Simplace Documentation

Old style - Deprecated:
You can visualize a list of numbers as a graph of an interpolation table, where x and y entries are alternating \(x_1 \quad y_1 \quad x_2 \quad y_2 \quad ... \quad x_n \quad y_n\). This is the way the FST-origined sim components (Lintul, Gecros ...) handle interpolation tables (implementing the FST-Function AFGEN).

New style:
You can give x and y entries by two arrays with \(x_1 \quad x_2 \quad ... \quad x_n\) and \(y_1 \quad y_2 \quad ... \quad y_n\).
Please enter the numbers separated by comma or space.

Formula

Given the numbers \(x_1 \quad y_1 \quad x_2 \quad y_2 \quad ... \quad x_n \quad y_n\) with \(x_1 \lt x_2 \lt ... \lt x_n \), the interpolation algorithm defines a piecewise linear function

LINEAR

Linear interpolation between given points - piecewise linear function

\[ f(x) := \begin{cases} y_1 & \text{for} & x \lt x_1 & \text{(extrapol.)}\\ \\ \dfrac{y_{k+1} - y_{k}}{x_{k+1} -x_{k}} (x-x_{k}) + y_{k} & \text{for} & x_k \leq x \lt x_{k+1}, \quad k=1 ... n-1 & \text{(interpol.)}\\ \\ y_n & \text{for} & x_n \leq x & \text{(extrapol.)} \end{cases} \]

CONSTANT_LEFT

Constant value of the preceding point - piecewise constant function

\[ f(x) := \begin{cases} y_1 & \text{for} & x \lt x_1 & \text{(extrapol.)}\\ \\ y_{k} & \text{for} & x_k \leq x \lt x_{k+1}, \quad k=1 ... n-1 & \text{(interpol.)}\\ \\ y_n & \text{for} & x_n \leq x & \text{(extrapol.)} \end{cases} \]

CONSTANT_MID

Constant value of the nearest point - piecewise constant function

\[ f(x) := \begin{cases} y_1 & \text{for} & x \lt \frac{x_1 + x_2}{2} & \text{(extrapol. / interpol.)}\\ \\ y_{k} & \text{for} & \frac{x_{k-1} +x_k}{2} \leq x \lt \frac{x_k + x_{k+1}}{2}, \quad k=2 ... n-1 & \text{(interpol.)}\\ \\ y_n & \text{for} & \frac{x_{n-1} + x_n}{2} \leq x & \text{(interpol. / extrapol.)} \end{cases} \]

CONSTANT_RIGHT

Constant value of the next point - piecewise constant function

\[ f(x) := \begin{cases} y_1 & \text{for} & x \lt x_1 & \text{(extrapol.)}\\ \\ y_{k+1} & \text{for} & x_k \lt x \leq x_{k+1}, \quad k=1 ... n-1 & \text{(interpol.)}\\ \\ y_n & \text{for} & x_n \leq x & \text{(extrapol.)} \end{cases} \]

NONE

Defined only for the given x-values - discrete function.
For x-values not given in the table, null is returned. Alternatively one can specify a default value, e.g. 0.0 for irrigation or fertilizer tables.

\[ f(x) := \begin{cases} y_k & \text{for} & x = x_k, \quad k=1 ... n & \text{(table value)}\\ \\ \emptyset \text{ or }y_{default} & \text{else} & & \text{(null, if default not explicitely defined)}\\ \\ \end{cases} \]

Example pitfalls

Here you see some possible pitfalls when using interpolation tables. Click on the small graph icon to visualize the graph and the xy-table. Red number sets represent wrong interpolation tables, green ones represent correct tables.

