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\def\editors	 {{\href{mailto:pbettinger@warnell.uga.edu}{Editor:~Pete Bettinger}}}
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\def\citename	{Søvde} 		%"Author" or "FirstAuthor et al."
\def\citeemail	{nils.egil.sovde@gmail.com} 	% Use later: {\href{mailto://\citeemail}{\citename}}
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\author{	{\href{mailto://\citeemail}{Nils Egil \citename}}
%}\affiliation{	\small\it{Professor, {\href{http://web.unbc.ca/~garcia/}{University of Northern British Columbia, Prince George, BC, Canada. Ph./FAX:\,250.960.5004/5539}}}
}\affiliation{
  \small\it{{\href{http://www.skogoglandskap.no}{Norwegian Forest and
        Landscape Institute, Ås, Norway. Molde University College, Molde, Norway}}}
}\def\yourtitle	{{Off road transportation cost calculations for ground based
  forest harvesting systems}} 				%need double {{ for \\ e.g.: {{Title \\ Subtitle}}
\def\yourkwords	{terrain transportation, terrain transportation cost,
  forest planning, forest operations, forest harvesting}
\def\yourabstract{
Ground based systems are the main approach used for off-road timber
transportation throughout the world. Estimates of terrain
transportation costs are required for several forest planning problems
and for assessment of harvesting contracts and forest land
values. Methods for these calculations can be categorized into two
groups. Methods based on average transportation distance predate
computers, are analytical, and based on manual calculations. Network
methods are based on weighted graph representations and are solved
with graph methods. Here, the two categories are compared and
linked. Analytical methods in the literature have been limited to flat
terrain and including detail is difficult. The network method
can be extended to include uneven terrain or detailed input data.
}%----------------------------------------------------------------------

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\title{\Large\bf\uppercase\yourtitle}
\begin{document} \markright{\hfil{{{\href{mailto://\citeemail}{\citename}\citeetal}}~(\issueyear)/\mcfnshead}} \twocolumn[ \begin{@twocolumnfalse}
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\begin{abstract}\yourabstract\\\\{\bf Keywords:}~\yourkwords \end{abstract}\hrule\vspace{.3truein}\end{@twocolumnfalse}]
%\numberwithin{figure}{section}


% Continue with the first Section:

\section{Introduction}

Ground based systems are the dominant systems used for off road timber
transportation throughout the world, either using forwarders or
skidders. A harvesting operation consists of several steps. The trees
have to be felled, limbed, cross cut and transported to roadside.
The terrain transportation cost (TTC) is a significant part of the
total harvesting cost. However,
there has not been much focus on the cost calculations of off
road driving in the research lately. Such calculations are
required for \eg planning of forest operations, forest planning in
general or environmental planning, but seldom discussed in
detail. Computational power, remote sensing techniques and
optimization techniques have significantly improved the past 25 years,
and a revisiting of this topic seems appropriate.

The TTC can be calculated analytically or numerically. Early methods
that predate computers are in general analytical and based on average
skidding distance (\asd ), sometimes referred to as average yarding
distance.  Although such \asd -methods also can be numerical,
computers and numerical methods can handle more complex and detailed
models. Such methods will in this work be referred to as the network
method and are the main focus of this work, as \asd -methods have been
discussed elsewhere
\citep[\eg][]{sundberg88, greulich03}.
In a real world case there are several factors that may influence the
TTC. Detours increasing the skidding distance may be necessary due
to \eg steep slopes, rock outcrops, soil types or environmental
values. Varying volume density may also affect the actual average
skidding distance. Another issue is that skidding time may be varying
due to terrain and soil type. \citet{nurminen06} found driving speeds from
$\SI{14.5}{\meter\per\minute}$ to $\SI{87.3}{\meter\per\minute}$
(loaded and empty), but did not relate this to terrain types.

In this work, the \asd -method and the network method are compared and
their limitations are analyzed in light of how they can be developed
further.
A forest parcel is a forest area which is meaningful to consider as a
whole. Reasons can be equal site index, uniform or uniformly aged
forest or simply that the parcel is a suitable silvicultural or
computational treatment unit. A forest parcel may thus be a stand, a
forest compartment, a grid point (or mesh) or other units used in
model formulations. It is noteworthy that \eg a mathematical model in
general will be valid for different sizes of forest parcels, but the
parcel size has impact on how the model parameters should be
calculated.

