Assuming I've understood your intention correctly, this would be my suggestion.
You want to offset your x-data by 1,4, and 8, so you have to add these values to your columns $2, $3, $4.
If I understood your sketch with 0 m/s correctly, you also want to subtract the first velocity value from it.
Please correct me if I'm wrong.
So, you take the pseudocolumn 0 (check help pseudocolumns), which is basically the line number starting from 0. If $0==0 (check ternary operator help ternary) you assign the current value (only the first value) of your column to a variable, e.g. xoff=$2and subtract it from all following values in your column. Since you have a header you have to skip one line, i.e. skip 1.
In case you want several x-axes on top all starting from 0, this could be either done with arrows and labels or depending on what exactly you need maybe with multiplots (check help multiplot).
Check the following as starting point:
Code:
### plotting with different offsets
reset session
$Data <<EOD
z(m) U_1(m/s)* U_4(m/s)* U_8(m/s)*
0.50 1.66 1.82 1.95
0.75 1.85 2.04 2.11
1.00 2.00 2.20 2.18
1.25 2.12 2.34 2.21
1.50 2.23 2.36 2.22
1.75 2.33 2.37 2.22
2.00 2.41 2.38 2.22
EOD
set xrange[0:10]
set xtics 1
set yrange [0.3:2.4]
set grid xtics, ytics
vs = "1 4 8"
set for [v in vs] arrow from v, graph 0 to v, graph 1 lw 1.5 lc "black" dt 2 nohead
plot $Data u ($0==0?xoff=$2:0, $2+1-xoff):1 skip 1 w lp ps 1.5 lw 2 pt 7 lc "red" ti "U_1", \
'' u ($0==0?xoff=$3:0, $3+4-xoff):1 skip 1 w lp ps 1.5 lw 2 pt 7 lc "green" ti "U_4", \
'' u ($0==0?xoff=$4:0, $4+8-xoff):1 skip 1 w lp ps 1.5 lw 2 pt 7 lc "black" ti "U_8"
### end of code
Result:

Addition:
Maybe now I have a better idea about what you probably want.
I guess you "simply" want to have subplots at defined locations within a "Main"plot.
gnuplot offers the feature of multiplot (check help multiplot) which will place several plots next to or on top of each other.
You can define the origin (check help origin) and the size (check help size) of each subplot.
However, the coordinates and sizes of the subplots are given in screen coordinates.
The problem is now: how to place the plot at the desired location relative to the first plot?
With labels, arrows, and objects you can use screen, graph and axes coordinates (check help coordinates), but I am not aware that you can use axis coordinates of the previous plot.
So, you have to calculate these positions yourself using some internal gnuplot variables GPVAL_.... It looks complicated but is just some coordinate transformation.
Code:
### plotting with different offsets
reset session
$Data <<EOD
z(m) U_1(m/s)* U_4(m/s)* U_8(m/s)*
0.50 1.66 1.82 1.95
0.75 1.85 2.04 2.11
1.00 2.00 2.20 2.18
1.25 2.12 2.34 2.21
1.50 2.23 2.36 2.22
1.75 2.33 2.37 2.22
2.00 2.41 2.38 2.22
EOD
set for [v in "1 4 8"] arrow from v, graph 0 to v, graph 1 lw 1.5 lc "black" dt 2 nohead
XFirstToGrphRel(x) = (x-MainXMin)/(MainXMax - MainXMin) # relative position to graph
YFirstToGrphRel(y) = (y-MainYMin)/(MainYMax - MainYMin)
PosX(x) = XminScrRel+GraphInScrSizeX*XFirstToGrphRel(x) # screen positon of the subplot
PosY(y) = YminScrRel+GraphInScrSizeY*YFirstToGrphRel(y)
SizeX(dx) = GraphInScrSizeX*dx/(MainXMax - MainXMin) # screen size of a subplot
SizeY(dy) = GraphInScrSizeY*dy/(MainYMax - MainYMin)
set multiplot
set xlabel "x / m"
set xrange[0:12]
set xtics 1
set ylabel "z / m"
set yrange [0.3:2.4]
set ytics 0.5
set grid xtics, ytics
plot 0 w p ps 0 notitle # plot nothing, just background plot
# store the current terminal values for later use
MainXMin = GPVAL_X_MIN
MainXMax = GPVAL_X_MAX
MainYMin = GPVAL_Y_MIN
MainYMax = GPVAL_Y_MAX
GraphInScrSizeX = real(GPVAL_TERM_XMAX-GPVAL_TERM_XMIN)/GPVAL_TERM_XSIZE*GPVAL_TERM_SCALE # real() to avoid integer division
GraphInScrSizeY = real(GPVAL_TERM_YMAX-GPVAL_TERM_YMIN)/GPVAL_TERM_YSIZE*GPVAL_TERM_SCALE
XminScrRel = real(GPVAL_TERM_XMIN)/GPVAL_TERM_XSIZE*GPVAL_TERM_SCALE
YminScrRel = real(GPVAL_TERM_YMIN)/GPVAL_TERM_YSIZE*GPVAL_TERM_SCALE
unset arrow
set margins 0,0,0,0
unset xlabel
unset xtics
set x2label "v / m/s"
set x2range [0:2.5]
set x2tics 0.5 offset 0.5
unset ylabel
set format y ""
set yrange [0.5:2.0]
set grid x2tics, ytics
set object 1 rect from graph 0, graph 0 to graph 1, graph 1 behind fc "white"
set size SizeX(2),SizeY(1.5)
set origin PosX(1),PosY(0.5)
plot $Data u 2:1 axes x2y1 w lp pt 7 lc "red" notitle
set origin PosX(4.0),PosY(0.5)
plot $Data u 3:1 axes x2y1 w lp pt 7 lc "green" notitle
set origin PosX(8.0),PosY(0.5)
plot $Data u 4:1 axes x2y1 w lp pt 7 lc "blue" notitle
unset multiplot
### end of code
Result:
