\section{Introduction}
Absolute ballistic gravimeters measure gravity acceleration by tracking the free motion of the test mass in the gravity field. Having the distances $\{S_1, ..., S_N\}$ covered by the test mass over the time intervals $\{T_1, ..., T_N\}$, the acceleration
is found as parameter of some trajectory model fitted to the data pairs $(T_i, S_i)$ by methods of the regression analysis. One of the commonly used models describes unperturbed vertical motion of the test mass in the gravity field
\begin{equation}
\label{eq_z_i}
z_i = z_0 + V_0\, T_i + g\, T_i^2/2.
\end{equation}
The expression (\ref{eq_z_i}) represents a linear regression model, because it's defined by a linear combination of the estimated parameters $z_0, V_0, g$. The estimates are usually found by minimising the sum of the squared discrepancies between the measured data and the model:
\begin{equation}
\label{eq_OLS}
(z_0, V_0, g): \;\; \sum  \epsilon_i^2 \rightarrow  \textrm{min}, \;\;\; \textrm{where} \;\;\; \epsilon_i = S_i - z_i.
\end{equation}
Minimised here is the square norm ($L_2$-norm) of the vector $\bepsilon=\{\epsilon_i\}$, which is a special case of the more general $L_p$-norm
\begin{equation}
\label{eq_p_norm}
\|\bepsilon\|_p = \left(\sum |\epsilon_i|^p\right)^{\rfrac{1}{p}}, \; p\ge 1.
\end{equation}
The tradition of minimising the sum of squares (i.e. finding the \emph{least squares}) has a long and reach history dating back to the beginning of the 19-th century. At that time the main reason for using the least squares was their computational simplicity, as for the $p$=2 the fitting of a linear model is reduced to finding a unique solution for a full-rank system of linear equations.  At any other $p$ the equations become non-linear in estimated parameters and require involved computations to find the solution.

Another reason for using the least squares relates to a different kind of linearity: linearity of the estimates with respect to the dependent variable ($S_i$ in our case), which implies no bias of the estimates.
Moreover, according to the Gauss-Markov theorem, of all the linear/unbiased estimates, the least-squares deliver the most precise one, in case the errors in the observations are uncorrelated, have zero mean and constant variance. This is especially important in metrology, where the bias may lead to an incorrect estimates of the measurand and its uncertainty.

Nowadays, when advanced computing is no longer a problem, the $L_p$-approximation with $p \ne 2$ is widely used \cite{pennecchi2006}--\cite{ scales1988a}.
Many researches have noticed that the value of $p$ delivering the lowest variance to the $L_p$-estimates decreases as the kurtosis\footnote{The kurtosis $\upbeta_2$ characterises the spread of the distribution over the possible values of the random variable. It's defined as $\upmu_4/\upsigma^4$, where $\upmu_4$ is the fourth central moment, $\upsigma$ is the standard deviation.}  of the error distribution grows; there are formulae suggested for this dependence \cite{money1982, sposito1983}.
Most results have been obtained for the values $1 < p < 2$, that provide better estimates when the noises have heavily-tailed distributions.
In trajectory tracking, however, more prevalent are limited-band noises, for which the properties of the $L_p$-approximation are less known.

The purpose of this work is to investigate the feasibility and possible advantages of approximating the gravimeter's test mass trajectory in different $L_p$-norms.

The paper has the following structure. The chapter \ref{sec_IRLS} discusses implementation of the Iteratively Re-weighted Least Squares (IRLS) for the $L_p$-norm approximation. The chapter \ref{sec_simulation} describes the Monte-Carlo simulation of gravity estimates in $L_p$-norm. A real-life example of improving the estimates by the $L_p$-norm approximation is discussed in the chapter \ref{sec_examples}. We summarise our study in the chapter \ref{sec_conclusions}.
\section{The $L_p$-norm approximation by the Iteratively Re-weighted least-squares}
\label{sec_IRLS}
We implement the $L_p$-norm approximation by performing the weighted least-squares (WLS) approximation, in which the weights are defined in a special way \cite{burrus2012}.
For the model (\ref{eq_z_i}), the WLS solution is determined by
\begin{equation}
\label{eq_WLS}
\bi{\bxi} = \bi{A}^+ \bi{S},
\end{equation}
where\\
$\bi{\bxi}=(z_0, V_0, g)^T$ is the vector of the unknown parameters,\\
$\bi{S}=(S_1, S_2, ..., S_N)^T$ is the vector of the observations,\\
$\bi{A}^+$ is the Moore-Penrose pseudo-inverse matrix calculated as
\begin{equation}
\label{eq_Aplus}
\bi{A}^+=(\bi{A}^T\bi{V}\bi{A})^{-1}\bi{A}^T\bi{V},
\end{equation}
where
\begin{equation}
\label{eq_A}
\bi{A} =
\left(
\begin{array}{ccc}
1       & T_1    & T_1^2/2 \\
1       & T_2    & T_2^2/2 \\
\vdots  & \vdots & \vdots  \\
1       & T_{N}  & T_{N}^2/2
\end{array}
\right)
\end{equation}
is the experiment design matrix,\\
\begin{equation}
\label{eq_V}
\bi{V}=\textrm{diag}(v_1, v_2, ..., v_N)
\end{equation}
is the diagonal matrix of weights.
The approximation in the $L_p$-norm is achieved when the weights $v_i$ are related to the discrepancies $\epsilon_i$ like \cite{burrus2012}
\begin{equation}
\label{eq_vi}
v_i = |\epsilon_i|^{p-2}.
\end{equation}
For $p$=2, the weights (\ref{eq_vi}) turn into units, yielding the standard least-squares solution.
The formula (\ref{eq_vi}) cannot be used directly to find the weights $v_i$, because the weights need to be known before the approximation, while the residuals $\epsilon_i$ are known only after the approximation is done. Because of that, the WLS is usually applied several times leading to the Iteratively Re-weighted Least Squares. The iterations start with the weights $v_i$=1, i.e. with the regular least-squares. The residuals $\epsilon_i$ are then used to build the new weights
\begin{equation}
\label{eq_V_by_e}
\bi{V}=\textrm{diag}(|\epsilon_1|^{p-2}, |\epsilon_2|^{p-2}, ..., |\epsilon_N|^{p-2}),
\end{equation}
which are then applied to approximate the original data. The iterations can be repeated, but the process is known to sometimes have a problematic convergence. Though different approaches exist to improve the convergence  \cite{burrus2012}, the simulations found that two iterations always produce a sufficiently accurate for our purposes approximation, with the bias not discernible in random noise.
For better computational stability we normalised the weights (\ref{eq_vi}), so they would range from 0 to 1:
\begin{equation}
v_i = \left(\frac{|\epsilon_i|}{M} \right)^{p-2}, \;\;\; p \ge 2,
\end{equation}
where
\begin{equation}
M = \max|\epsilon_i|.
\end{equation}
If $p<2$, the powers in (\ref{eq_vi}) become negative, and to prevent zero division errors another modification of the weights was necessary:
\begin{equation}
\label{eq_threshold}
v_i = \left(\frac{\max (|\epsilon_i|, rM)}{rM}\right)^{p-2} ,\qquad p<2,
\end{equation}
where $r=0.001$ is a regularisation parameter establishing the threshold of smallness as fraction $r$ of the maximum residual $M$. According to (\ref{eq_threshold}), the residuals below the threshold get replaced with the threshold value $rM$. A similar way of regularisation is implemented in \cite{scales1988a}.
\section{Modelling the properties of the absolute gravity estimates in the $L_p$-norm}
\label{sec_simulation}
\subsection{Organisation of the simulation experiment}
\label{ssec_org_exp}
For the simulations, we added random perturbations to the parabolic trajectory (\ref{eq_z_i}) with known acceleration $g=$ 9.8 ms$^{-2}$\footnote{Other parameters were $z_0$ = 1 nm and $V_0$ = 0 ms$^{-1}$. Zero initial velocity was chosen to maximise the difference between the experiment designs.}, and then recovered the acceleration from the perturbed parabola using the $L_p$-norm approximation (\fref{fig_SIMULATION}). The values of $p$ ran from 1 to 6 with the increment of 0.1, so the standard least-squares solution ($p$=2) was also available from the simulation and was used for comparison.
\begin{figure}[ht]
\centering
\includegraphics[height=100mm]{fig_SIMULATION.png}
\caption{Numerical simulation of the properties of the ballistic trajectory approximation in the $L_p$-norm.}
\label{fig_SIMULATION}
\end{figure}
To build the parabola, 700 data points $\{T_1, ..., T_{700}\}$ representing 699 time intervals with the  common start at $T_1$ were used, spanning the total time $T_{700}$ = 0.22 s. We considered 3 distributions of the $T_i$-points (experiment designs) found in different types of absolute gravimeters (\fref{fig_PLANS}). The EST (equally spaced in time) design has the uniform distribution of the $T_i$-points and is used in both free-fall and rise-and-fall gravimeters. The ESD$_{_\textrm{\tiny{FF}}}$ design has the minimal density at the start, linearly increasing towards the end of the interval. The ESD$_{_\textrm{\tiny{RF}}}$ design has the minimal density in the centre, linearly increasing towards the start and the end of the interval. These two designs are found in free-fall (FF) and rise-and-fall (RF) gravimeters with the levels equally spaced in distance (ESD).
\begin{figure}[ht]
\centering
\includegraphics[height=65mm]{fig_PLANS.pdf}
\caption{Experiment designs used in the simulation:\\
EST -- equally spaced in time. Time intervals do not depend on the trajectory, can be used in both free-fall and rise-and-fall gravimeters;\\
ESD$_{_\textrm{\tiny{FF}}}$ -- equally spaced in distance for the free-fall trajectory;\\
ESD$_{_\textrm{\tiny{RF}}}$ -- equally spaced in distance for the rise-and-fall trajectory.
}
\label{fig_PLANS}
\end{figure}
For every type of perturbation we simulated 3000 drops and, after approximating the trajectory in different $L_p$-norms, analysed the results. Every realisation of the simulated noise was used in all three experiment designs by assigning either EST, ESD$_{_\textrm{\tiny{FF}}}$, or ESD$_{_\textrm{\tiny{RF}}}$ realisations of the vector $\bi{T}=\{T_1, ..., T_{700}\}$ to the same noise vector $\bepsilon=\{\epsilon_1, ... , \epsilon_{700} \}$.

The simulations and data analysis were performed with the MATLAB$^{\textregistered}$ software.
\subsection{Simulation results for the random uncorrelated perturbations}
The trajectory noise in absolute gravimeters comes from a number of sources, such as ground vibrations, laser stabilisation, digital signal acquisition,  fringe counting, etc., so the combined noise most often is of the Gaussian type. Sometimes, however, individual sources can dominate, creating other noise distributions. We did the simulations for several noise types, all having the standard deviation of 1 nm.