\section{Early methods based on average skidding distance}

The calculation of harvesting and forwarding costs in forestry
literature has been limited by the technology and techniques of each
era. Advances in maps and surveying techniques, and in software and
hardware have improved calculations. Early approaches predate
computers and were designed for calculation by
humans. \citet{greulich03} describes some of the early literature
refering to the \asd .

\citet{matthews42} treats \asd -estimates analytically.
His work is a commonly cited reference for harvesting cost
calculations. It describes cost calculations and optimization for
varying cases (\eg skidding and yarding, uphill and downhill,
different road and landing layouts) and problems (\eg road and landing
location, choice of equipment).  In this paper the focus is on TTC, and
the method for calculating TTC described by
\citet{matthews42} will be referred to as the Matthews' method.
The Matthews' method is simple in the sense that it
is designed for hand calculations and relies on the geometric shape of
the forest parcel under consideration. The mean unit TTC for a parcel,
$\bar{c}$, is given by
\begin{equation}
  \bar{c} = c_d d_c , \label{matthews.eq}
\end{equation}
where $c_d$ is the mean unit distance dependent cost and $d_c$ is
Matthews' estimate of the \asd .
The calculations of $d_c$ vary according to the assumptions made. If
all the wood is assumed to be transported to one landing,
$d_c = d(x_c, y_c)$ where the function
$d(x, y) = \sqrt{ \left( x -   x_l \right)^2 + \left( y -
y_l \right)^2 }$
is the distance from $(x, y)$ to a landing $(x_l, y_l)$.
If continuous landings are used, $d_c$ is found by dividing the parcel
in smaller parts according to the shape of the parcel.

\citet{suddarth64} derived another estimate of \asd\ by integrating
the function $d(x, y)$ across the parcel. If $A_p$ is the area of the
parcel $p$, the \asd\ is given by
\begin{equation}
  \asd = \frac{1}{A_p} \iint_p d(x, y) \ud A . \label{asd.def.eq}
\end{equation}
The \asd\ is in general not the same as the $d_c$ in
Equation~\eqref{matthews.eq}.
By replacing $d_c$ in Equation~\eqref{matthews.eq} with the \asd ,
$\bar{c}$ is given by
\begin{equation}
  \bar{c} = c_d \cdot \asd . \label{asd.cost.eq}
\end{equation}
The total TTC of harvesting a parcel is
\begin{equation}
  c = \bar{c} \cdot V , \label{asd.total.cost.eq}
\end{equation}
where $V$ is the timber volume of the parcel.

The integral in Equation~\eqref{asd.def.eq} can be formulated for any
parcel shape. Analytical solutions have been reported for some shapes
\citep[\eg][]{suddarth64, peters78}, but the derivations are in
general cumbersome or lengthy or both. Equation~\eqref{asd.def.eq}
has also been further extended for side slope
\citep{greulich80, greulich87, greulich89}, for rectilinear thinnings
\citep{greulich94} and continuous landings
\citep{greulich94a, greulich94b}.
The basis for TTC in the above references is
Equation~\eqref{asd.cost.eq}, but \citet{greulich87} formalized the
calculations further by introducing regression analysis. The harvest
of an area was assumed to consist of a number of turns with a distance
$\rho$ to the landing, and the \asd\ is the expected value of the
random variable $\rho$.
The calculations were further refined by \citet{greulich91}, who
calculated the expected yarding cost to be
%modified the TTC by replacing Equation~\eqref{asd.cost.eq} with
\begin{equation}
  E\{YC\} = \beta_0 + \beta_1 w E\{\rho\} + \beta_2 w^{2}E\{\rho^2\}
  . \label{greulich91.eq}
\end{equation}
The right hand side of Equation~\eqref{greulich91.eq} is a truncated
power series of order 2, approximating an unknown function. The latter
two terms can be considered the expected TTC. Likewise,
Equations~\eqref{asd.cost.eq}
%% ~and~\eqref{asd.total.cost.eq}
can be
considered the TTC part of a truncated power series of order 1 for the
same unknown function, and thus, Equation~\eqref{greulich91.eq} will
be more accurate.

