#### 3.1. Estimation Error

From the “Introduction” and the results reported in previous works [

34,

35], it is clear that both the number of frames (which depends on the acquisition time selected) and the dimension of the interrogation area (on which the extension of the data sample depends) could influence the accuracy of the results. The measurement points are nodes of a regular mesh whose dimension depends on the dimension of the interrogation area and a high dimension of data sample could determine modeling processes too much time-consuming.

In order to perform the sensitivity analysis of the estimated surface velocity values to the number of pairs of frames, for all the recorded images, the velocity components

v_{x}(

t_{f}) and

v_{y}(

t_{f}) have been calculated in each node of the rectangular mesh, corresponding to a fixed dimension of the interrogation area, by considering six values of the total number of pairs of frames (i.e., of time instants) to be processed (

n_{1} = 100,

n_{2} = 500,

n_{3} = 1000,

n_{4} = 1500,

n_{5} = 2000,

n_{6} = 2400). The standard deviation (estimation error) of the velocity components has been determined for each

i-th node of the mesh as:

where

p_{i,j} and

${\overline{p}}_{{n}_{j},i}$ are, respectively, the examined component

p [

p =

${v}_{x}\left({t}_{f}\right)$,

${v}_{y}\left({t}_{f}\right)$] and the corresponding time-averaged value,

N is the number of nodes. From the error theory [

42], the measured averaged value of the examined velocity component

p falls in the range

${\overline{p}}_{{n}_{j},i}$ ±

${\sigma}_{{n}_{j},i}$ and the uncertainty in averaging the component

p can be estimated as

σ_{E} =

${\sigma}_{{n}_{j},i}/\sqrt{{n}_{j}}$ [

42].

Figure 4 reports, for each component

p, the highest value of the estimation error obtained by Equation (1) for each

${n}_{j}$. It can be observed that, especially for the component

p =

${v}_{x}\left({t}_{f}\right)$,

${\sigma}_{{n}_{j},i}$ tends to decrease as the number

${n}_{j}$ increases; for

${n}_{j}$ > 1200 it assumes the lower values. Thus, the maximum value of the uncertainty

σ_{E} in averaging the component

p is obtained for

${n}_{j}$ = 100. In this case, it has been estimated that the uncertainty

σ_{E} is less than 2.4% in the

x direction and less than 4% in the

y direction. For

${n}_{j}$ > 1200 the uncertainty

σ_{E} becomes less than 1% in both the directions

x and

y. This means that a large sample size is necessary to obtain the same uncertainty in averaging the component

p.

The deviation of the time-averaged velocity components, estimated for

${n}_{j}$ = 100, 500, 1000, 1500, 2000, from those evaluated for the highest number values of the total number of pairs of frames (

${n}_{j}$ = 2400) has been obtained as:

Figure 5 reports the estimated values of

${\sigma}_{{n}_{j}}\left(\overline{p}\right)$ against

${n}_{j}$. From this figure it can be observed that

${\sigma}_{{n}_{j}}\left(\overline{p}\right)$ exponentially decreases as

${n}_{j}$ increases. Thus, for the examined velocity range, low and almost constant values of

${\sigma}_{{n}_{j}}\left(\overline{p}\right)$ are obtained for

${n}_{j}$ ≥ 1200, i.e., for a number of processed frames equal to or greater than half of the available number of pair of frames.

This behavior can be also observed from

Figure 6 where the values of the time-averaged velocity components (

$\overline{p}$ =

$\overline{{v}_{x}}$ and

$\overline{p}$ =

$\overline{{v}_{y}}$) estimated for

${n}_{j}$ = 100, 500, 1000, 1500, 2000, have been compared with those evaluated for

${n}_{j}$ = 2400. It can be seen that for all the considered images, and for both the velocity components, the points well fit the bisector line when

${n}_{j}$ > 1000.

Then, the sensitivity of surface velocity to the dimension of the interrogation area has been performed. The values of the component

$\overline{p}$ have been estimated by considering different values of the interrogation area (

IA_{k}). In this case, it has been assumed a constant value

${n}_{j}$ = 2400. According to the literature [

30], in order to keep the background noise in the correlation matrix low [

42,

43], the interrogation area was reduced in such a way as to obtain a minimum size of the interrogation area equal to one quarter of the initial size. This is in accordance with literature [

42,

43,

44] which shows that the size of the interrogation area should be not less than 4 times of the maximum displacement. Thus, the size of the interrogation area of the first pass in the processing PivLab code was defined taking into account that the value of the maximum displacement

lmax =

Umax *

dt (where

Umax is the maximum measurable velocity and

dt is the time step). The minimum size of the interrogation area was assumed equal to 8 × 8 pixels that is greater than 4 *

lmax = 5.6 pixels. The maximum size of the interrogation area was determined on the basis of the ROI dimension in the PivLab code.

Thus, the sensitivity analysis has been conducted for five (k = 5) values of the spatial resolution of the interrogation area: IA_{1} = 32 px; IA_{2} = 24 px; IA_{3} = 16 px; IA_{4} = 12 px; IA_{5} = 8 px. It is clear that as the size of IA_{k} decreases, the number of nodes N_{k} of the regular mesh increases. But, it should be considered that a high number of nodes determines modeling processes too much time-consuming.