For all noise distributions, the distribution of the $g$-estimates was normal, for any $p$.
\begin{figure}[ht]
\centering
\includegraphics[height=80mm, angle=90]{fig_3_PLOTS_R.png}
\caption{Three ways of graphical representation of the simulation results (noise: Uniform, trajectory sampling: EST)\\
a -- mean gravity estimates shown relatively to the known value of $g$ = 9.8 ms$^{-2}$ (1 $\upmu$Gal = $10^{-8}$ ms$^{-2}$) and their standard deviations as error bars, for different $L_p$-norms;\\
b -- variance of the $L_p$-norm estimates relative to the variance of the standard least-squares solution;\\
c -- relative efficiency of the estimates as ratio of the variance of the $L_2$-estimate to the variance of the $L_p$-estimate -- shows the number of times the variance has decreased compared to the standard least-squares.
}
\label{fig_3_PLOTS}
\end{figure}
For every noise distribution there existed an optimal value of $p$ delivering minimal variance of the estimate (\fref{fig_3_PLOTS}a). The optimal $p$ ranged from 1.4 to 3.3 depending on the noise distribution. This result deviates from the theory \cite{nyquist1983} predicting no optimal $p$ for at least the Uniform distribution.

For the numeric characteristic of the distribution shape, we used  anitkurtosis\footnote{
The antikurtosis $\upchi$ relates to the kurtosis $\upbeta_2$ as $\upchi=1/\sqrt{\upbeta_2}$ \cite{novitskiy1991}.
}, because it can only assume values from 0 to 1 for any distribution. We found it more convenient than the traditionally used kurtosis, which can range from 1 to infinity.
The optimal $p$ increased with antikurtosis (\tref{tab_RESULTS}), but not always the variance at the optimal $p$ improved substantially. We visualised the improvement by plotting the variance for every $p$ (\fref{fig_3_PLOTS}b) divided by the variance of the standard least-squares. The factor of the variance improvement is shown on the plot of the inverse ratio (\fref{fig_3_PLOTS}c), which is equivalent to the relative efficiency of the  $L_p$-estimate compared to the standard least-squares one.
The relative efficiency plots for all simulated noises and experiment designs are presented on the \fref{fig_6_PLOTS}.
As seen on the figure and at the \tref{tab_RESULTS}, significant improvement of the precision is achieved for the Uniform and Arcsine distributions, if the norm  $L_{3.3}$  is used for the parabola fitting instead of the $L_2$.

As non-linearity of the $L_p$-estimates can entail bias, we did the drop-to-drop comparison of the $L_2$-estimates (theoretically unbiased) with the $L_p$-estimates at the optimal $p$. No statistically significant divergence of the two estimates was detected. We thus confirmed no bias for our approximation procedure, the fact known for the $L_p$-estimates in case of the symmetric noise distributions \cite{money1982, nyquist1983}.
\begin{figure}[ht]
\centering
\includegraphics[height=80mm, angle=90]{fig_6_PLOTS_REV.png}
\caption{
Relative efficiency of the $L_p$-norm estimates (Y-axis) for different noise types. The efficiency is calculated as ratio of the $L_2$-estimate variance for the EST-design to other $L_p$-estimates for all designs.\\
\full EST, \dotted ESD$_{_\textrm{\tiny{RF}}}$,
\longbroken ESD$_{_\textrm{\tiny{FF}}}$.

}
\label{fig_6_PLOTS}
\end{figure}
\begin{table}
\caption{Results of the simulation of the absolute gravity estimates by approximating the trajectory in different $L_p$-- norms. The case of random uncorrelated perturbations of the trajectory.}
\begin{indented}
\item[]\begin{tabular}{@{}lccccc}
\br
&Anti-&&\centre{3}{Rel. eff.$^{\rm a}$ for the best $p$}\\
\ns
Noise & kurtosis &  Best &\crule{3}\\
Distribution & $\upchi$ & $p$  & EST & ESD$_{_\textrm{\tiny{FF}}}$ & ESD$_{_\textrm{\tiny{RF}}}$ \\
\mr
Laplace &   0.41 & 1.4 & 1.3 & 1.3 & 1.3\\
Normal &   0.58 &  2.0 & 1.0 & 1.0 & 1.0\\
Triangle &  0.65 &  2.5 & 1.1 & 1.1 & 1.1\\
Uniform &  0.75 &  3.3 & 2.8 & 2.8 & 2.8\\
Arcsin &  0.82 & 3.3 & 7.0 & 7.0 & 7.1\\
\br
\end{tabular}
\item[] $^{\rm a}$Determined as ratio of the variance of the $L_2$-estimate to the variance of the $L_p$-estimate for the same experiment design.
\end{indented}
\label{tab_RESULTS}
\end{table}
\subsection{Simulation results for the ``harmonic in the Gaussian noise'' perturbations}
According to the \tref{tab_RESULTS}, the $L_p$--estimates with $p>2$ can provide a significant gain in the efficiency when the errors are distributed by the Uniform or Arcsine law. In trajectory tracking such errors are often caused by harmonic perturbations of the reference frame. We modelled the perturbations by the sum of a sinusoid and Gaussian noise. In every simulated drop, a sinusoid with a given frequency $f$ and amplitude of 1.41 nm was assigned an initial phase randomly taken from the uniform distribution on $[0, 2\pi]$, and then sampled according to the experiment design. The sinusoid samples were added by the Gaussian noise, one noise realisation for all three experiment designs. The signal-to-noise ratio (SNR) is defined as quotient of the standard deviations of the sinusoid and the noise, so the infinite SNR means no random noise.

The results of the simulation for the frequencies of 17 Hz, 35 Hz, 55 Hz and the SNR values of $\infty$, 100, 10, 2 are shown in the \tref{tab_RESULTS_SIN}. The antikurtosis $\upchi$ of the perturbation was from 0.70 to 0.82, typical for the range occupied by the Uniform and Arcsine distributions. The resulting $g$--estimates have passed the normality test (Lilliefors , 5\%) only for some low SNR values. The optimal $p$ and relative efficiencies  for which the normality was observed are shaded in the \tref{tab_RESULTS_SIN}. Like for uncorrelated noises (\tref{tab_RESULTS}), there always existed an optimal $p$ delivering the lowest value to the variance of the $L_p$-estimate. The optimal $p$ was between 2.8 and 3.8, yielding the variance decrease of the estimate between 1.2 and 30.6 times. The values of $p$ and the efficiency gain have a complicated dependence on the signal frequency, the experiment design, and the SNR, so the values have to be evaluated experimentally in every practical case.

\begin{table}
\caption{Results of the simulation of the absolute gravity estimates by approximating the trajectory in different $L_p$-- norms. The trajectory perturbations are ``harmonic in the Gaussian noise.''}
\begin{indented}
\lineup
\item[]\begin{tabular}{@{}ccccccccccc}
\br
&&\centre{3}{EST}&\centre{3}{ESD$_{_\textrm{\tiny{FF}}}$}&\centre{3}{ESD$_{_\textrm{\tiny{RF}}}$}\\
\ns
$f$/Hz & SNR & \crule{3}&\crule{3}&\crule{3}\\
 & & $\upchi$ & $p$ & rel.eff.$^{\rm a}$& $\upchi$ & $p$ & rel.eff.$^{\rm a}$& $\upchi$ & $p$ & rel.eff.$^{\rm a}$\\
\mr
17 & $\infty$ & 0.81 & 3.6 & 5.5 & 0.81 & 3.5 & 17.5 & 0.78 & 3.4 &\01.5\\
   & 100      & 0.81 & 3.6 & 5.8 & 0.81 & 3.5 & 16.9 & 0.78 & 3.4 &\01.5\\
   & \010     & 0.80 & 3.6 & 5.5 & 0.80 & 3.5 & 17.7 & 0.78 & 3.4 &\01.6\\
   & \0\02    & 0.70 &\cc 3.6 &\cc 2.7 & 0.70 &\cc 3.8 &\cc \07.4 & 0.68 & 3.3 &\01.2\\
\\\ns\ns
35 & $\infty$ & 0.82 & 3.5 & 29.3 & 0.82 & 3.4 & 22.2 & 0.82 & 3.6 & 14.9\\
   & 100      & 0.82 & 3.5 & 29.5 & 0.82 & 3.4 & 23.2 & 0.82 & 3.5 & 15.1\\
   & \010     & 0.81 & 3.5 & 22.0 & 0.81 & 3.4 & 19.6 & 0.81 & 3.6 & 13.7\\
   & \0\02    & 0.70 &\cc 3.5 &\cc\02.7 & 0.70 &\cc 3.4 &\cc\03.3 & 0.70 &\cc 3.5 &\cc\03.3 \\
\\\ns\ns
55 & $\infty$ & 0.82 & 3.3 & 16.0 & 0.82 & 3.3 & 28.5  & 0.82 & 3.3 & 25.8\\
   & 100      & 0.82 & 3.3 & 15.8 & 0.82 & 3.3 & 28.8 & 0.82 & 3.3 & 25.6\\
   & \010     & 0.80 &\cc 3.3 &\cc 15.6 & 0.80 &\cc 3.3 &\cc 30.6 & 0.81 & 3.3 & 25.3\\
   & \0\02    & 0.70 &\cc 2.8 &\cc\01.3 & 0.70 &\cc 3.1 &\cc\02.3  & 0.70 &\cc 3.1 &\cc\02.2 \\

\br
\end{tabular}
\item[] $^{\rm a}$Determined as ratio of the variance of the $L_2$-estimate to the variance of the $L_p$-estimate for the same experiment design.
\end{indented}
\label{tab_RESULTS_SIN}
\end{table}
\section{Example of using the $L_p$-estimates on real data}
\label{sec_examples}
The field measurement of absolute gravity are often performed under the harsh geophysical conditions that deteriorate the performance of the instruments. Oftentimes it's not possible to repeat the measurements or improve the conditions, so the researches have to make the best possible gravity estimates using the available data.
As example, we consider a series of measurements taken by the TAG-1 absolute gravimeter developed by the Earthquake Research Institute of the University of Tokyo. The instrument features a laser interferometer with built-in accelerometer for the independent measurement and correction of the ground vibrations, and a homodyne quadrature fringe signal detector \cite{araya2014, svitlov2014}. The free-fall trajectory of about 12 cm is sampled with the frequency of 10 MHz (EST-design). A typical drop-to-drop scatter of the $g$-values at the station ‘Esashi’ is about 20 $\upmu$Gal\footnote{1 $\upmu$Gal = $10^{-8}$ ms$^{-2}$}, so the standard deviation of the mean below 1 $\upmu$Gal is reached in several hundreds drops. The bandwidth of the built-in accelerometer is up to 4 Hz, so the increased intensity of the high-frequency seismic noise could significantly disturb the measurements.
The figure \ref{fig_S_L2}(a) shows one such noisy
set of gravity estimates taken on December 23, 2011 at the `Esashi' gravity station, after applying standard tidal, atmospheric, and polar motion corrections. To obtain the $L_p$--estimates, the trajectory was decimated to 2 kHz, the standard speed-of-light and gradient corrections were applied to the time and distance coordinates respectively, then the gravity was found using the standard least-squares approximation.
The set fails the normality test, making the standard `three-sigma' rejection too restrictive for the obtained heavily-tailed distribution of the $L_2$-estimates. However, neither four, nor five sigmas made much difference to the mean $g$-value or its standard deviation, which remained at about 3 $\upmu$Gal (\tref{tab_appxs}).

