2.1 Undistorted and distorted models of urban flooding

Laboratory models have been used for studying urban flooding at various levels, ranging from limited spatial extents (e.g. a single storm drain) up to the level of a whole urban district. Some laboratory models focussing on a limited spatial extent were constructed at the prototype scale, i.e. with a scale factor equal to unity (Djordjevic et al., 2013; Lopes et al., 2013, 2017). At intermediate levels, such as when a single street or single crossroads are represented, scale factors of the order of 10–20 were used (Lee et al., 2013; Mignot et al., 2013; Rivière et al., 2011) which are generally deemed not to lead to excessive scale effects (Chanson, 2004). In contrast, when it comes to the experimental analysis of flooding at the level of an entire urban district, the spatial extent of the prototype to be represented becomes considerably larger (∼ 10²–10³ m), as summarized in Table 1 and sketched in Supplement 1, so that scale factors reach values as high as 100–200.

LaRocque et al. (2013) focussed on a limited portion of an urbanized floodplain (280 m × 124 m) and used a scale factor of 50, leading to a model Reynolds number of the order of $\sim \mathrm{7}\times {\mathrm{10}}^{\mathrm{4}}$. Testa et al. (2007) considered a scale factor of 100 for studying transient flooding of a group of buildings extending over 160 m × 120 m. This led nonetheless to model Reynolds numbers exceeding 10⁵ because extreme flooding scenarios were tested (dam-break-induced flood). To analyse river flooding at the level of an entire urban district (1 km × 2 km), Ishigaki et al. (2003) also used a scale factor of 100; but in this case, despite a particularly large experimental facility (10 m × 20 m), the model Reynolds number was of the order of 7×10³, with water depth lower than 1 cm and being even lower than this amount in some streets. Ishigaki et al. (2003) reported that the observed flow became “laminar” in some parts of the model, hence exacerbating scale effects. This questions the validity of the upscaled lab observations and highlights a difficulty in the design of experimental models for analysing urban flooding, namely the substantial difference between the characteristic length in the horizontal direction (e.g. street width) and in the vertical direction (e.g. water depth). This issue is similar to the case of laboratory models of large rivers and coastal systems (Wakhlu, 1984).

To partly overcome this difficulty, so-called distorted laboratory models have also been used. They consist of applying distinct scale factors, respectively, e_H and e_V, along the horizontal and vertical directions:

where H_p and H_m are characteristic heights in the prototype and in the model, respectively. Since L_p is always much higher than H_p, using a horizontal scale factor e_H larger than the vertical scale factor e_V enables preserving higher water depths in the laboratory model compared to an equivalent undistorted model requiring the same horizontal space in the laboratory. This approach offers several advantages: (i) inaccuracies in water depth measurement become smaller in relative terms, and (ii) the Reynolds number is higher so that some artefacts (e.g. viscous effects) are minimized due to a change in the turbulence regime. By using a distorted model, it is even possible to keep both Reynolds and Froude numbers identical in the prototype and in the model, with the same fluid (Finaud-Guyot et al., 2018). Distorted models have been used for a broad range of applications in fluvial and coastal hydraulics. Among others, Jung et al. (2012) applied scale factors e_H=120 and e_V=50 to study a floating island in a river; Wakhlu (1984) obtained comparable results on an undistorted ($e_{\mathrm{H}}=e_{\mathrm{V}}=\mathrm{36}$) and a distorted model (e_H=100; e_V=17) of a river division weir. However, since a distorted laboratory model corresponds to a representation of the prototype shrunk differently along the horizontal and the vertical directions, it may also lead to specific artefacts in the laboratory observations. For instance, Sharp and Khader (1984) highlighted distortion effects by comparing an undistorted ($e_{\mathrm{H}}=e_{\mathrm{V}}=\mathrm{20}$) and a distorted model (e_H=400; e_V=100) to study wave transmission and assess stone stability in a harbour.

Figure 1Recent laboratory models of urban flooding at the district level as a function of the horizontal and vertical scale factors, e_H and e_V. The grey shading qualitatively reveals the possible magnitude of (a) scale effects and (b) distortion effects; a range of maximum scale factors (purple lines) and distortion ratio (orange lines) were recommended by Chanson (2004).