Linear interpolation for isolated events (irrigation, fertilizer tables, other look-up tables)

Some older SimComponents use linear interpolation for all tables. If you want to irrigate only on DOY 12 and 31, then providing the values only for these days in the table might give you unexpected results: for the other days values are interpolated, giving you irrigation on all days.
12 61 31 28

You have to provide y-values of 0 for the preceding and following days, as you can see here:
11 0 12 61 13 0 30 0 31 28 32 0

Notice: If x values are real numbers and not integer, then you might add 0 y-values to \(x_k - \delta \) and \(x_k + \delta \), with \(0 \lt \delta \ll 1\) (i. e. \(\delta\) is small positive number).
1.39999 0 1.4 61 1.40001 0 3.4999 0 3.5 28 3.50001 0

Unordered x-Values in interpolation table

The table has to be sorted by x-values. See what happens if not:
30 0 31 28 32 0 11 0 12 61 13 0
It is not a graph of a function anymore, so interpolation is not well defined (for days 12 and 31 you have 2 possible values!). Interpolated results might be arbitrary and not what you expect.

Adding some jitter on the y-values make the problem even more obvious
30 0 31 28 32 2 11 2 12 61 13 4

Ordering the table by x-values gives a proper graph where unambigous interpolation is possible
11 0 12 61 13 0 30 0 31 28 32 0

Repeated x-values

x-values have to be different - providing two y-values to same x-value makes the interpolation at this point ambigous:
0 5 1 2 1 3 2 4

If your function has "jumps", then just change one of the repeated x-value by a small amount
0 5 1 2 1.05 3 2 4

It could be a really small amount
0 5 1 2 1.000000000001 3 2 4

Wrong shaped tables

The number of entries has to be even (as many x- as y-values). Otherwise the last entry is discarded.
0 10 2 7 3 8 4

If you provide only y-values, then interpolation will for sure not do what you expect, e.g. LUE dependend on DVS:
Right: 0 3.2 1 3.1 1.5 3.15 2.0 3.03
Wrong: 3.2 3.1 3.15 3.03

Interpolation tables in Simplace

New style

An interpolation table is provided as two SimVariables of datatype DOUBLEARRAY.

The naming of the two SimVariables follows the scheme VariablenameTableXVariable VariablenameTableYVariable.

The literal "Table" indicates that the variables define interpolation tables,
the Variablename which is the same for both SimVariables indicates that they belong together.
The XVariable and YVariable are different - which is the X (driving) and which is the Y (response) variable is normally explained in the description (or can be derived from the context).

Example: cLeavesPartitioningTableDVS and cLeavesPartitioningTableFraction define the interpolation table for leaves partitioning. From the description of the second variable "Fraction of above-ground dry matter to leaves as function of DVS" one can see that Fraction depends on DVS, so DVS is the X value and Fraction the Y value.

Notice: for backward compatibility, most of the older SimComponents support both styles. That means one could use either one variable with alternating values, or a pair of variables. The description of the new variables indicates which is the old (alternating) variable. However, you should not use both styles at the same time, as one may get confused. If you set old and new variables at the same time, then the new ones will be used.

   <parameter id="crop_key" datatype="CHAR">wheat</parameter>
   <parameter id="cLeavesPartitioningTableDVS" datatype="DOUBLEARRAY">
     <value>0</value> 
     <value>1</value> 
     <value>1.1</value> 
     <value>2</value> 
   </parameter>
   <parameter id="cLeavesPartitioningTableFraction" datatype="DOUBLEARRAY">
     <value>.4</value> 
     <value>.3</value>
     <value>.0</value> 
     <value>.0</value> 
   </parameter>

In CSV files, you can provide the table in long format:

crop_key	cLeavesPartitioningTableDVS	cLeavesPartitioningTableFraction
wheat	0	.4
wheat	1	.3
wheat	1.1	.0
wheat	2	.0
barley	0	.33
barley	1	.2
barley	2	.0

Old style - DEPRECATED

An interpolation table is provided as a SimVariable of datatype DOUBLEARRAY. You can use CSV files as well as XML files to provide the table.

Although you can write the values in the XML file in any shape (linebreaks, spaces etc.), I recommend to put always two values side by side, so that the reader can quickly recognize the x/y-value pairs:

   <parameter id="crop_key" datatype="CHAR">wheat</parameter>
   <parameter id="Irrigation" datatype="DOUBLEARRAY">
     <value>12</value> <value>0</value>
     <value>13</value> <value>61</value>
     <value>14</value> <value>0</value>
   </parameter>