\begin{figure}[ht]
\centering
\includegraphics[height=75mm]{S_L2.pdf}
\caption{Improving precision of the absolute gravity estimates by using the $L_p$-norm approximation of the free-fall trajectory:\\
(a) standard least-squares approximation ($p$=2), the bars are for the 3$\upsigma$, 4$\upsigma$, and 5$\upsigma$; \\
(b) approximation for $p$=3.5, the bars are for the 3$\upsigma$, 4$\upsigma$, and 5$\upsigma$;\\
(c) mixed $p$=2/$p$=3.5 approximation based on the antikurtosis of the residuals: black dots -- Gaussian group ($p$ = 2), grey dots -- Arcsine group ($p$ = 3.5).
}
\label{fig_S_L2}
\end{figure}
We then approximated the trajectory in the $L_p$-norm for $p$ ranging from 1 to 6 (see section \ref{sec_IRLS}). For each $p$ we obtained the variance of the set gravity estimate and plotted the relevant relative efficiency (\fref{fig_EX_EFF}). According to the plot (\fref{fig_EX_EFF}(a), curve~1), the variance at $p$=3.5 decreased several times w.r.t. $p$=2, reducing the standard deviation of the mean to about 1 $\upmu$Gal (\tref{tab_appxs}).
\begin{figure}[ht]
\centering
\includegraphics[height=60mm]{EX_EFF.pdf}
\caption{Relative efficiency of the gravity estimates with the $L_p$-norm approximation of the trajectory:\\
(a), 1: The entire set with the $5\upsigma$ censoring. The variance of the $L_p$-approximation for $p$=3.5 decreases 10 times.\\
(a), 2: The portion of the set with high antikurtosis values. The variance of the $L_p$-approximation for $p$=3.5 decreases 12.5 times. $3\upsigma$ censoring applied.\\
(b): The Gaussian portion of the set. The standard least-squares ($p=2$) are close to optimal. $3\upsigma$ censoring applied.
}
\label{fig_EX_EFF}
\end{figure}
Based on the simulations (\sref{sec_simulation}), the peak of the efficiency at $p>2$ is suggestive of the residuals with non-Gaussian distributions. We investigated the residuals and found  the values of antikurtosis ranging from 0.487 to 0.813, which corresponds to the wide range of distributions -- from Laplace to Arcsine (\fref{fig_AK_SET}).
\begin{figure}[ht]
\centering
\includegraphics[height=100mm]{AK_SET.pdf}
\caption{Antikurtosis composition of the noise in the data set.\\
The histogram on the right-hand side shows the distribution of the antikurtosis for the noises found in the set's drops. The histograms in the middle and the plots on the left-hand side show examples of the noises present in the set. Due to the inclement seismic conditions, the noises cover a wide range of distributions, from Laplace to Arcsine.
}
\label{fig_AK_SET}
\end{figure}
The likely reason for the non-Gaussian errors was resonant oscillations of the reference mirror caused by the excessive seismic noise.
The most prevalent in the set were residuals with the Gaussian and Arcsine distributions. According to the simulations (\sref{sec_simulation}), the most efficient $L_p$-approximations for the Arcsine-noised trajectory were achieved for the values of $p$ between 3 and 4, which explains the decrease of the uncertainty. However, for the Gaussian noises the approximation with $p$ in that range can be several times less efficient than the standard least-squares (\fref{fig_6_PLOTS}, second from the top). This led us to the idea of splitting the set into two parts with predominantly Gaussian and Arcsine noises, and independent optimising the $L_p$-norm on each set. We have split the groups at the antikurtosis $\upchi$  = 0.65, about in the middle of two peaks at the \fref{fig_AK_SET}. Gravity estimates in each group have passed the normality test (Lilliefors , 5\%), so further processing was carried out with the $3\upsigma$ rejection, each group with its own $\upsigma$.  The splitting has increased the efficiency of the estimate for the Arcsine group with $p$=3.5 from 10.0 to 12.5 (\fref{fig_EX_EFF}(a), 2). For the Gaussian group we used the standard least-squares with  $p$=2.0, which helped to avoid the decrease of the efficiency at $p$=3.5 for that group (\fref{fig_S_L2}(b)). Combining the estimates for both groups has reduced the standard deviation of the weighted mean to 0.76 $\upmu$Gal (\fref{fig_S_L2}(c)).
The gravity estimates obtained with different approximations are summarised in the \tref{tab_appxs}.
\begin{table}
\caption{Estimates of the absolute gravity value obtained with different modes of the approximation of the free-fall trajectory }
\begin{tabular}{@{}lccccc}
\br
Mode of approximation & Rejection & Accepted & Set mean & Set std.dev.& Mean's std.dev.\\
of the trajectory & limits & drops & $\bar{g}$ / $\upmu$Gal & $\bar{\upsigma}$ / $\upmu$Gal & $\bar{\upsigma}$ / $\upmu$Gal \\
\mr
Whole set, $L_2$ & $ 3\upsigma$ & 784 & 980 121 363.52 & 80.44  & 2.87  \\
                 & $ 4\upsigma$ & 803 & 980 121 362.38 & 95.24 & 3.36  \\
                 & $ 5\upsigma$ & 811 & 980 121 363.96 & 107.31 & 3.77  \\
\\\ns\ns
Whole set, $L_{3.5}$ & $ 3\upsigma$ & 802 & 980 121 362.70 &  29.51 & 1.04 \\
                     & $ 4\upsigma$ & 817 & 980 121 363.31 & 32.67 & 1.14 \\
                     & $ 5\upsigma$ & 820 & 980 121 362.76 & 33.88 & 1.18 \\
\\\ns\ns
Split groups, $L_2$/$L_{3.5}$
& $ 3\upsigma/3\upsigma$ & 396/406 & 980 121 364.50$^{\rm a}$ & 16.37/38.18   & 0.76$^{\rm b}$  \\
\br
\end{tabular}
$^{\rm a}$ Weighted average of two groups, the weights are the reciprocals of the groups' variances\\
$^{\rm b}$ Standard deviation of the weighted average

\label{tab_appxs}
\end{table}

\section{Conclusions}
\label{sec_conclusions}
We have simulated the approximation of the ballistic trajectory in different $L_p$- norms, in view of using it to improve the absolute gravity estimates. To perform the approximation, the Iteratively Re-weighted Least Squares method was implemented. The estimates were simulated for three different plans of the experiment corresponding to two types of the ballistic trajectory (free-fall, rise-and-fall) and two schemes of data location (equally spaced in time or in distance). Each plan was applied to several perturbation types, including the cases of random uncorrelated noises and a harmonic in the Gaussian noise. As an application example, the simulation results were used to improve the absolute gravity estimates obtained at the `Esashi' gravity station by the TAG-1 gravimeter.

The following conclusions can be drawn from the study.
\begin{enumerate}
\item The Iteratively Re-weighted least-squares with two iterations approximate the ballistic trajectory in the $L_p$-norm with the accuracy sufficient for the purposes of absolute gravimetry. No statistically significant bias of the estimate was found for the wide range of the trajectory noises.
\item The $L_p$-norm approximation of the ballistic trajectory with $p>2$ estimates gravity more precisely than the standard least-squares ($p$=2), if the trajectory perturbations deviate from the Gaussian towards more platykurtic distributions, i.e. those with flattish or concave shape.
\item The observed efficiency maximum for certain values of $p$ on platykurtic distributions disagrees with the prediction of the work \cite{nyquist1983}, where the monotonous increase of the efficiency with the norm power $p$ is obtained for the Uniform distribution. This could possibly be explained by a limited iterations of our IRLS procedure, resulting in a ``quasi'' rather than the ``exact'' $L_p$-approximation.
\item For the same measurement interval $T$ and fixed number of data points $N$, the EST sampling of the free-fall trajectory makes the gravity estimates about twice as efficient as those with the data equally spaced in distance, for any $p$. This points to an obvious way of improving the modern free-fall absolute gravimeters, which predominantly use the ESD scheme. For the symmetric rise-and-fall trajectory, the efficiency of the EST and ESD sampling is about the same.
    %
\end{enumerate}
The presented study is a first attempt to obtain the gravity estimates by approximating the ballistic trajectory in the $L_p$-norm with $p\ne 2$. Many important questions are yet to be answered by future work. The questions include, but are not limited to
\begin{enumerate}
\item The scope of the applicability of the $L_p$-norm approximation. For the purposes of absolute gravimetry we investigated only the estimates of the quadratic coefficient of the second-order polynomial model. The results can be applied to other models to build more efficient estimates in wider range of applications.
\item Auto-correlation of the residuals. The main source of the platykurtic distributions is the low-frequency noises. Our simulations of the correlated Arcsine noises and processing of the real data suggest that the efficiency of the $L_p$-approximation is defined mostly by the noise distribution and is not deteriorated by the auto-correlation of the residuals. Still, the influence of the auto-correlation calls for more thorough investigation, including the comparison to other algorithms of the low-frequency noise filtering.
\item Contaminated distributions within a drop.  Further investigations are desirable to better understand the properties of the $L_p$-norm approximation for the mixed noise distributions, such as harmonics in the Gaussian noise.
\item Contaminated distributions of the set and set splitting. When a set has drops with different noise types, a single value of $p$ may not yield the best $L_p$-approximations for all drops. In the example (\sref{sec_examples}) we have split the set in two at the centre of the antikurtosis histogram. Though we gained certain uncertainty decrease, there may exist better ways of set splitting, worth to be investigated.
\item The reasons of the departure of the simulated results from those predicted in \cite{nyquist1983} have to be investigated to better understand the properties of the IRLS algorithm of the $L_p$-approximation.
\end{enumerate}

\section*{References}


\section{Introduction}

Generalized parton distributions (GPDs) encode the informations about the three dimensional spatial structure of the proton as well as the spin and orbital angular momentum of the constituents.
The GPDs(see \cite{rev} for reviews on GPDs)  are off-forward matrix elements and appear in the exclusive processes like Deeply Virtual Compton Scattering (DVCS) or vector meson productions.  The GPDs being functions of three variables namely, longitudinal momentum faction $x$ of the parton, square of the total momentum transferred $t$  and  the longitudinal momentum transferred $\zeta$ so called skewness in the process contains more informations than the ordinary parton distribution functions(PDFs).