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2.2 Recent studies based on distorted models of urban flooding and potential artefacts

Figure 1 shows the horizontal and vertical scale factors used in recent laboratory studies of urban flooding at the district level. The grey shading in Fig. 1a suggests conceptually that the larger the scale factors, the greater the expected scale effects, while Fig. 1b indicates that using a strongly distorted model (i.e. a large ratio e_H∕e_V) also leads to specific artefacts, which we refer to hereafter as distortion effects. Based on experience, Chanson (2004) suggests keeping the ratio e_H∕e_V below 5–10 (orange dotted lines in Fig. 1b).

An outdoor distorted model was used by Smith et al. (2016) to represent pluvial flooding in an urban district of relatively limited extent (Table 1). The scale factors were as low as e_H=30 and e_V=9 (distortion ratio: $e_{\mathrm{H}}/e_{\mathrm{V}}=\mathrm{3.3}$), thus minimizing potential scale effects. Similarly, Güney et al. (2014) used an outdoor distorted model of a large urban district (Table 1), with e_H=150 and e_V=30 (distortion ratio: $e_{\mathrm{H}}/e_{\mathrm{V}}=\mathrm{5}$), to represent dam-break-induced flood waves. The distortion ratio of these two models remains below the upper bound of 5–10, as recommended by Chanson (2004). Lipeme Kouyi et al. (2010), Araud (2012) and Finaud-Guyot et al. (2018) considered the same geometric configuration (Supplement 1) involving seven streets aligned along one direction, crossing seven other streets. The scale factors used by Lipeme Kouyi et al. (2010) were e_H=100 and e_V=25. The set-up of Araud (2012) and Finaud-Guyot et al. (2018) contains substantial improvements compared to the initial set-up of Lipeme Kouyi et al. (2010), mainly regarding the control of the inflow in each street separately, but it uses a considerably higher horizontal scale factor (e_H=200) and, simultaneously, a smaller vertical scale factor (e_V=20). This leads to a particularly high ratio e_H∕e_V, equal to the upper limit of 10 suggested by Chanson (2004). If the model of Araud (2012) and Finaud-Guyot et al. (2018) had not been distorted, the Reynolds numbers would have been about 30 times lower ($R\sim \mathrm{1}\times {\mathrm{10}}^{\mathrm{2}}-\mathrm{1}\times {\mathrm{10}}^{\mathrm{3}}$) than they actually are (Table 1).

2.3 Specific objective of the present study

While the motivations for using a large distortion ratio between the horizontal and vertical scale factors are not arguable (fit the model within a limited laboratory space, improve the accuracy of water depth measurement and maintain a sufficiently high Reynolds number), the assumption of having no artefacts in experimental observations performed on a strongly distorted model may legitimately be questioned. Among other aspects, the complex three-dimensional flow structures observed in individual crossroads (Mignot et al., 2008, 2013; Rivière et al., 2011, 2014) suggest that “shrinking” the model vertically is likely to alter these flow structures and hence also impair the representation of flow partition in-between the streets. The influence of strong distortion in laboratory-scale models was investigated for some specific applications, such as in coastal engineering (Ranieri, 2007; Sharp and Khader, 1984), but it has not been analysed to date in the context of laboratory models of urban flooding.

Therefore, in this paper, we aim to evaluate the artefacts arising from the use of distorted laboratory models in experimental studies of urban flooding at the district level. We base our assessment on the reanalysis of two recent datasets, presented, respectively, by Araud (2012) and by Velickovic et al. (2017). The latter does not represent a realistic urban district but solely a regular grid of obstacles, which is similar to a network of streets to some extent. We focus on the influence of model distortion on the observed water depths and discharge partition in-between the streets.

Table 2Initial interpretation of the laboratory model runs as various flooding scenarios represented with fixed scale factors (Araud, 2012; Finaud-Guyot et al., 2018) vs. interpretation in the present reanalysis, involving a single flood scenario represented with various vertical scale factors. Notations Q_m and Q_p refer to the total inflow in the laboratory model and in the prototype urban district, respectively.

