoid level the Seawater Surface height can be achieved to Reference Ellipsoid as follow:

Important points: as in this study we deal with tide modeling of waters; therefore, in the modeling calculations the corrected value from this phenomenon would not be applied on observed height. Therefore, to prepare data and to model calculations the entire stated corrections were conducted except tide corrections applied in observations.

Formation of times series of observing satellite altimetry and coastal tide gauges:

Position of tide gauge stations in which time series were formulated via such stations as follow (

The position of time series in the cross over points of satellite altimetry are documented below (

According to sampling theorem to rebuild a signal from its sample the signal with a higher cost twice more than its most frequency should be sampled [

Duration (time) | Longitude | Latitude | Location |
---|---|---|---|

21/3/2005-20/3/2014 | 49.4623˚ | 37.478˚ | Anzai |

21/3/2006-21/3/2014 | 51.5047˚ | 36.6584˚ | Noshahr |

1/1/2000-31/8/2012 | 53.3656˚ | 36.8502˚ | Neka |

Considerations | Duration (time) | Position | Name of station |
---|---|---|---|

Cross over of passes 16 and 92 | 1992-2014 | Lat: 39.20˚ Lon: 49.59˚ | Anzali-Cross Over |

Cross over of passes 16 and 32 | 1992-2014 | Lat: 37.09˚ Lon: 51.02˚ | Noshahr-Cross Over |

Cross over of passes 19 and 107 | 1992-2014 | Lat: 37.09˚ Lon: 53.85˚ | Neka-Cross Over |

frequency should be _{w} period can be totally rebuild from value of samples that might have been obtained

in the intervals less than_{w} period will be converted to a Ta period that is called Aliasing period.

Aliasing period can be calculated by the following equation:

Here, f is the frequencies of tide components,

For example, for the component of half daily month (M2) that the main frequency is 1.9305 cycles per day according to stated alias frequency 0.0161 cycles per day corresponds to 62/107 days.

For satellite Topex, Jason 1, Jason 2 to divide the two great components of half days M2, S2 with period of 62 days and 59 days, at least we need 3 years of data for this satellite. But for separating components of

The following table shows periods and alias frequencies related to the main components of half days, dailies, half months, monthly, biannual and annual of tide (40 components of tide) about observations of T/P satellite (

The least squares’ spectral analysis is a modeling method for approximate behavior of natural and physical behaviors [

Here

That here

Vectors

Therefore, (Vanicek, 1969) is spectral value for each frequency that can be calculated via the following equation:

Or

Finally, a set of entire spectral values for the entire frequencies can be achieved.

Alias period (day) | Frequency (cpd) | Period (day) | Frequency (cpt) | Components |
---|---|---|---|---|