 The first moments of GPDs give the form factors accessible in exclusive processes whereas they reduce to PDFs in the forward limit.
 At leading twist,  we can define three generalized distributions in parallel to
  three PDFs, namely, the unpolarized,
 helicity, and transversity distributions.  Similar to transversity distribution, the generalized  transversity distribution  $F_T$ is also chiral-odd. In the most general way, $F_T$ is parametrized in terms of four chiral-odd GPDs, namely $H_T$ , $\widetilde{H}_T$ , $E_T$ , and $\widetilde{E}_T$~\cite{diehl01,Pasquini1,Diehl05,Burk3}. The chiral-odd GPDs give information on the correlation between the spin and angular momentum of quarks inside the proton.
 At zero skewness, by performing a Fourier transform (FT) of the GPDs with respect to the momentum transfer in the transverse direction $\Delta_{\perp}$, one obtains the impact parameter dependent parton distributions, which provide us the picture that how the partons of a given longitudinal momentum fraction ($x$) are distributed in impact parameter ($b_{\perp}$) or transverse position space. Unlike the GPDs themselves, impact parameter dependent parton distributions have probabilistic interpretation and satisfy the positivity condition~\cite{burk1,burk2,Burk3}. In the $t\to 0$ limit, the second
moment of the GPDs are related to the angular momentum contribution to the nucleon by the quark or gluon~\cite{ji}. The impact parameter dependent PDFs are transversely distorted when one considers transversely polarized nucleons. The transverse distortion can also be connected with Ji's angular momentum relation. An interesting interpretation of Ji's angular momentum sum rule~\cite{ji} for transversely polarized state was obtained in terms of the impact parameter dependent PDFs in~\cite{Burk3}. For the unpolarized quark, transverse distortion arises due to the chiral-even GPD $E$ which is related to the anomalous magnetic moment of the quarks. As far as the transverse distortion of transversely polarized quark distributions is concerned, the linear combination of chiral-odd GPDs ($2\widetilde{H}_T+E_T$) plays a  role similar to the GPD $E$ as for the unpolarized quark distributions. In the forward limit, a relation between the transverse total angular momentum of the quarks and a combination of second
moments
of $H_T$, $\widetilde{H}_T$ and $E_T$ has been proposed in~\cite{Burk3}, in analogy with Ji's relation.  $\widetilde{E}_T$ being an odd function of  $\zeta$, does not contribute at $\zeta=0$. For nonzero skewness one can also represent the GPDs in the longitudinal position space by taking FT of the GPDs with respect to $\zeta$~\cite{CMM1,CMM2,Dahiya07,BDHAV,chandan,Kumar1}.

Unlike the chiral-even GPDs, it is very difficult to measure chiral-odd GPDs. In a  very recent  COMPASS experiment \cite{Adolph},  exclusive production of  $\rho^0$ mesons by scattering muons off transversely polarized proton was measured. The target spin asymmetries measured in the experiment agree well with GPD-based model calculations which indicate the first experimental evidence  of chiral-odd GPDs, especially the transversity GPD $H_T$. There has been  proposals to get access to the chiral-odd GPDs through diffractive double meson production~\cite{Yu,Enberg}.
The role of transversity GPDs in leptoproduction of vector mesons~\cite{Goloskokov1} as well as in hard exclusive electroproduction of pseudoscalar mesons~\cite{Goloskokov2} have been investigated within the framework of the handbag approach. A simple model for the dominant transversity GPD $H_T$ based on the concept of double distribution has been proposed and has been used to estimate the unpolarized differential cross section for this process in the kinematics of the Jlab and COMPASS experiments in~\cite{Beiyad}. The chiral-odd GPDs in a constituent quark model have been studied for nonzero skewness using the overlap representation in terms of light-front wave functions (LFWFs) in~\cite{Pasquini1}. The general properties of the chiral-odd GPDs in a QED
model have been investigated in both momentum and transverse position as well as longitudinal position spaces~\cite{CMM1}; the impact parameter representation of the GPDs have been studied in a QED model of a dressed electron~\cite{Dahiya07} and  in a quark-diquark model~\cite{kumar} for $\zeta=0$. The Mellin moments of the transverse GPDs have been evaluated on lattice~\cite{gock,Hagler:2009,Hagler:2004,Bratt:2010}.

There have been numerous  attempts to gain insight into the  hadron structure by studying QCD inspired models as nonperturbative properties of hadrons are always very difficult to  evaluate from QCD first principle
 In this work, we consider  a phenomenological light front quark-diquark model recently proposed by Gutsche et. al~\cite{Gut} where
 the LFWFs  are modeled by the wave functions obtained from a soft-wall model in light front AdS/QCD correspondence~\cite{BT1,BT}.
This model is consistent with Drell-Yan-West relation which relates the high $Q^2$ behavior of the nucleon form factors and  the large $x$ behavior of the structure functions.

The paper is organized in the following way. In Section \ref{model}, a brief introductions about the nucleon LFWFs of quark-diquark model has been given. We present the overlap formalism of the chiral-odd GPDs and show the results for proton GPDs of $u$ and $d$ quarks in momentum space in Section \ref{chiral_gpds}. The GPDs in the transverse as well as the longitudinal impact parameter space are presented in Sections \ref{chiral_impact} and \ref{chiral_longi_impact}. Finally we summarize all the results in Section \ref{summary}.

\section {Light-front quark-diquark model for the nucleon}\label{model}
Here, we consider the quark-diquark model with a scalar  diquark.
The 2-particle Fock-state expansion for $J^z = + \frac{1}{2}$ and   $J^z = - \frac{1}{2}$  are then written as
 \begin{eqnarray}
  |P;+\rangle
& =& \sum_q \int \frac{dx~ d^2\textbf{k}_{\perp}}{2(2\pi)^3\sqrt{x(1-x)}} \bigg[ \psi^{+}_{+q}(x,\textbf{k}_{\perp})|+\frac{1}{2},0; xP^+,\textbf{k}_{\perp}\rangle \nonumber \\
 &+& \psi^{+}_{-q}(x,\textbf{k}_{\perp})|-\frac{1}{2},0; xP^+,\textbf{k}_{\perp}\rangle\bigg],\\

  |P;-\rangle
& =& \sum_q \int \frac{dx~d^2\textbf{k}_{\perp}}{2(2\pi)^3\sqrt{x(1-x)}} \bigg[\psi^{-}_{+q}(x,\textbf{k}_{\perp})|+\frac{1}{2},0; xP^+,\textbf{k}_{\perp}\rangle \nonumber \\
 &+& \psi^{-}_{-q}(x,\textbf{k}_{\perp})|-\frac{1}{2},0; xP^+,\textbf{k}_{\perp}\rangle\bigg],
  \end{eqnarray}
 where the $|\lambda_q,\lambda_s ; xP^+, \textbf{k}_{\perp} \rangle $ represents a two particle state with a quark spin $\lambda_q = \pm $, longitudinal momentum $xP^+$ and a spectator of spin $\lambda_s=0$ (scalar diquark). The states are   normalize as:
\begin{eqnarray}
\langle \lambda'_q,\lambda'_s; x'P^+,\textbf{k}'_{\perp}|\lambda_q \lambda_s;xP^+,\textbf{k}_{\perp}\rangle = \prod^2_{i=1} 16\pi^3 p^+_i \delta(p'^+_i-p^+_i)\delta^2(\textbf{k}'_{\perp i}-\textbf{k}_{\perp i}) \delta_{\lambda'_i \lambda_i},
\end{eqnarray}
and $\psi^{\lambda_N}_{\lambda_q q}$ are the light-front wave functions with nucleon helicities $\lambda_N = \pm $ and quark helicities $\lambda_q = \pm $. We adopt the generic ansatz for the quark-diquark model of the valence Fock state of the nucleon LFWFs at an initial scale $\mu_0=313$~MeV as proposed in \cite{Gut} :
\begin{eqnarray}\label{wf}
\psi^{+}_{+q}(x,\textbf{k}_{\perp})&=&\varphi^{(1)}_q(x,\textbf{k}_{\perp}),\nonumber\\
\psi^{+}_{-q}(x,\textbf{k}_{\perp})&=&-\frac{p^1+ip^2}{xM}\varphi^{(2)}_q(x,\textbf{k}_{\perp}),\nonumber\\
\psi^{-}_{+q}(x,\textbf{k}_{\perp})&=&\frac{p^1-ip^2}{xM}\varphi^{(2)}_q(x,\textbf{k}_{\perp}),\\
\psi^{-}_{-q}(x,\textbf{k}_{\perp})&=& \varphi^{(1)}_q(x,\textbf{k}_{\perp}),\nonumber
\end{eqnarray}
where $\varphi_q^{(1)}(x,{\bf k}_{\perp}) $ and $\varphi_q^{(2)}(x,{\bf k}_{\perp}) $ are the wave functions predicted by soft-wall AdS/QCD\cite{BT}
\begin{eqnarray}
\varphi_q^{(i)}(x,{\bf k}_{\perp})=N_q^{(i)}\frac{4\pi}{\kappa}\sqrt{\frac{\log(1/x)}{1-x}}x^{a_q^{(i)}}(1-x)^{b_q^{(i)}}\exp\bigg[-\frac{{\bf k}_{\perp}^2}{2\kappa^2}\frac{\log(1/x)}{(1-x)^2}\bigg],
\end{eqnarray}

