Differences of Calculated Results

Introduction

For data generated by

• mathematical models (model results), or
• analysis of calculated results (characteristic numbers), or
• measured data (observational data)

Various differences can be computed. Input data can be typically categorized as follows:

• Category K0: [math]\displaystyle{ f(x,y,z) }[/math], time-independent quantities;
• Category K1: [math]\displaystyle{ f(x,y,z,t_1) }[/math], time-dependent quantities, one time step;
• Category KC: [math]\displaystyle{ f(x,y,z,t_i) }[/math], time-dependent quantities, several discrete time steps, constant time step [math]\displaystyle{ \Delta_t }[/math];
• Category KN: [math]\displaystyle{ f(x,y,z,t_i) }[/math], time-dependent quantities, several discrete time steps, varying time step [math]\displaystyle{ \Delta_t(i) }[/math].

For geophysical data categories K1, KC and KN are of significance. Examples:

• Category K1: topography/bathymetry [math]\displaystyle{ h(x,y,z,t_1) }[/math] for a specific instant in time;
• Category KC: water level [math]\displaystyle{ \eta(x,y,z,t_i) }[/math] at discrete times [math]\displaystyle{ t_i }[/math] with constant time step, e. g. computed by a mathematical model;
• Category KN: tidal high water [math]\displaystyle{ \eta^{\rm{HW}}(x,y,z,t_i) }[/math] for times [math]\displaystyle{ t_i }[/math] at non-equidistant time intervals, e.g. derived from a water level time serie.

Definitions

• reference data [math]\displaystyle{ r }[/math]: with respect to [math]\displaystyle{ r }[/math] various deviations for [math]\displaystyle{ f }[/math] can be evaluated. Typical data are either observational data or computational as well as analysis results for a specific (reference) state (situation);
• variant data [math]\displaystyle{ f }[/math]: can be also either observational data or computational as well as analysis results, for which deviations shall be computed with respect to the reference state. Typically variant data are given for a different period in time (natural variation) or a different state of the system under study.
• valid operator 1: [math]\displaystyle{ V(r_i) }[/math] returns .T. or .F., in dependence whether [math]\displaystyle{ r_i }[/math] is valid or invalid. Can be also applied to [math]\displaystyle{ f_i }[/math].
• valid operator 2: [math]\displaystyle{ V(r_I,f_i) }[/math] returns .T. or .F., in dependence whether [math]\displaystyle{ V(r_i)\land V(f_i) }[/math] is valid or invalid.

Requirements for the computation of differences

The following requirements must be fulfilled by [math]\displaystyle{ r }[/math] and [math]\displaystyle{ f }[/math]:

1. [math]\displaystyle{ r }[/math] and [math]\displaystyle{ f }[/math] must belong to the same category (see above);
2. the number of times [math]\displaystyle{ t_i }[/math] must be identical for [math]\displaystyle{ r }[/math] and [math]\displaystyle{ f }[/math];
3. for data belonging to category KC constant time steps must coincide [math]\displaystyle{ \Delta t }[/math] for [math]\displaystyle{ r }[/math] and [math]\displaystyle{ f }[/math];
4. (physical) dimension as well as meaning must be equivalent for [math]\displaystyle{ r }[/math] and [math]\displaystyle{ f }[/math];
5. [math]\displaystyle{ r_i }[/math] (short for [math]\displaystyle{ r(x,y,z,t_i) }[/math]) as well as [math]\displaystyle{ f_i }[/math] (short for [math]\displaystyle{ r(x,y,z,t_i) }[/math]) must be valid data for the same instant [math]\displaystyle{ i }[/math] in time; otherwise the dervied results will become invalid.

Computational rules

The following computational rules are implemented in NCDELTA.

Simple difference

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Overview