86.59748212 | 0.01154768 | 0.498634738 | 2.005476 | K2 |

20.63624862 | 0.04845842 | 0.507984175 | 1.96856526 | L2 |

62.10599902 | 0.016101504 | 0.517524947 | 1.932274 | M2 |

49.52825362 | 0.020190496 | 0.527431168 | 1.895982 | N2 |

53.08128696 | 0.018839031 | 0.516792787 | 1.935011527 | Ma2 |

69.99909071 | 0.0142859 | 0.499316491 | 2.00273778 | R2 |

58.74170616 | 0.01702368 | 0.5 | 2 | S2 |

50.60354889 | 0.01976146 | 0.500685383 | 1.99726222 | T2 |

32.76785475 | 0.030517713 | 0.962436532 | 1.039029553 | J1 |

173.1949642 | 0.00577384 | 0.997269476 | 1.002738 | K1 |

23.59880154 | 0.042375033 | 1.035050102 | 0.966136807 | M1 |

45.7135667 | 0.021875344 | 1.075805563 | 0.929536 | O1 |

88.89015367 | 0.01124984 | 1.002745517 | 0.997262 | P1 |

69.36421348 | 0.014416656 | 1.119514937 | 0.893244 | Q1 |

117.4834123 | 0.00851184 | 1 | 1 | S1 |

38.05995355 | 0.026274336 | 0.345016751 | 2.89841 | M3 |

39.16113744 | 0.02553552 | 0.333333333 | 3 | S3 |

31.05396383 | 0.032202008 | 0.258762541 | 3.864547 | M4 |

29.37085308 | 0.03404736 | 0.25 | 4 | S4 |

98.3000804 | 0.010172932 | 0.207010026 | 4.8306839 | M5 |

20.70242825 | 0.048303512 | 0.172508346 | 5.796821 | M6 |

20.08833063 | 0.049780144 | 0.166666667 | 6 | S6 |

27.43697439 | 0.036447167 | 0.12938127 | 7.729094 | M8 |

30.52832512 | 0.032756465 | 0.125 | 8 | S8 |

36.16743265 | 0.027649184 | 13.66082894 | 0.073202 | Mf |

27.55428194 | 0.036292 | 27.55428194 | 0.036292 | Mm |

30.18851156 | 0.033125184 | 14.76537814 | 0.067726 | Msf |

29.91959654 | 0.033422911 | 0.929419758 | 1.075940113 | Oo1 |

182.6150475 | 0.005476 | 182.6150475 | 0.005476 | Ssa |

1083.937532 | 0.000922562 | 0.254305803 | 3.932273613 | Ms4 |

244.5339491 | 0.004089412 | 0.26121558 | 3.82825558 | Mn4 |

96.83175341 | 0.010327191 | 0.340714164 | 2.935011527 | Mk3 |

365.230095 | 0.002738 | 365.230095 | 0.002738 | Sa |

26.33230385 | 0.037976168 | 0.34942929 | 2.86180932 | Mo3 |

593.6403785 | 0.001684522 | 0.353917447 | 2.825517673 | No3 |

22.53825457 | 0.044369008 | 0.537723937 | 1.85969032 | 2N2 |

206.1273043 | 0.004851371 | 0.341351019 | 2.929535707 | So3 |

43.86424117 | 0.022797613 | 0.333029399 | 3.002737907 | Sk3 |

25.7077979 | 0.038898731 | 0.934174083 | 1.070464293 | SO1 |

219.6229789 | 0.004553258 | 0.253952167 | 3.937749433 | Mk4 |

The significance of the spectral values can be statistically tested that is the most important advantages to the least square spectral analysis method [

It is possible to define the approving or rejecting criterion for primary hypothesis

Here,

Spectral analysis with approach LSSA on time series of altimeter cross over points and tide gauge stations was done. As show spectral analysis on time series of altimeter cross over near tide gauge station of Anzali (

To analyze and predict the tide the following equation was used [

Tide components

Here:

Period | (Cycle per day) frequency | Name of station |
---|---|---|

9.7 years 3.6 years 5.4 years 2.4 years 1.6 years 1 years 6 month 5.7 month 3 month 1.9 month 1.1 month | 0.00028 0.00075 0.00050 0.0011 0.0017 0.0027 0.0054 0.0058 0.011 0.017 0.028 | Anzali-Cross Over |

3 years 1.5 years 1 years 9.1 month 6 month 4.5 month 4 month 1.9 month | 0.00091 0.0018 0.0027 0.0036 0.0054 0.0074 0.0082 0.017 | Anzali-Tide Gauge |

9.7 years 3.6 years 5.4 years 2.4 years 1.6 years 1 years 4.3 month 2.5 month 1.3 month 1.1 month | 0.00028 0.00075 0.00050 0.0011 0.0017 0.0027 0.0077 0.013 0.024 0.028 | Noshahr-Cross Over |

4 years 2.6 years 2 years 1 years 9 month 6 month 3 month | 0.00068 0.0010 0.0013 0.0027 0.0037 0.0054 0.011 | Noshahr-Tide Gauge |

9.7 years 4.2 years 1 years 6 month 3.2 month 2.3 month 1.7 month 1.4 month | 0.00028 0.00064 0.0027 0.0055 0.010 0.014 0.019 0.023 | Neka-Cross Over |

4.2 years 2.5 years 1.2 years 1 years 10.2 month 6 month 4 month 3.2 month | 0.00064 0.0010 0.0021 0.0027 0.0032 0.0054 0.0082 0.01 | Neka-Tide Gauge |

After spectral analysis for extraction effective frequency, modeling of time series was performed with using Equation (13). The amplitude and phase of each frequency were determined and In the following with calculated variables of equation thirteen modeling was done. Figures 9-14 show time series modeling and residuals.

Using the least square spectral analysis on time series the tide gauge stations and time series by observing satellite altimetry of annual periods, biannual and monthly in the Caspian Sea were discovered and spectral analyses of time series for tide gauge observations and altimetry observations are corresponding. But, it was totally clear that because tide gauge data existed in a lower tide gauge interval don’t show several years’ periods and a series of periods are seen in spectral analyses observed via time series of satellite altimetry that are not seen in the spectral analyses of time series of tide gauges.

By least square spectral analysis on time series observations for tide gauge stations and time series by observations of satellite altimetry annual periods, biannual and monthly periods were discovered in Caspian Sea which showed several local influential tide components via global model of tide achieved by satellite, like global models of Got and… that existed in corrections of SSH which are not able to be elicited well.

The tide analysis of 40 components of time series by altimetry observations is approximately corresponding but, their gratitude scale is different and influence of each component was similar in comparison to tide analyses of tide gauge stations in the annual components and biannual that were corresponding but, the effect of the entire components was not similar. After comparing tide analysis of the time series of observations for satellite altimetry observation with tide gauge stations, the tide analysis of tide gauge stations showed lack of influence of tide gauge components except annual and biannual component showed controversy to time series altimetry observations, but in tide analysis of time gauge stations there were several components with small values that in this way, the tide analysis of altimetry observation of time series showed that other components existed in addition to annual, biannual and monthly components. The prominent problem of tide gauge data for the coast of the north of Caspian Sea was lack of collecting and appropriate gathering remaining still with many gaps. Also, sampling distances of Anzali tide gauge daily and tide gauge of Noshahr once every three hours have influences on tide modeling and decrease calculating precision.

The results shows that the annual (Sa) and semi-annual Solar (Ssa) constituents on all of the ports listed have the highest amplitude in comparison with the other constituents which are respectively 16 cm, 18 cm and 15 cm for annual constituent and 2.8 cm, 5.4 cm and 3.7 cm for semi-annual constituent.

Mahmoud Pirooznia,Sayyed Rouhollah Emadi,Mehdi Najafi Alamdari, (2016) The Time Series Spectral Analysis of Satellite Altimetry and Coastal Tide Gauges and Tide Modeling in the Coast of Caspian Sea. Open Journal of Marine Science,06,258-269. doi: 10.4236/ojms.2016.62021