where $\kappa$ is the AdS/QCD scale parameter which is taken to be $0.4$ GeV \cite{CM1,CM2}.
 The   parameters $a^{(i)}_q$ and $b^{(i)}_q$ with the constants $N^{(i)}_q$  are fixed by fitting the electromagnetic properties of the nucleons:
$a^{(1)}_u  = 0.020,~  a^{(1)}_d= 0.10,~
b^{(1)}_u = 0.022,~b^{(1)}_d=0.38,~
a^{(2)}_u=  1.05,~ a^{(2)}_d=  1.07,~
b^{(2)}_u= -0.15, ~b^{(2)}_d= -0.20,
N^{(1)}_u = 2.055,~ N^{(1)}_d = 1.7618,
N^{(2)}_u= 1.322, N^{(2)}_d = -2.4827 $.
\section{Chiral-odd generalized parton distributions}\label{chiral_gpds}
The chiral-odd GPDs are defined as off-forward matrix elements of the bilocal operator of light-front correlation functions of the
tensor current~\cite{diehl01}
\begin{eqnarray}
\label{gpd_eq}
& &\frac{1}{2}\int\frac{dz^-}{2\pi}e^{i \bar xP^+z^-}\langle
p', \lambda'|\bar{\psi}(-{z}/2)\sigma^{+i}\gamma_5
\psi({z}/2)|p, \lambda\rangle_{|_{z^+ = 0, \vec z_\perp=
0}}
\nonumber\\
& &= \frac{1}{2P^+}\bar{u}(p',
\lambda')\bigg[H^q_T\sigma^{+i}\gamma_5 +
\widetilde{H}^q_T\frac{\epsilon^{+i\alpha\beta}\Delta_\alpha
P_\beta}{M^2} +
E^q_T\frac{\epsilon^{+i\alpha\beta}\Delta_\alpha\gamma_\beta}{2M}
+
\widetilde{E}^q_T\frac{\epsilon^{+i\alpha\beta}P_\alpha\gamma_\beta}{M}\bigg]u(p,
\lambda),
\end{eqnarray}
where $i=1,2$ is a transverse index. $p$ $(p')$ and $\lambda$ $(\lambda')$ denote the proton momenta and the helicity of the initial (final) state of proton, respectively.
In the symmetric frame, the kinematical variables are
\begin{eqnarray}
P^\mu=\frac{(p+p')^\mu}{2}, \quad\quad \Delta^\mu=p'^\mu-p^\mu, \quad\quad \zeta=-\Delta^+/2P^+,
\end{eqnarray}
and $t=\Delta^2$. We choose the light-front gauge $A^+=0$, so that the gauge link appears in between the quark fields in Eq. (\ref{gpd_eq}) is unity. The GPDs which involve the quark helicity flip can be related to the following matrix elements \cite{diehl01,Pasquini1}
\begin{eqnarray}
  \label{amplitude}
A_{\lambda'+, \lambda-} &=&
\int \frac{d z^-}{2\pi}\, e^{i\bar x P^+ z^-}
  \langle p',\lambda'|\, {\cal O}_{+,-}(z)
  \,|p,\lambda \rangle \Big|_{z^+=0,\, \vec{z}_\perp=0} \, ,
\nonumber \\
A_{\lambda'-, \lambda+} &=&
\int \frac{d z^-}{2\pi}\, e^{i\bar x P^+ z^-}
  \langle p',\lambda'|\, {\cal O}_{-,+}(z)
  \,|p,\lambda \rangle \Big|_{z^+=0,\, \vec{z}_\perp=0} \, ,
\end{eqnarray}
with the operators $O_{+,-}$ and $O_{-,+}$ defined by
\begin{eqnarray}
{\cal O}_{+,-}
&=& \frac{i}{4}\,
  \bar{\psi}\, \sigma^{+1} (1-\gamma_5)\, \psi \,,\nonumber\\
{\cal O}_{-,+}
&=& - \frac{i}{4}\, \bar{\psi}\, \sigma^{+1} (1+\gamma_5)\, \psi.
\hspace{2em}
\label{eq:op_o}
\end{eqnarray}
Using  the reference frame where  the momenta $\vec p$ and $\vec p\,'$ lie in the $x-z$ plane, one can explicitly derive the following relations~\cite{diehl01}
\begin{eqnarray} \label{relations}
A_{++,+-} &=&   \epsilon\, \frac{\sqrt{t_0-t}}{2m} \left( \widetilde{H}^q_T
              + (1-\zeta)\, \frac{E^q_T + \widetilde{E}^q_T}{2} \right) ,
\nonumber \\
A_{-+,--} &=&   \epsilon\, \frac{\sqrt{t_0-t}}{2m} \left( \widetilde{H}^q_T
              + (1+\zeta)\, \frac{E^q_T - \widetilde{E}^q_T}{2} \right) ,
\nonumber \\
A_{++,--} &=& \sqrt{1-\zeta^2} \left(H^q_T + \
              \frac{t_0-t}{4 m^2}\, \widetilde{H}^q_T -
              \frac{\zeta^2}{1-\zeta^2}\, E^q_T +
              \frac{\zeta}{1-\zeta^2}\, \widetilde{E}^q_T \right) ,
\nonumber \\
A_{-+,+-} &=& - \sqrt{1-\zeta^2}\; \frac{t_0-t}{4 m^2}\, \widetilde{H}^q_T,
\end{eqnarray}
where, $\epsilon =
\mathrm{sgn}(D^1)$, where $D^1$ is the $x$-component of $D^\alpha =
P^+ \Delta^\alpha - \Delta^+ P^\alpha$ and $D^1=0$ corresponds to $t=t_0$. The minimum value of $-t$ for given $\zeta$ is $- t_0 = 4 m^2 \zeta^2/(1-\zeta^2)$.  Due to parity invariance one has the relation
$A_{-\lambda'-,-\lambda +}=(-1)^{\lambda'-\lambda}A_{\lambda'+,\lambda -}$.

The chiral-odd GPDs are off-diagonal in the quark helicity basis but they can also be calculated in the transversity basis~\cite{Pasquini1} which is more useful for the overlap formalism used in this work. Here we  briefly discuss the transformation of matrix elements defining chiral-odd GPDs from helicity basis to transversity basis \cite{Pasquini1}.
Consider  the operators $
\mathcal{O}_{+,-} + \mathcal{O}_{-,+}=
-\frac{i}{2}\bar{\psi}\sigma^{+1}\gamma_5\psi$
and $
\mathcal{O}_{+,-} - \mathcal{O}_{-,+} = \frac{i}{2}\bar{\psi}\sigma^{+1}\psi$  in the transversity basis  i.e.
\begin{eqnarray}
\label{T}
T^{q}_{\lambda'_t\lambda_t} &=& \langle p',
\lambda'_t|\int\frac{dz^-}{2\pi}e^{i \bar xP^+z^-}
\bar{\psi}(-{z}/2)\gamma^+\gamma^1\gamma_5
\psi({z}/2)|p, \lambda_t\rangle,
\\
\label{Ttilde}
\widetilde{T}^{q}_{\lambda'_t\lambda_t} &=&\langle p',
\lambda'_t|\int\frac{dz^-}{2\pi}e^{i\bar xP^+z^-}
\frac{i}{2}\bar{\psi}(-{z}/2)\sigma^{+1}\psi({z}/2)
|p, \lambda_t\rangle,
\end{eqnarray}
where $\lambda_t$ ($\lambda'_{t}$) labels the transverse polarization of the initial (final) nucleon polarized along +ve $x$($\uparrow$) or -ve $x$($\downarrow$) direction and
the transverse basis states are defined as
\begin{eqnarray}
\mid p,\uparrow\rangle &=& \frac{1}{\sqrt{2}}(\mid p,+\rangle+\mid p,-\rangle),\\
\mid p,\downarrow\rangle &=&  \frac{1}{\sqrt{2}}(\mid p,+\rangle-\mid p,-\rangle).
\end{eqnarray}
These matrix elements obey the following relations as a result of  parity invariance
\begin{eqnarray}
T^{q}_{\uparrow\uparrow}&=&-T^{q}_{\downarrow\downarrow}\,,\quad\quad
T^{q}_{\uparrow\downarrow}=T^{q}_{\downarrow\uparrow}\,,\nonumber\\
\widetilde T^{q}_{\uparrow\uparrow}&=&\widetilde T^{q}_{\downarrow\downarrow}\, ,~~\quad\quad
\widetilde T^{q}_{\uparrow\downarrow}=-\widetilde T^{q}_{\downarrow\uparrow}\, .
\end{eqnarray}
We can now express them in terms of the matrix elements in the helicity basis as
\begin{eqnarray}
T^q_{\uparrow\uparrow} =A_{++,--} + A_{-+,+-}, &\quad&
T^q_{\uparrow\downarrow} =A_{++,+-} - A_{-+,--},\nonumber\\
\widetilde{T}^q_{\uparrow\uparrow} =
A_{++,+-}+A_{-+,--}, &\quad&
\widetilde{T}^q_{\downarrow\uparrow} =
A_{++,--}-A_{-+,+-}.
\end{eqnarray}
Finally, one can obtained the chiral-odd GPDs from the transverse matrix elements through the relations
\begin{eqnarray}
H^{q}_T &=& \frac{1}{\sqrt{1 - \zeta^2}}T^q_{\uparrow\uparrow} -
\frac{2M\zeta} {\epsilon\sqrt{t_0 - t}(1 - \zeta^2)}T^q_{\uparrow\downarrow},\label{HT}
\\
E_T^{q} &=& \frac{2M}{\epsilon\sqrt{t_0 - t}(1 -\zeta^2)}
\bigg(\zeta T^q_{\uparrow\downarrow}
+ \widetilde{T}^q_{\uparrow\uparrow}\bigg)
- \frac{4M^2}{(t_0 - t)\sqrt{1 - \zeta^2}(1 - \zeta^2)}
\bigg(\widetilde{T}^q_{\downarrow\uparrow} - T^q_{\uparrow\uparrow}\bigg).\label{ET}
\\\nonumber
\\
\widetilde{H}_T^{q} &=& \frac{2M^2}{(t_0 - t)\sqrt{1 -\zeta^2}}
(\widetilde{T}^q_{\downarrow\uparrow} - T^q_{\uparrow\uparrow}),\label{HtT}
\\
\widetilde{E}^{q}_T &=& \frac{2M}{\epsilon\sqrt{t_0 - t}(1 -\zeta^2)}
\bigg(T^q_{\uparrow\downarrow}
+ \zeta\widetilde{T}^q_{\uparrow\uparrow}\bigg)
- \frac{4M^2\zeta}{(t_0 - t)\sqrt{1 - \zeta^2}(1 - \zeta^2)}
\bigg(\widetilde{T}^q_{\downarrow\uparrow} - T^q_{\uparrow\uparrow}\bigg).\label{EtT}
\end{eqnarray}

