Elementary Aspects of Data Analyses

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. high water level, low water level and tidal range;
• objective description of specific characteristics for dynamic systems, e.g. 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 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).

Graphical representation for a sequence of synoptic data sets. Also available as 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 can be created (example: 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 input data are available in the form of 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: 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 ebb period, flood period, ebb current duration and flood current duration. Drying and wetting of tidal flats is taken into account. Examples can be found for characteristic numbers of water level, characteristic numbers of current velocity, characteristic numbers of salinity as well as characteristic numbers of bed load transport.

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 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.

Graphical representation of characteristic numbers which are typically calculated for a serie of periodic events (example: 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 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 (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.

Graphical representation for the computation of differences between characteristic numbers. Figure shows computed maxima for two different series of periodic events. Also available as Encapsulated PostScript file. Graphics available with German text only.

Precision of Computed Results

Discrete 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.

Graphical representation of the improved calculation of values for characteristic numbers by means of interpolation from neighbouring data. Also available as Encapsulated PostScript file. Graphics available with German text only.

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 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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Overview