The $w$ in Equation~\eqref{greulich91.eq} is a wander factor. An
extraction trail rarely follows the straight line to a landing, and
throughout the forestry literature a wander factor ($w$) has been used
for correcting the relation between TTC and \asd . Although
\citet{vonsegebaden64} included the wander factor and is frequently
credited the invention, the concept was mentioned earlier by
\citet{hughes30}. Usually, it is assumed that the harvest area is
divided in parcels with uniform wander factor.

Another approach for corrections to skidding distance is to include
obstacles when calculating TTC. This approach was used by
\citet{gibson75}, who found the skidding distance as the distance of
line segments avoiding the obstackle. A similar approach was used by
\citet{perkins79}, who included a sub-origin some distance from the
landing. The \asd\ was calculated either to the landing or the
suborigin, depending on the layout, and in the latter case the
distance from the sub-origin to the landing was included in the
\asd .


\section{The network method}

Models of terrain transportation based on \asd\ are in genral
analytical and possible to do by hand calculations. Such methods suits
well to productivity studies based on work cycles. Likewise, \asd
-methods are well adapted to forest parcels, the commonly used unit
both in forestry and forest research. Most forest planning problem
formulations require that input parameters (\eg TTC) are supplied by
experts. \citet{tan92} introduced the network method for TTC
calculations. The forest was modeled as a weighted graph, \ie a
regular network of grid points independent of forest parcels. Instead
of calculating the \asd\ of each grid point (the \asd -method), the
wood was assumed to be transported to one of the neighboring grid
points and recursively through grid point until a landing was
reached. For a network, there is a large number of possible paths
between two grid points, but finding the cheapest one is referred to
as the shortest path problem.

More precisely, the problem can be defined as follows.  Let the
terrain be given by a weighted graph $G=(V, E)$, where each vertex
(grid point) $v_i \in V$ represents a point in the terrain. The edges
$E$ link each vertex with its neighbors, and the unit cost of
transport between the neighbors $v_i$ and $v_j$ is $c(v_i, v_j)$.  A
path from vertex $v_0$ to vertex $v_n$ is given by $p = \langle v_0,
v_1, \dots, v_n \rangle$, and the unit cost of transporting timber on
this path is the sum of its constituent edges, given by
Equation~\eqref{sp_cost}.
\begin{equation}
  c(p) = \sum_{i=1}^n c(v_{i-1}, v_i) \label{sp_cost}
\end{equation}
The mean unit TTC of grid point $v_j$ can be found by minimizing
Equation~\eqref{cj.cost.eq}, \ie the shortest/cheapest path from
$v_j$ to a vertex $v_0$ that is a landing.
\begin{equation}
  c_j = \left\{
    \begin{array}{ll}
      \min \left\{ c(p) : v_j \stackrel{p}{\leadsto} v_0 \right\} &
        \begin{array}{l}
          \text{if there is a path} \\
          \quad \text{from $v_j$ to $v_0$}
        \end{array} \\
      \infty &
        \begin{array}{l}
          \text{otherwise} \\
        \end{array} \\
    \end{array}
  \right. \label{cj.cost.eq}
\end{equation}
This problem formulation can be solved fast by standard combinatorial
mathematics (\eg by Dijkstra's shortest path
algorithm \citep{dijkstra59}).

The total TTC of harvesting a parcel is given by
\begin{equation}
  c = \sum_j V_j c_j , \label{p.ttc.eq}
\end{equation}
where $V_i$ is the timber volume at grid point $i$.
There are some studies in the literature using the shortest path
formulation given by Equation \eqref{sp_cost}--\eqref{cj.cost.eq}.
Although the method is largely the same, there are some variations of
how the cost $c(v_i, v_j)$ of transport to a neighbor
(Equation~\eqref{sp_cost}) is calculated.
\citet{tan92} calculated the cost as a function of distance and
terrain class.
\citet{contreras07} used a skidding cost derived from the regression
model by \citet{han05}, including distance and slope gradient uphill
or downhill.
In a more recent paper, \citet{contreras11} refined the skidding model
to also include a maximum roll limit.
\citet{chung08} used a cost based on distance.
\citet{sovde13} used a cost based on distance and penalties for roll
and pitch.