\subsection{Overlap formalism}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.6cm,height=5.5cm,clip]{HTq_vs_xt_z0.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.6cm,height=5.5cm,clip]{HtTq_vs_xt_z0.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.6cm,height=5.5cm,clip]{ETq_vs_xt_z0.pdf}
\end{minipage}
\caption{\label{gz0}(Color online) Plots of the chiral-odd GPDs for zero skewness vs $x$ and different values of $-t$ in $\rm{GeV}^2$ for $u$ and $d$ quarks.}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTq_vs_xt_z15.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTq_vs_xt_z15.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETq_vs_xt_z15.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTq_vs_xt_z15.pdf}
\end{minipage}
\caption{\label{gz15}(Color online) Plots of chiral odd GPDs for the nonzero skewness vs $x$ and different values of $-t$ in $GeV^2$, for fixed value of $\zeta=0.15$. (a) $H^q_T$, (b) $\widetilde{H}^q_T$ and (c) $E^q_T$, (d) $\widetilde{E}^q_T$ ; $q$ stands for $u$ and $d$ quark.}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTq_vs_xz_t7.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTq_vs_xz_t7.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETq_vs_xz_t7.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTq_vs_xz_t7.pdf}
\end{minipage}
\caption{\label{gt7}(Color online) Plots of chiral odd GPDs for the nonzero skewness vs $x$ and different values of $\zeta$, for fixed value of $t=0.7$ $GeV^2$. (a) $H^q_T$, (b) $\widetilde{H}^q_T$ and (c) $E^q_T$, (d) $\widetilde{E}^q_T$ ; $q$ stands for $u$ and $d$ quark.}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTu_vs_zt_x8.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTd_vs_zt_x8.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETu_vs_zt_x8.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETd_vs_zt_x8.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(e)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTu_vs_zt_x8.pdf}
\hspace{0.1cm}%
\small{(f)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTd_vs_zt_x8.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(g)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTu_vs_zt_x8.pdf}
\hspace{0.1cm}%
\small{(h)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTd_vs_zt_x8.pdf}
\end{minipage}
\caption{\label{gx8}(Color online) Plots of chiral odd GPDs for the nonzero skewness vs $\zeta$ and different values of $-t$ in $GeV^2$, for fixed value of $x=0.6$. Left pannel is for $u$ quark and the right pannel is for $d$ quark.}
\end{figure}

Using the overlap representation of light front wave functions, we evaluate the chiral-odd GPDs in light front quark-diquark model. We restrict our discussion to the DGLAP domain, i.e., $\zeta<x<1$ where $\zeta$ is the skewness and $x$ is the light front longitudinal momentum fraction carried by the struck quark. This kinematical domain describes the diagonal $n\rightarrow n$ overlaps where the particle number remain conserved.  This region corresponds to the situation where one removes a quark from the initial proton  with light-front longitudinal momentum $(x+\zeta)P^+$ and re-insert it into the final proton with longitudinal momentum $(x-\zeta)P^+$. The diagonal $2\rightarrow 2$ overlap representation of the matrix elements $T^{q}_{\lambda\lambda'}$ and $\widetilde{T}^{q}_{\lambda\lambda'}$ in terms of light-front wave functions in the quark-diquark model are given by
\begin{eqnarray}
T^q_{\uparrow\uparrow}&=& \int \frac{d^2{\bf k}_{\perp}}{16\pi^3}~\bigg[\psi_{+q}^{+*}(x',{\bf k}_{\perp}')\psi_{-q}^-(x'',{\bf k}_{\perp}'')
+\psi_{+q}^{-*}(x',{\bf k}_{\perp}')\psi_{-q}^+(x'',{\bf k}_{\perp}'')\bigg],\label{T1}\\
T^q_{\uparrow\downarrow}&=& \int \frac{d^2{\bf k}_{\perp}}{16\pi^3}~\bigg[\psi_{+q}^{+*}(x',{\bf k}_{\perp}')\psi_{-q}^+(x'',{\bf k}_{\perp}'')
-\psi_{+q}^{-*}(x',{\bf k}_{\perp}')\psi_{-q}^-(x'',{\bf k}_{\perp}'')\bigg],\label{T2}\\
\widetilde{T}^q_{\uparrow\uparrow}&=& \int \frac{d^2{\bf k}_{\perp}}{16\pi^3}~\bigg[\psi_{+q}^{+*}(x',{\bf k}_{\perp}')\psi_{-q}^+(x'',{\bf k}_{\perp}'')
+\psi_{+q}^{-*}(x',{\bf k}_{\perp}')\psi_{-q}^-(x'',{\bf k}_{\perp}'')\bigg],\label{T3}\\
\widetilde{T}^q_{\uparrow\downarrow}&=& \int \frac{d^2{\bf k}_{\perp}}{16\pi^3}~\bigg[\psi_{+q}^{+*}(x',{\bf k}_{\perp}')\psi_{-q}^-(x'',{\bf k}_{\perp}'')
-\psi_{+q}^{-*}(x',{\bf k}_{\perp}')\psi_{-q}^+(x'',{\bf k}_{\perp}'')\bigg],\label{T4}
\end{eqnarray}
where, for the final struck quark
\begin{eqnarray}
x'=\frac{x-\zeta}{1-\zeta}, \quad\quad\quad {\bf k}_{\perp}'={\bf k}_{\perp}+(1-x')\frac{\bf{\Delta}_{\perp}}{2},
\end{eqnarray}
and for the initial struck quark
\begin{eqnarray}
x''=\frac{x+\zeta}{1+\zeta}, \quad\quad\quad {\bf k}_{\perp}''={\bf k}_{\perp}-(1-x'')\frac{\bf{\Delta}_{\perp}}{2}.
\end{eqnarray}
The explicit calculation of the matrix elements $T^{q}_{\lambda\lambda'}$ and $\widetilde{T}^{q}_{\lambda\lambda'}$ using the light front wave functions of the quark-diquark model given in Eq.(\ref{wf}) gives
\begin{eqnarray}
T^q_{\uparrow\uparrow}&=&\frac{1}{\kappa^2}\Big[{\frac{\log x'\log x''}{(1-x')(1-x'')}}\Big]^{1/2}\bigg[(N_q^{(1)})^2(x'x'')^{a_q^{(1)}}\{(1-x')(1-x'')\}^{b_q^{(1)}}\frac{1}{A}\nonumber\\
&-&(N_q^{(2)})^2\frac{1}{M_n^2}(x'x'')^{a_q^{(2)}-1}\{(1-x')(1-x'')\}^{b_q^{(2)}}
\bigg(\frac{B^2}{4A^2}-\frac{1}{4}(1-x')(1-x'')\nonumber\\
&+&\frac{B}{4A}(x''-x')\Big)\frac{Q^2}{A}\Big]
\exp\bigg[Q^2\Big(C-\frac{B^2}{4A}\Big)\bigg] ,\label{T1F}\\
\nonumber\\
T^q_{\uparrow\downarrow}&=&-\frac{N_q^{(1)}N_q^{(2)}}{\kappa^2}\Big[{\frac{\log x'\log x''}{(1-x')(1-x'')}}\Big]^{1/2}\frac{1}{M_n}\bigg[(x')^{a_q^{(1)}}(1-x')^{b_q^{(1)}}(x'')^{a_q^{(2)}-1}(1-x'')^{b_q^{(2)}}\nonumber\\
&\times&\bigg(\frac{BQ}{2A^2}-\frac{Q}{2A}(1-x'')\Big)+(x')^{a_q^{(2)}-1}(1-x')^{b_q^{(2)}}(x'')^{a_q^{(1)}}(1-x'')^{b_q^{(1)}}\nonumber\\
&\times&\bigg(\frac{BQ}{2A^2}+\frac{Q}{2A}(1-x'')\Big)\Big]
\exp\bigg[Q^2\Big(C-\frac{B^2}{4A}\Big)\bigg] \label{T2F},
\end{eqnarray}
\begin{eqnarray}
\widetilde{T}^q_{\uparrow\uparrow}&=&-\frac{N_q^{(1)}N_q^{(2)}}{\kappa^2}\Big[{\frac{\log x'\log x''}{(1-x')(1-x'')}}\Big]^{1/2}\frac{1}{M_n}\bigg[(x')^{a_q^{(1)}}(1-x')^{b_q^{(1)}}(x'')^{a_q^{(2)}-1}(1-x'')^{b_q^{(2)}}\nonumber\\
&\times&\bigg(\frac{BQ}{2A^2}-\frac{Q}{2A}(1-x'')\Big)-(x')^{a_q^{(2)}-1}(1-x')^{b_q^{(2)}}(x'')^{a_q^{(1)}}(1-x'')^{b_q^{(1)}}\nonumber\\
&\times&\bigg(\frac{BQ}{2A^2}+\frac{Q}{2A}(1-x'')\Big)\Big]
\exp\bigg[Q^2\Big(C-\frac{B^2}{4A}\Big)\bigg] ,\label{T3F}\\
\nonumber\\
\widetilde{T}^q_{\uparrow\downarrow}&=&\frac{1}{\kappa^2}\Big[{\frac{\log x'\log x''}{(1-x')(1-x'')}}\Big]^{1/2}\bigg[(N_q^{(1)})^2(x'x'')^{a_q^{(1)}}\{(1-x')(1-x'')\}^{b_q^{(1)}}\frac{1}{A}\nonumber\\
&+&(N_q^{(2)})^2\frac{1}{M_n^2}(x'x'')^{a_q^{(2)}-1}\{(1-x')(1-x'')\}^{b_q^{(2)}}
\bigg(\frac{B^2}{4A^2}-\frac{1}{4}(1-x')(1-x'')\nonumber\\
&+&\frac{B}{4A}(x''-x')\Big)\frac{Q^2}{A}\Big]
\exp\bigg[Q^2\Big(C-\frac{B^2}{4A}\Big)\bigg] ,
\label{T4F}\end{eqnarray}
where ${{\Delta}}_{\perp}^2=Q^2=-t(1-\zeta^2)-4M_n^2\zeta^2$. $A$, $B$ and $C$ are  functions of $x'$ and $x''$,
\begin{eqnarray}
A&=&A(x',x'')=-\frac{\log x'}{2\kappa^2(1-x')^2}-\frac{\log x''}{2\kappa^2(1-x'')^2},\nonumber\\
B&=&B(x',x'')=\frac{\log x'}{2\kappa^2(1-x')}-\frac{\log x''}{2\kappa^2(1-x'')},\\
C&=&C(x',x'')=\frac{1}{4}\Big[\frac{\log x'}{2\kappa^2}+\frac{\log x''}{2\kappa^2}\Big].\nonumber
\end{eqnarray}
Using the matrix elements calculated in Eqs.(\ref{T1F}-\ref{T4F}) we evaluate the chiral-odd GPDs in Eqs.(\ref{HT}-\ref{EtT}). All the GPDs are suitably scaled by the flavor factors $P_q$ where $P_u=\frac{4}{3}$, and $P_d=-\frac{1}{3}$ are dictated by SU(6) spin-flavor symmetry \cite{Karl}.