Figure 7 plots, for all the images, the percent area

IA_{k}% against of the number of nodes

N_{k}.

Figure 7 indicates that the number of nodes increases with a logarithm law as the percent area decreases. Thus, a reduction of the interrogation area implies a remarkable increase of the number of nodes.

For each

IA_{k}, the cross-correlation has been determined with the PivLab tool and the velocity components

v_{x}(

t_{f}) and

v_{y}(

t_{f}) have been estimated in each node of the corresponding rectangular mesh. Then, the estimation error of the velocity values has been determined as:

where

N_{k} indicates the number of nodes corresponding to the

k-th interrogation area,

IA_{k}.

Figure 8 reports the highest value,

${\sigma}_{k}\left(\overline{p}\right)$, of the estimation error obtained varying the interrogation area

IA_{k}.From

Figure 8 it can be seen that

${\sigma}_{k}\left(\overline{p}\right)$ tends to decrease as the size of the interrogation area increases until that it assumes the lower values for 16 px <

IA_{k} < 20 px; then it increases as the size of the interrogation area increases. It should be noted that for

IA_{3} = 16 px the size of the interrogation area is reduced of one half compared to the original size.

Thus, the values of each time-averaged velocity component,

$\overline{p}$, estimated for the interrogation areas corresponding to

k = 1, 2, 4 and 5 have been compared with those evaluated by assuming

k = 3. This comparison is reported in

Figure 9 which shows that for

k > 2 the points tend to arrange around the bisector line, for

k ≤ 2 they tend to move away from the bisector line. This behavior suggests that, for the examined case, a reduction of the size of interrogation area of one half compared with the initial size represents a good compromise between the extension of the data sample and the accurate estimation of flow velocity. This result could also be consistent with previous works [

45] identifying a limit of the maximum recoverable spatial displacement in any sampling directions to half the window size in that direction.

Then, in order to verify the reliability of the estimated velocity measurements along the channel, the flux of mass in adjoining regions of consecutive images was also verified. As an example,

Figure 10 reports the comparison between the profiles of the specific longitudinal,

mx/

ρ =

h_{tr}$\overline{{v}_{x}}$ (h_{tr} = local water depth), and transversal,

my/

ρ =

h_{tr}$\overline{{v}_{y}}$, fluxes of mass estimated along two adjacent sections.

Figure 10 shows that the profiles of the specific flux of mass compare well both in longitudinal and in transversal directions.

#### 3.2. Discharge Estimation

Based on the results presented in the previous section, the discharge has been estimated by assuming

IA_{3} = 16 px and

${n}_{j}$ = 1500 and the surface velocity vectors have been determined, as explained in

Section 2.3, at nodes of the corresponding rectangular mesh.

Then, the transversal sections reported in

Figure 1c have been considered and the time-averaged velocity components along the transversal and the stream-wise directions (

$\overline{{v}_{{t}_{r}}}$ and

$\overline{{v}_{l}}$ respectively—see

Figure 11) have been estimated at the nodes included in each transversal section. For the aims of the present work, only the stream-wise velocity component

$\overline{{v}_{l}}$ has been considered.

According to reference [

10], it has been assumed that the shape of the vertical profile of the stream-wise velocity component is the same at each node of the transversal section. Furthermore, from the literature it is clear that the depth-averaged velocity can be estimated as a function of the free-surface velocity [

46] through a coefficient equal to 0.85, which is the value generally used with other measurement techniques. Recently, Termini and Moramarco [

6] by applying the entropy-based approach, verified that the maximum velocity is linearly related to the mean flow velocity through a dimensionless parameter

Φ(M) (M is the entropic parameter) which assumed a value of around 0.8 along the curved stretches of the channel. Other authors (see in [

47]) found a value of such a coefficient equal to 0.9.

In the present work, by taking into account that the water depths are very low (order of magnitude of 1 cm) the difference between the depth-averaged and the surface velocities has been assumed almost null so that the coefficient has been assumed equal to 1. The discharge has been determined as the sum of the elementary stream-wise flux per unit time estimated as:

$\partial q={\widehat{v}}_{l}{h}_{{t}_{r}}\partial {t}_{r}$ (

$\partial {t}_{r}$ = width of the elementary area in the transversal direction

t_{r}—see

Figure 11). In

Figure 12 the estimated values of the discharge (

qe) have been compared with those measured (

q_{m}) in sections 4, 5, 11, 12. From the

Section 2.1 it is clear that the value of flow discharge measured by the ultrasonic instrument at sections 4 and 5 is equal to QI1 = 0.004 m

^{3}/s and that measured at sections 11 and 12 is equal to QI1 + QI2 = 0.00537 m

^{3}/s. Furthermore,

Table 1 reports the values of the normalized error in the discharge estimation:

Figure 11 shows that the points arrange around the line of perfect agreement demonstrating a good agreement between the estimated and the measured values of the discharge. From

Table 1 it can be seen that the highest value of error is of 22.0 (%).