% grid size gets smaller

One advantage of the network method is that as long as the distance
between neighboring grid points is small, there is no need for
introducing a wander factor.


\section{Discussion}

Although the network method has been in use for more than 20 years,
the method has not been analyzed in depth. The topics of the cited
literature are: road planning \citep{tan92, chung08}, landing location
\citep{contreras07} and extraction trail layout
\citep{contreras11, sovde13}. As the calculation of TTC is only
a part of the studied problems, the network method is not discussed
much.

\subsection{The impact of uneven terrain}

%The \asd -method and the network method can be compared and to some
%extent evaluated by looking at how a double integral is defined.
Let $f(x, y)$ be a function of the TTC per area, incorporating all
the irregularities that may influence the TTC and also the timber
volume.
Without loss of generality, the parcel $p$ is assumed to be
rectangular. The parcel can be partitioned into $n \times m$
sub-rectangles in the $x$ and $y$ direction.
The TTC of a parcel is given by the sum of the cost of harvesting
each sub-rectangle,
\begin{equation}
  \hat{c} = \sum_{k=1}^n \sum_{l=1}^m f(x_k^*, y_l^*) \Delta A , \label{r.sum.eq}
\end{equation}
where $(x^*, y^*)$ is the center point of each sub-rectangle. By
increasing the number of sub-rectangles, the double sum approaches the
integral.
\begin{align}
  c &= \lim_{n, m \rightarrow \infty} \sum_{k=1}^n \sum_{l=1}^m
       f(x_k^*, y_l^*) \Delta A \label{c.r.sum.eq} \\
    &= \iint_p f(x, y) \ud A \label{c.int.eq}
\end{align}
Equation~\eqref{asd.total.cost.eq} follows directly from
Equation~\eqref{c.int.eq} if $f(x, y) = V \cdot c_d / A_p \cdot d(x,
y)$.  Finding the analytical function $f(x, y)$ is not
straightforward.  The timber from a point $(x, y)$ can be transported
along many extraction trails, maybe also to different landings. One
possibility is that the value of $f(x, y)$ is the cost of using the
cheapest extraction trail to the point $(x, y)$.  The \asd -method
assumes that the skidding distance is the straight line to a
landing. If there are obstacles, and the trail follows a known curve
$(x(t), y(t))$, the length of the curve can be found by integrating
along the curve. Also, finding the shortest curve of all the possible
extraction trail curves is possible. However, if the driving speed
varies, the problem of finding the fastest path turns into \emph{the
problem of the brachistochrone}, a more difficult problem of
variational calculus. The interested reader may find more details in
\citet{troutman96}. Whether the integral in
Equation~\eqref{c.int.eq} can be solved easily is a matter of the
function $f(x, y)$ and the shape of the parcel.

Equation~\eqref{p.ttc.eq} is the same as Equation~\eqref{r.sum.eq} if
the sums and indices are rearranged and $f(x_k^*, y_l^*)\Delta A = V_j
c_j$. If $c_j$ in Equation~\eqref{cj.cost.eq} is a good estimate of
$f(x, y)$, the problem of finding the TTC in uneven terrain can
readily be solved with the network method.


\subsection{Input data}

A model formulation or a method such as the \asd -method or the
network method relies on the input data.  There are numerous ground
based harvesting systems operating around the world and the
productivity varies. Traditionally, productivity studies in forestry
are based on time studies. A harvesting operation is observed and the
operation at hand is partitioned into part operations. Typically, the
harvesting operation is repetitive and the data is analyzed as
cycles. Time studies are time consuming and has some limitations,
though. There is a limit to how much information a person can
register, as well as to how detailed the data can be. Although data
such as position or terrain can be registered or maybe assessed
afterwards, a study relies on the predefined areas of interest. Recent
productivity studies of terrain transport
\citep[\eg][]{han05, nurminen06} still report cycle times. Such data
are suitable for the \asd -method, but not directly applicable for the
network method. Cycle times are average values where the observed
distances are larger than the raster size used in some of the later
publications using the network method and the estimates of driving
speed may be smoothed.