In Fig.\ref{gz0} we show the $t$ dependence chiral-odd GPDs $H^q_T$, $\widetilde{H}^q_T$ and $E^q_T$ for up and down quarks in the quark-diquark model when the skewness $\zeta=0$ . Being an odd function of $\zeta$, the GPD $\tilde E^q_T$ vanishes at $\zeta=0$ in this model. The similar behavior of $\tilde E^q_T$ has been reported in~\cite{diehl01, Pasquini1}. One can notice that the sign of all three GPDs for $u$ quark are opposite with respect to $d$ quark and $\widetilde{H}^q_T$ shows opposite sign of ${H}^q_T(x,0,t)$ as expected from $SU(6)$ symmetry. The peaks of all the distributions move to higher values of $x$ as $-t$ increases. For $\zeta\ne 0$, all the four GPDs are shown in Fig.\ref{gz15} and \ref{gt7}. In Fig.\ref{gz15} the GPDs are shown for fixed value of $\zeta=0.15$ but for different values of $-t$. In Fig.\ref{gt7}, we plot the GPDs for fixed value of $-t=0.7~ \rm{GeV^2}$ and different values of $\zeta$. One can notice that the height of the peaks of the distributions increase  and shift to
higher $x$
with increasing $\zeta$ for fixed $-t$. In all cases, the GPDs vanish at $x=\zeta$. The reason is that in our approach we consider only the contribution from the valence quarks. In this model we can not evaluate the total (sea+valence) GPDs as the model itself depends only on the valence quarks.
The similar behavior of the chiral-odd GPDs has been found in the relativistic constituent quark model calculated in~\cite{Pasquini1}. Also, the region for $x<\zeta$, the so called ERBL region, where quark-antiquark pair creation/annihilation are involved are not included in this model.
In Fig.\ref{gx8}, we have shown the GPDs as functions of $\zeta$ for different values of $-t$ and fixed $x$. It can be noticed that only $\widetilde{E}^q_T(x,\zeta,t)$ ( Fig.\ref{gx8}(g) and  Fig.\ref{gx8}(h)) shows markedly different behavior from the other GPDs. $\widetilde{E}^q_T(x,\zeta,t)$ rises smoothly from zero as $\zeta$ increases for all $t$ values whereas the other GPDs have   different values at $\zeta=0$ for different values of $-t$. The similar behaviors  of the chiral-odd GPDs  have been observed  \cite{CMM1} in a QED model.

\subsection{Mellin moments of chiral-odd GPDs}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{gTu.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{gTd.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{kTu.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{kTd.pdf}
\end{minipage}
\caption{\label{tensor}(Color online) Tensor form factors for $u$ and $d$ quarks are compared with the lattice~\cite{gock} and chiral quark-soliton model (xQSM)~\cite{xQSM1,xQSM2} results.}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTun_vs_q.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTdn_vs_q.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETun_vs_q.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETdn_vs_q.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(e)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTtun_vs_q.pdf}
\hspace{0.1cm}%
\small{(f)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTtdn_vs_q.pdf}
\end{minipage}
\caption{\label{moments}(Color online) Plots of first three moments of the chiral odd GPDs for zero skewness vs $\sqrt{-t}$ in $GeV$. Left pannel is for $u$ quark and the right pannel is for $d$ quark.}
\end{figure}

The Mellin moments of the valence GPDs are defined as
\begin{eqnarray}
H^q_{Tn0}(t)=\int_0^1~dx x^{n-1}H^q_{T}(x,0,t),\label{moment_formula}
\end{eqnarray}
where the index $n=1,2,3$ etc., and the second subscript indicates that the moments are evaluated at zero skewness. The moments of  the other GPDs, $E^q_{Tn0}(t)$ and $\widetilde{H}^q_{Tn0}(t)$ can also be defined in the same way as (\ref{moment_formula}).
The first moments of chiral-odd GPDs give the tensor form factors. The forward values, $t = 0$, of the form factor $g_T=H_{T10}(t=0)$ can be identified as the tensor charge~\cite{Hagler:2009}. The combination of tensor form factors $\bar{E}^q_{T10}=(E^q_{T10}+2\widetilde{H}^q_{T10})$ in the forward limit plays a role very similar to that of anomalous magnetic moment $\kappa^q$ and therefore may be identified with a tensor magnetic moment, $\kappa^q_T=\bar{E}^q_{T10}(t=0)$~\cite{Burk3}. In Fig.(\ref{tensor}), we have compared our result for the tensor form factors with the corresponding results from lattice~\cite{gock} and chiral quark soliton model \cite{xQSM1,xQSM2}. The tensor form factors for $u$ quark in this model  agrees well with the chiral quark-soliton model ($\chi$QSM) but both(this model and $\chi$QSM model) deviate from lattice results for both $u$ and $d$ quarks.
The second moments of these GPDs correspond to the gravitational form factors of quarks with transverse spin in an unpolarized nucleon.  A linear combination of $H^q_{T20}(t)$, $E^q_{T20}(t)$ and $\widetilde{H}^q_{T20}(t)$ gives the angular momentum carried by quarks with transverse spin in an unpolarized nucleon\cite{Burk3}, in analogy to Ji's angular momentum sum rule. The third moments of the GPDs generate form factors of a twist-two operator having
two covariant derivatives~\cite{rev} and the higher order moments give the form factors of higher-twist operators.

In Fig.\ref{moments} we show the first three moments of the chiral-odd GPDs $|t|H^q_{Tn0}(t)$, $|t|E^q_{Tn0}(t)$ and $|t|\widetilde{H}^q_{Tn0}(t)$ as functions of $\sqrt{-t}$ for $u$ and $d$ quarks. We find a strong decrease in the  magnitudes of the moments with increasing  $n$.
This can be understood from the behavior of the GPDs with $x$ as shown in Fig.\ref{gz0}. Higher moments involve higher power of $x$ and hence the dominant contributions come from the large $x$ region($x\to1$), but  the GPDs decrease rapidly as $x$ increases, and hence the higher moments become smaller.
We also observe that as the index $n$ increases, the decrease of the moments becomes slower with increasing $-t$. This again can be  explained in terms of the decrease of the GPDs with momentum fraction $x$, which results in a weaker $t$ slope for the higher moments. The similar behavior has been observed in lattice QCD calculations of the moments of chiral-odd GPD~\cite{gock}.

\section{Impact parameter representation of chiral-odd GPDs}\label{chiral_impact}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTu_vs_xb_z2_3D.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTd_vs_xb_z2_3D.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETu_vs_xb_z2_3D.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETd_vs_xb_z2_3D.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(e)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTu_vs_xb_z2_3D.pdf}
\hspace{0.1cm}%
\small{(f)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTd_vs_xb_z2_3D.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(g)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTu_vs_xb_z2_3D.pdf}
\hspace{0.1cm}%
\small{(h)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTd_vs_xb_z2_3D.pdf}
\end{minipage}
\caption{\label{giz2}(Color online) Plots of chiral odd GPDs for the nonzero skewness in impact space vs $x$ and $b=|\bf b|$ for fixed value of $\zeta=0.2$. Left pannel is for $u$ quark and the right pannel is for $d$ quark.}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTu_vs_zb_x6_3D.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTd_vs_zb_x6_3D.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETu_vs_zb_x6_3D.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETd_vs_zb_x6_3D.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(e)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTu_vs_zb_x6_3D.pdf}
\hspace{0.1cm}%
\small{(f)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTd_vs_zb_x6_3D.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(g)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTu_vs_zb_x6_3D.pdf}
\hspace{0.1cm}%
\small{(h)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTd_vs_zb_x6_3D.pdf}
\end{minipage}
\caption{\label{gix6}(Color online) Plots of chiral odd GPDs for the nonzero skewness in impact space vs $\zeta$ and $b=|\bf b|$ for fixed value of $x=0.6$. Left pannel is for $u$ quark and the right pannel is for $d$ quark.}
\end{figure}
GPDs in transverse impact parameter space are defined by a two-dimensional Fourier transform in $\Delta_{\perp}$ as follows~\cite{burk1,burk2}:
\begin{eqnarray}
{H}_T(x,\zeta,b_{\perp})&=&{1\over (2 \pi)^2} \int d^2 \Delta_{\perp} e^{-i \Delta_{\perp} \cdot b_{\perp}}
H_T(x,\zeta,t),\\
{E}_T(x,\zeta,b_{\perp})&=&{1\over (2 \pi)^2} \int d^2 \Delta_{\perp} e^{-i \Delta_{\perp} \cdot b_{\perp}}
E_T(x,\zeta,t),
\\
\widetilde{{H}}_T(x,\zeta,b_{\perp})&=&{1\over (2 \pi)^2} \int d^2 \Delta_{\perp} e^{-i \Delta_{\perp} \cdot b_{\perp}}
\widetilde{H}_T(x,\zeta,t),\\
\widetilde{{E}}_T(x,\zeta,b_{\perp})&=&{1\over (2 \pi)^2} \int d^2 \Delta_{\perp} e^{-i \Delta_{\perp} \cdot b_{\perp}}
\widetilde{E}_T(x,\zeta,t).
\end{eqnarray}
Here, $b_{\perp}$ is the transverse impact parameter conjugate to $\Delta_{\perp}$.
For zero skewness, $\bfb$ gives a measure of  the transverse distance between the struck parton and the center of momentum of the hadron. $b_{\perp}$ satisfies the condition $\sum_i x_i b_{\perp i}=0$, where the sum is over the number of partons. The relative distance between the struck parton and the center of momentum of the spectator system is given by  ${\mid \bfb \mid\over 1-x}$, which provides us an estimate of the size of the bound state \cite{diehl}.
In the DGLAP region $x>\zeta$, the impact parameter $b_{\perp}$ implies the location where the quark is pulled out and re-insert to the nucleon. In the ERBL domain $x<\zeta$, $b_{\perp}$ provides the transverse distance of the quark-antiquark pair inside the nucleon. For zero skewness, the chiral-odd GPDs also have a density interpretation in transverse impact parameter space like chiral-even GPDs depending on the polarization of both the active quark and the nucleon. A combination of ${E}_T(x,b_{\perp})$ and $\widetilde{{H}}_T(x,b_{\perp})$ i.e. $({E}_T+2\widetilde{{H}}_T)$ is responsible for a deformation in the transversity asymmetry quarks in an unpolarized target~\cite{Burk3,Diehl05,Dahiya07}. This is similar to the role played by ${E}(x,b_{\perp})$ for both unpolarized active quark and the nucleon. On the other hand a combination of ${H}_T(x,b_{\perp})$ and $\widetilde{{H}}_T(x,b_{\perp})$ provides a distortion in the transverse spin density when the active quark and the nucleon are transversely polarized~\cite{Diehl05,Pasquini2}. Note
that the density interpretation is possible only in the limit $\zeta=0$, but in most experiment $\zeta$ is nonzero. So, it is interesting to investigate the chiral-odd GPDs in the impact parameter space with nonzero $\zeta$.

