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.
• integer operator 1: [math]\displaystyle{ P(r_i) }[/math] returns 1 if [math]\displaystyle{ V(r_i) }[/math] else 0. Similar for [math]\displaystyle{ f_i }[/math].
• integer operator 2: [math]\displaystyle{ P(r_i,f_i) }[/math] returns 1 if [math]\displaystyle{ V(r_i)\land V(f_i) }[/math] .T. else 0.

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 results

Program NCDELTA can be used to compute all subsequent results. Locations of [math]\displaystyle{ r }[/math] are not required to coincide with those of [math]\displaystyle{ f }[/math]. Values [math]\displaystyle{ r }[/math] are interpolated to locations of [math]\displaystyle{ f }[/math], as long as the geographical distance between the different locations does not exceed [math]\displaystyle{ R^\max }[/math]. In case the distance exceeds that limit, no results will be computed. In such a situation an invalid result value will be generated. The follwing results can be computed using NCDELTA.

Ordinary differences

Difference

A result is computed for all times (one value for time-independent data) at all locations [math]\displaystyle{ (x,y,z) }[/math]:

1. The difference between [math]\displaystyle{ f_i }[/math] and [math]\displaystyle{ r_i }[/math] is calculated in case [math]\displaystyle{ V(r_i,f_i) }[/math] returns .T.:

   [math]\displaystyle{ d_i = f_i - r_i }[/math], if [math]\displaystyle{ V(r_i,f_i) }[/math];

2. Result will be invalid, if [math]\displaystyle{ V(r_i,f_i) }[/math] returns .F.:

   [math]\displaystyle{ d_i = \rm{invalid} }[/math] if [math]\displaystyle{ \lnot V(r_i,f_i) }[/math].

Results are computed for data belonging to categories K0, K1, KC und KN, which means for all types of data.

Maximum difference

Maximum difference is determined using absolute value in combination with sign preservation:

1. At first all differences [math]\displaystyle{ d_i }[/math] will be computed as indicated above;
2. Out of all valid data index [math]\displaystyle{ i^\max }[/math] is determined in such a way that [math]\displaystyle{ \left|d_i\right| }[/math] is maximal

   [math]\displaystyle{ d^\max = d_{i^\max} }[/math]

   is equal to the maximum difference according to this definition; this value can be negative, positive or zero;

3. In case all values [math]\displaystyle{ d_i }[/math] are invalid [math]\displaystyle{ d^\max = \rm{invalid} }[/math] will be set.

Computation is performed for categories KC and KN. A valid [math]\displaystyle{ d^\max }[/math] is obtained as long there exists at least one valid difference [math]\displaystyle{ d_i }[/math]. During visualization NCPLOT enables filtering using ancillary variable Number of valid differences.

Minimum difference

Minimum difference is determined using absolute value in combination with sign preservation:

1. At first all differences [math]\displaystyle{ d_i }[/math] will be computed as indicated above;
2. Out of all valid data index [math]\displaystyle{ i^\min }[/math] is determined in such a way that [math]\displaystyle{ \left|d_i\right| }[/math] is minimal

   [math]\displaystyle{ d^\min = d_{i^\min} }[/math]

   is equal to the minimum difference according to this definition; this value can be negative, positive or zero;

3. In case all values [math]\displaystyle{ d_i }[/math] are invalid [math]\displaystyle{ d^\min = \rm{invalid} }[/math] will be set.

Computation is performed for categories KC and KN. A valid [math]\displaystyle{ d^\min }[/math] is obtained as long there exists at least one valid difference [math]\displaystyle{ d_i }[/math]. During visualization NCPLOT enables filtering using ancillary variable Number of valid differences.

Mean Difference

Mean value is computed for all valid differences:

1. At first all differences [math]\displaystyle{ d_i }[/math] will be computed as indicated above;
2. From all valid differences the mean value is computed

   [math]\displaystyle{ d^{\rm{mit}}=\frac{\sum_{i\in I}P(d_i)d_i}{\sum_{i\in I}P(d_i)} }[/math];

3. In case all [math]\displaystyle{ d_i }[/math] are invalid [math]\displaystyle{ d^{\rm{mit}} = \rm{invalid} }[/math] will be set.

Computation is performed for categories KC und KN. A valid [math]\displaystyle{ d^{\rm{mit}} }[/math] is obtained as long there exists at least one valid difference [math]\displaystyle{ d_i }[/math]. During visualization NCPLOT enables filtering using ancillary variable Number of valid differences.

Percentiles

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Overview