View source for Elementary Aspects of Data Analyses

[[de: Grundlegende Aspekte einer Berechnung von Kennwerten]]
For a '''better understanding of natural processes''' just watching them is normally by far not sufficient. It may be a much better approach to take, in a first step, some precise and objective measurements. Afterwards, with some [[characteristics]] of the processes in mind, different analyses may yield characteristic numbers for each of them.

This can be also expresssed in a different way: sitting close to an [[estuary]], looking at the rise and fall of the [[water surface]], may be pleasant, interesting and even recreative. Any better understanding of the natural processes may be only achieved if we are able to grasp the '''temporal variation''' of the [[water surface]] precisely (time-serie of [[water level]] where numerical values prescribe the position of the [[water level]] surface). Thereafter data analyses may be carried through to obtain values for different well defined '''characteristic attributes''' (e.g. period, highest and lowest [[water level]]). These derived data can be easily used to describe local temporal changes (e.g. differences between two successive days or years) or to compare results for different places around the world. Characteristic differences become precisely quantifiable in that way.

Many methods are available at BAW to carry through data analyses (see [[Analysis of Calculated Results]]). Application of methods is not restricted to one or some locations along an [[estuary]]. The methods are designed to produce area-wide results '''for the overall [[estuary]]'''. Results may even vary with [[water depth]] if three-dimensional data have been used as input. Present methods for data analyses generate a great variety of characteristic numbers. They are intimately related to observations: [[water level]] elevation, current velocity, salinity etc. Therefore the methods contribute to an improved understanding of natural processes.

Area-wide computation of characteristic numbers is a step forward to achieve one the following aims:

* reduction of the amount of data required for a reasonable description of a dynamic system, compared with a situation where synoptic data sets are exclusively used;
* quantitative description of the dynamics of a system using a small set of characteristic quantities, e.g. [[Tidal Characteristic Numbers of Water Level#Definition of the Tidal Characteristic Numbers of Water Level|high water level]], [[Tidal Characteristic Numbers of Water Level#Definition of the Tidal Characteristic Numbers of Water Level|low water level]] and [[Tidal Characteristic Numbers of Water Level#Definition of the Tidal Characteristic Numbers of Water Level|tidal range]];
* objective description of specific [[characteristics]] for dynamic systems, e.g. [[Harmonic Analysis of Water Level#Shallow Water Constituents of Tides|tidal asymmetry]] between ebb and [[flood]] tide;
* evaluation of physical reasons for phenomena observed in synoptic data sets, e.g. baroclinic circulation;
* representation of differences for the dynamics of a natural system, e.g. variation of tidal range due to alterations of bathymetry. 

The following sections highlight some aspects which are of practical importance during the computation of characteristic numbers.
==Input Data==
Normally we start from '''synoptic data sets''' before we can later derive some characteristic numbers. A synoptic data set consists of physical data valid for a specific moment and describes the spatial variation of a physical quantity. [[Water level]] elevation as well as current velocity are typical synoptic data sets for estuaries and coastal seas.

For large estuaries synoptic data sets are frequently generated by means of [[Mathematical Models for Coastal Areas and Estuaries|mathematical models]] which are able to simulate relevant physical processes. Models are constructed upon basic physicals laws, e.g. mass conservation and results are therefore consistent with elementary physical principles.

Area-wide (overall [[estuary]]) synoptic measurements cannot be achieved for a reasonable amount of money. Therefore numerical models are mainly used today to provide us with area-wide synoptic data sets for different physical quantities.

Numerically generated data are normally only available at a larger number of discrete points in space and time ('''discretisation''').

[http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/pdf/diskret.pdf Graphical representation] for a sequence of synoptic data sets. Also available as [http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/eps/diskret1.eps Encapsulated PostScript file]. Graphics available with German text only.

The '''output time-step''' of a numerical [[model]] normally exceeds the respective computational time step. Typical output time-steps amount to e.g. ten minutes for estuarine applications.

'''Time-series''' are generated out of synoptic data sets for each point of the computational grid. Therefore time-series are available area-wide in the overall [[estuary]] for different physical quantities of interest.
==Usage of Synoptic Data Sets==
The '''spatial variation''' of a physical quantity can be clearly expressed using synoptic data.

Out of a larger set of synoptic data an '''[[ANIMATION|animation]]''' can be created (example: [[Example: Animation of Salinity in the Außenweser Estuary|tidal variation of salinity]] in the Weser [[estuary]], Germany). But different viewers may get very different subjective impressions about the processes animated.
==Computation of Characteristic Numbers==
If suitable [[Elementary Aspects of Data Analyses#Input Data|input data]] are available in the form of [[Elementary Aspects of Data Analyses#Input Data|time-series]], characteristic numbers can be easily computed for different physical quantities, e.g. for

* near surface wind velocity,
* [[water level]],
* total [[water depth]],
* current velocity,
* salinity,
* [[suspended sediment concentration]],
* [[bed load]] transport or
* bathymetric depth. 