An interesting question is whether $c_j$ in
Equation~\eqref{cj.cost.eq} is a good estimate of $f(x, y)$. This is
in fact an open research question. Neither \citet{contreras11}
nor \citet{sovde13} found studies on how driving speed is affected by
the micro terrain. Hopefully, such studies will appear soon. Sensors
(\eg accelerometers, gyros and gps) are available at budget prices,
and high accuracy DTMs and inventories are widely used in forestry.
Some other data that may
influence the TTC (\eg soil data) is not available at the same
level of accuracy, but nevertheless, the prospects for future
improvements are good.

\subsection{Computational complexity}

A model is a simplification of the real world, and may be arbitrarily
close to the real world. In addition, some models are itractable by
computers \citep{gandj79}, and can not be solved to optimality for
large problem instances. The computational complexity of the Matthews'
method is in general $\mathcal{O}(n)$. If an integral for the \asd
-method can be calculated, this method is also $\mathcal{O}(n)$. In
contrast, if the network method is implemented using Dijkstra's
algorithm with Fibonacci heaps, the computational complexity is
$\mathcal{O}(n \log(n))$ \citep{fredman87}. This may limit the problem
instance size solvable by the method.

In most publications, the TTC is needed as parameters in other
models. However, it is possible to include analytical TTC calculations
in larger models. \citet{greulich12} found the near-optimal location
of two landings using a combination of continuous and numerical
techniques.



\section{Conclusions}

The Matthews' method predates computers and was designed for hand
calculations. It is simple to use and understand, and is occasionally
the basis of harvesting cost calculations in the literature. Also, the
method is probably the preferred method by forest managers, forest
owners or contractors when estimating harvesting costs or negotiating
contracts.

The \asd -method often yields lengthy derivations and adaptations
according to landing layout and parcel shape
\citep[\eg][]{greulich89}. Improving the method to include uneven
terrain may be difficult or impossible in practice, at least there
seems to be no attempts at this in the literature. However, once a
formula is derived, it can be used in the same manner as the Matthews'
method. Also, cumbersome integral calculations can be estimated
numerically (\ie by Equation~\eqref{r.sum.eq} on a grid of the
parcel). This approach was suggested by \citet{suddarth64}, who also
found that dividing the parcel in eight parts yielded estimates
deviating less than $1 \%$ from the \asd .

Another reason for using Matthews' method or the \asd -method is that
most of the productivity studies in the literature are reporting cycle
times, suitable for the methods.

The network method is a promising method for the calculation of
TTC. Once implemented, different parcel shapes can readily be
calculated. In addition, the method is more flexible than the \asd
-method. More complex TTC-functions $f(x, y)$ in
Equation~\eqref{c.int.eq} can be handled without affecting the
computation time much.


\subsection{Future research}

All TTC models discussed here are simplifications of a real world
problem, and thus incorrect. Barring \citet{contreras07}, who included
a comparison of their method to the method of \citet{greulich91} for
some polygons, there are few publications discussing errors in the
models or how well they describe the real world.

The network method has been applied to several forestry problems in
the cited literature, even though studies correlating driving speed
with \eg micro topography and detailed inventories are lacking in the
literature.

Hopefully, future productivity studies can shed light to which method
is yielding better solutions for different problems, and also
supplying better input data for the models.

\section*{Acknowledgments}
Thanks to the Norwegian Research Council (grant NFR186912/I30)
for funding this study. Also thanks to two anonymous reviewers for their time and comments, which
improved the manuscript.

%\addtolength{\textheight}{-.35truein}
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%\bibliography{ref_ground_ttc}

\input{References.bbl}

%% \vspace{.2in}
%% \section*{Biographical Note}

\label{docend}
\end{document}