We show the skewness dependent chiral-odd GPDs in transverse impact parameter space for fixed $\zeta=0.2$ as functions of $b=|b_{\perp}|$ and $x$ for $u$ and $d$ quark in Fig.\ref{giz2}. Similarly, all the chiral-odd GPDs as functions of $\zeta$ and $b$ for a fixed value of $x=0.6$ are shown in Fig.\ref{gix6}.
The peak of the distribution $H_T(x,\zeta,b_{\perp})$ for fixed $\zeta$ appears at lower $x$ for $d$ quark and shifts to higher $x$ for $u$ quark while for $\widetilde{H}_T(x,\zeta,b_{\perp})$ we get the peak at lower $x$ for both $u$ and $d$ quarks. For both $E_T(x,\zeta,b_{\perp})$ and $\widetilde{E}_T(x,\zeta,b_{\perp})$, the peaks arise at lower $x$ for both $u$ and $d$ quarks but we also get an oscillatory behavior  for both the GPDs of $d$ quark. This is due to the fact that  $E_T^d(x,\zeta,t)$ and $\widetilde{E}_T^d(x,\zeta,t)$  have slight oscillatory behavior as can be seen in  Fig. \ref{gz15}(c)-(d). Except $\widetilde{H}_T(x,\zeta,b_{\perp})$, the peak of $u$ quark in all other distributions are sufficiently large compare to $d$ quark. For $\widetilde{H}_T(x,\zeta,b_{\perp})$, the peak of $u$ quark is slightly large compare to $d$ quark. For small $b$, $\widetilde{E}_T(x,\zeta,b_{\perp})$ falls off slowly at large $x$ for $u$ quark compare to $d$ quark.
With increasing $x$, the width of all the distributions in transverse impact parameter space  decreases, which implies that the distributions are more localized near the center of momentum for higher values of $x$. Substantial differences is observed in $\widetilde{E}_T(x,\zeta,b_{\perp})$ from other GPDs when the GPDs are plotted against $\zeta$ and $b$ for fixed values of $x$ in Fig.\ref{gix6}. $\widetilde{E}_T(x,\zeta,b_{\perp})$
increases with increasing $\zeta$.
Another interesting behavior of all the GPDs is that the peaks of all the distributions become broader as $\zeta$ increases for a fixed value of $x$.  This  means that as the  momentum transfer in the longitudinal direction increases the probability of hitting the transversely polarized active quark  at a larger transverse impact parameter $b$  increases.

\subsection{GPDs in longitudinal impact parameter space}\label{chiral_longi_impact}
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTu_vs_s_x_fixed.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTd_vs_s_x_fixed.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETu_vs_s_x_fixed.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETd_vs_s_x_fixed.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTu_vs_s_x_fixed.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTd_vs_s_x_fixed.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTu_vs_s_x_fixed.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTd_vs_s_x_fixed.pdf}
\end{minipage}
\caption{\label{glx6}(Color online) Plots of the chiral-odd GPDs in longitudinal impact space vs $\sigma$ and different values of $-t$ in $\rm{GeV}^2$, for fixed value of $x=0.3$. Left pannel is for $u$ quark and the right pannel is for $d$ quark.}
\end{figure}
The boost invariant longitudinal impact parameter is defined as $\sigma=\frac{1}{2}b^-P^+$ which is conjugate to $\zeta$, the measure of longitudinal momentum transfer. The parameter $\sigma$ was first introduced in \cite{BDHAV}. The DVCS amplitude in a QED model of a dressed electron  shows an interesting diffraction pattern in the longitudinal impact parameter space analogous to diffractive scattering of a wave in optics~\cite{BDHAV}.
In analogy with optics, the finite size of the $\zeta$ can be interpreted  as a slit of finite width and produces the diffraction pattern.  It should be mentioned here that the FT with a finite range of $\zeta$ of any function does not show the diffraction pattern~\cite{CMM2}. The pattern depends on the behavior of the function. The chiral-odd GPDs calculated in \cite{CMM1} for
a simple relativistic spin half system of an electron dressed  with a photon exhibit the similar diffraction pattern in the longitudinal impact parameter space. A phenomenological model for proton GPDs show the similar diffraction pattern \cite{CMM2}. Similar diffraction patterns are also observed for the chiral-even GPDs  in this light front quark-diquark model~\cite{chandan} as well as  in QED model~\cite{Kumar1}.
The GPDs in longitudinal position space are defined as:
\begin{eqnarray}
H_T(x,\sigma,t)&=&{1\over 2 \pi} \int_0^{\zeta_f} d \zeta e^{i\zeta P^+b^-/2}H_T(x,\zeta,t),\nonumber\\
&=&{1\over 2 \pi} \int_0^{\zeta_f} d \zeta e^{i\zeta \sigma}H_T(x,\zeta,t),\\
E_T(x,\sigma,t)&=&{1\over 2 \pi} \int_0^{\zeta_f} d \zeta e^{i\zeta P^+b^-/2}E_T(x,\zeta,t),\nonumber\\
&=&{1\over 2 \pi} \int_0^{\zeta_f} d \zeta e^{i\zeta \sigma}E_T(x,\zeta,t).
\end{eqnarray}
Similarly one can obtain $\widetilde{H}_T(x,\sigma,t)$ and $\widetilde{E}_T(x,\sigma,t)$ as well.
Since we are considering the region $\zeta<x<1$, the upper limit of $\zeta$ integration $\zeta_f$ is given by $x$ if $x$ is smaller than $\zeta_{max}$, otherwise by $\zeta_{max}$ if $x$ is larger than $\zeta_{max}$ where the maximum value of $\zeta$ for a fixed $-t$ is given by
\begin{eqnarray}
\zeta_{max}=\sqrt{\frac{(-t)}{(-t+4M_n^2)}}.
\end{eqnarray}
We show the Fourier spectrum of all the chiral-odd GPDs for $u$ and $d$ quarks in longitudinal position space as a function of $\sigma$ for fixed $x=0.3$ and different values of $-t$ in Fig.\ref{glx6}. ${H}_T^q$, ${E}_T^q$ and $\widetilde{H}_T^q$ display a diffraction pattern in the $\sigma$ space as observed for the DVCS amplitude in~\cite{BDHAV},
but $\widetilde{E}_T^q(x,\sigma,t)$ does not show the same pattern. This is due to the fact that the distinctly different behavior of $\widetilde{E}_T^q(x,\zeta,t)$ with $\zeta$ compared to the other GPDs. This again shows that the diffraction pattern is not solely due to the finiteness of $\zeta$ integration, the functional form of the GPDs are also crucial.  For all the diffraction patterns the first minima appears at the same values of $\sigma$. We also show the chiral-odd GPDs in $\sigma$ space for fixed $-t=0.4$ $GeV^2$ and different values of $x$ in Fig.\ref{glt5}. In analogous to the single slit optical diffraction pattern, here $\zeta_{max}$ plays the role of the slit width.  Since the position of the minima are inversely proportional to the slit width, the minima move towards the center of the diffraction pattern as the slit width $\zeta_{max}$ increases.
\begin{figure}[htbp]
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}
\includegraphics[width=7.5cm,height=5.15cm,clip]{HTu_vs_s_t_fixed.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HTd_vs_s_t_fixed.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETu_vs_s_t_fixed.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{ETd_vs_s_t_fixed.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(a)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTu_vs_s_t_fixed.pdf}
\hspace{0.1cm}%
\small{(b)}\includegraphics[width=7.5cm,height=5.15cm,clip]{HtTd_vs_s_t_fixed.pdf}
\end{minipage}
\begin{minipage}[c]{0.98\textwidth}
\small{(c)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTu_vs_s_t_fixed.pdf}
\hspace{0.1cm}%
\small{(d)}\includegraphics[width=7.5cm,height=5.15cm,clip]{EtTd_vs_s_t_fixed.pdf}
\end{minipage}
\caption{\label{glt5}(Color online) Plots of the chiral-odd GPDs in longitudinal impact space vs $\sigma$ and different values of $x$, for fixed value of $-t=0.4$ $GeV^2$. Left pannel is for $u$ quark and the right pannel is for $d$ quark.}
\end{figure}

\section{Summary}\label{summary}
we have investigated the chiral-odd GPDs for $u$ and $d$ quark in proton for both zero and nonzero skewness in the light front quark-diquark model predicted by the soft-wall AdS/QCD. We have found that the $\widetilde{E}^q_T(x,0,t)$ is zero in this model due to odd function in nature with $\zeta$. $\widetilde{H}^q_T$ shows opposite sign of ${H}^q_T$ for both $u$ and $d$ quark as expected from $SU(6)$. For zero skewness, all the chiral-odd GPDs for $u$ quark are opposite with respect to $d$ quark .
We have calculated the GPDs for nonzero skewness in the DGLAP region i.e., for ($x>\zeta$).  The peaks of the distributions move to higher values of $x$ for fixed $\zeta$ with increasing of $-t$ similar as the nature of $\zeta=0$. The height of the peaks increases and also shift to higher values of $x$ as $\zeta$ increases for fixed  $-t$. We observed markedly different behavior for $\widetilde{E}^q_T$ from the other chiral-odd GPDs when we plot the GPDs against $\zeta$ for fixed $x$ and different $-t$. It shows that with increasing $\zeta$, $\widetilde{E}^q_T$ started to increase smoothly from zero but other GPDs rise from different values at $\zeta=0$ for different values of $-t$.

We have also presented all the chiral-odd GPDs in the transverse position or impact parameter($b$) as well as longitudinal position($\sigma$) spaces by taking FT of the GPDs with respect to transverse momentum transfer($\Delta_{\perp}$) and $\zeta$ respectively. The impact parameter $b$ gives a measure of  the transverse distance between the struck parton and the center of momentum of the hadron.
In this model, except $\widetilde{H}^q_T$, the behavior of the GPDs in the transverse impact parameter space for $u$ and $d$ quarks are quite different when plotted in $x$ and $b$. Except the magnitude, the nature of $H^q_T$, $E^q_T$ and $\widetilde{H}^q_T$ are more or less same when plotted against $\zeta$ and $b$ but $\widetilde{E}^q_T$ shows a different behavior. The width of the all distributions increase with increasing $\zeta$ as well as $x$ decreases. We found that the GPDs $H_T$, $E_T$ and $\widetilde{H}_T$ for $u$ and $d$ quarks in $\sigma$ space show diffraction patterns analogous to diffractive scattering of a wave in optics. A similar diffraction pattern also has been  observed  in some other models. The qualitative nature of the diffraction patterns for all three chiral-odd GPDs are same for both $u$ and $d$ quarks.
The general features of this pattern are mainly depends on the finiteness of $\zeta$ integration as well as the dependence of GPDs on $x$, $\zeta$ and $t$. Like other GPDs, $\widetilde{E}_T$ does not show the diffraction pattern. This is due to a different nature of $\widetilde{E}_T$ with $\zeta$ from the other GPDs.  It also indicates that the diffraction pattern is  not solely  due to finiteness of $\zeta$ integration and the functional behaviors of the GPDs are  important to have the diffraction pattern.