In addition, the computation of characteristic numbers can be also performed for different derived physical quantities, e.g.

* flow rate (computed from total [[water depth]] and current velocity) or
* salt transport rate (computed from total [[water depth]], current velocity and salinity), 

to mention only a few of the possible combinations.

Characteristic numbers are automatically computed using various computer programs which apply well defined '''computational rules''' on different physical quantities. For all points of a computational grid the following values can be calculated for each time-serie of a physical quantity:

* extreme values (maximum, minimum),
* difference between extreme values (amplitude, variation),
* mean value and
* integral. 

At the end of the computational process characteristic numbers are available area-wide, e.g. for the overall [[estuary]]. Results display the spatial variation of different [[characteristics]] of a physical quantity (example: [[Example figures: tidal variation of salinity|tidal variation of salinity]]).

The number of '''computational results''' does strongly depend on the physical quantity as well as the type of the available time serie.

In situations where a '''periodic time-serie''' is available, which contains a sequence of similar events (e.g. tides), the characteristic numbers are calculated first for every event, e.g. for every tide or tidal phase like [[Tidal Characteristic Numbers of Water Level#Definition of the Tidal Characteristic Numbers of Water Level|ebb period]], [[Tidal Characteristic Numbers of Water Level#Definition of the Tidal Characteristic Numbers of Water Level|flood period]], [[Tidal Characteristic Numbers of Current#Definition of Tidal Characteristic Numbers of Current|ebb current duration]] and [[Tidal Characteristic Numbers of Current#Definition of Tidal Characteristic Numbers of Current|flood current duration]]. Drying and wetting of tidal flats is taken into account. Examples can be found for [[Tidal Characteristic Numbers of Water Level|characteristic numbers of water level]], [[Tidal Characteristic Numbers of Current|characteristic numbers of current velocity]], [[Tidal Characteristic Numbers of Salinity|characteristic numbers of salinity]] as well as [[Tidal Characteristic Numbers of Bed Load Transport|characteristic numbers of bed load transport]].

[http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/pdf/ereignis.pdf Graphical representation] for the computation of characteristic numbers (e.g. maxima) for a series of events where drying and wetting is taken into account. Also available as [http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/eps/ereignis1.eps Encapsulated PostScript file]. Graphics available with German text only.

Having calculated characteristic numbers for all events during the [[period of data analysis]] also values for the (absolute) maximum, the (absolute) minimum as well as the mean value are determined.

[http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/pdf/ergebnis.pdf Graphical representation] of characteristic numbers which are typically calculated for a serie of periodic events (example: [[Tidal Characteristic Numbers of Water Level |characteristic numbers of water level]]). Characteristic numbers can be used to describe the behaviour of the system for every single event as well as, in a statistical sense, for the overall [[period of data analysis]]. Also available as [http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/eps/ergebnis1.eps Encapsulated PostScript file]. Graphics available with German text only.

For '''non-periodic time-series''' solely characteristic numbers which are representative for the overall [[period of data analysis]] are evaluated. As an example see [Characteristic Numbers of [[Water level|Water Level]] (independent of tides)|characteristic numbers of [[water level]]] which are independent of tides.

'''differences''' may be calculated from characteristic numbers which have been obtained for different situations. The influence of changes in bathymetry on tidal dynamics are a typical example. For each event within the [[period of data analysis]] the differences are determined together with some statistical quantities for the whole period, e.g.

* maximum difference (for all events),
* minimum difference (for all events) and
* significance for the difference of the mean values. 

Computed differences enable a detailed and objective description for many changes of the dynamics in natural systems.

[http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/pdf/diff_ampli.pdf Graphical representation] for the computation of differences between characteristic numbers. Figure shows computed maxima for two different series of periodic events. Also available as [http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/eps/diff_ampli1.eps Encapsulated PostScript file]. Graphics available with German text only.
==Precision of Computed Results==
Discrete [[Elementary Aspects of Data Analyses#Input Data|input data]] are used by the analyses programs. Therefore '''discretisation errors''' are present which influence computed values for the characteristic numbers. Interpolation and extrapolation are therefore used to improve calculated results.

[http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/pdf/interpol.pdf Graphical representation] of the improved calculation of values for characteristic numbers by means of '''interpolation''' from neighbouring data. Also available as [http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/eps/interpol1.eps Encapsulated PostScript file]. Graphics available with German text only.

[http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/pdf/extrapol.pdf Graphical representation] of improvements for the calculation of the times for either drying or wetting (on e.g. tidal flats) by means of '''extrapolation'''. Data from neighbouring time-steps are used. Also available as [http://www.baw.de/downloads/wasserbau/mathematische_verfahren/pre_postpr/eps/extrapol1.eps Encapsulated PostScript file]. Graphics available with German text only.

The influence of discretisation errors on computed values for characteristic numbers is reduced by means of interpolation and extrapolation - but of [[course]] they are still there.
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