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3.1 Datasets

Two datasets were reanalysed, corresponding, respectively, to an entire urban district and to a group of buildings. The former dataset was collected by Araud (2012) in the ICube laboratory in Strasbourg (France) and was also presented by Arrault et al. (2016) and Finaud-Guyot et al. (2018) (Fig. S3a in Supplement 2). The experimental model (5 m × 5 m) represents an idealized urban district of 1 km × 1 km at the prototype scale. It contains a total of 14 streets of various widths (0.05–0.125 m) and 49 intersections (crossroads). The inflow discharge was controlled in each street individually, the uncertainty of the inflow discharge is estimated at about 1 % (Fiaud-Guyot et al., 2018) and the outflow discharges were monitored downstream of each street. The uncertainties in the estimation of the outflow discharges are discussed in Sect. 4.1. For several steady inflow discharges, the water depths along the centreline of streets were measured using an optical gauge (1 mm accuracy; Figs. S4–S7). Hereafter, we consider the experimental runs performed with a total inflow discharge Q_m of 20, 60, 80 and 100 m³ h⁻¹ in the laboratory model (Table 2). In each test, 50 % of the total inflow discharge was fed to the west face of the model and 50 % to the north face of the model. The specific inflow discharge was kept the same for each street of a given face, the outflow discharge was estimated from a calibrated rating curve (Araud, 2012). As detailed in Supplement 2–5, several measurements were repeated, which allows appreciating the reproducibility of the tests.

Table 3Interpretation of the dataset of Velickovic et al. (2017) in the present reanalysis. Notations Q_m and Q_p refer to the total inflow discharge in the laboratory model and in the prototype, respectively.

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The second dataset was collected by Velickovic et al. (2017) in the Hydraulic Laboratory of Université Catholique de Louvain, Belgium. A group of 5×5 square obstacles (buildings) of 0.30 m × 0.30 m were installed in a horizontal flume 36 m × 3.6 m. Several layouts of obstacles were considered, and we analyse three of them here (aligned with the channel; Fig. S3b). For each of these layout, between four and six experimental runs (Table 3) were conducted with various steady inflow discharges (accuracy of flowmeters: ∼ 1 L s⁻¹). The three layouts differ by the distance in-between the obstacles (i.e. the street widths) in the direction normal to the main flow (0.0675, 0.10 and 0.135 m). Profiles of water depth (accuracy: 0.1 mm) were measured with movable ultrasonic probes along the centreline of the streets aligned with the main flow direction. In contrast with the dataset of Araud (2012), the flow partition in-between the streets was not measured by Velickovic et al. (2017).

Figure 2(a) Sketch of the initial interpretation of the laboratory model runs as various flooding scenarios represented with fixed scale factors. (b) Sketch of the interpretation in the present reanalysis, involving a single flood scenario represented with various vertical scale factors e_V.

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3.2 Method

As sketched in Fig. 2a, the initial interpretation of the various experimental runs of Araud (2012) and Velickovic et al. (2017) was that each model run corresponds to a different flooding scenario (i.e. a different total inflow discharge Q_m into the urban district) represented with fixed scale factors e_H and e_V. In contrast, in the present reanalysis of the laboratory dataset, we propose considering that a single flooding scenario was represented in each laboratory model but that the various experimental runs actually correspond to different vertical scale factors e_V. This new perspective is sketched in Fig. 2b.

As detailed hereafter, we followed a three-step procedure for the reanalysis. For each model, the procedure was as follows:

1. Select one experimental run and assign to it plausible scale factors e_H and e_V.

2. Estimate the scale factor e_V corresponding to each of the other experimental runs, assuming that e_H remains unchanged and that all experimental runs simulate the same flood scenario.

3. Upscale the experimental observations of each run to the prototype scale and compare them to each other.

4.1 Experimental uncertainties

The measurement accuracy and the fluctuations of water surface are considered the two main sources of uncertainty in the experimental records of water depths. For the dataset of Araud (2012), the accuracy of the optical gauge is about 1 mm. As shown in Supplement 2, the water depths were measured twice at some locations for Run 3 (60 m³ h⁻¹) and Run 2 (80 m³ h⁻¹). The repeatability of the tests is evaluated by comparing the repeated measurements. As detailed in Supplement 3, the difference in water depths between two measurements remains below 2 mm for 90 % of the dataset. Therefore, we estimate here the water depth measurement uncertainty at about 2 mm. Representative profiles of measured water depths (including repetitions) are displayed in Supplement 4, and they confirm the validity of this uncertainty estimate. Uncertainties arising from inaccuracies in the inflow discharge measurement are also lumped into this uncertainty estimate.

The discharge at the outlet of each street was estimated from a rating curve corresponding to a weir at the downstream end of dedicated measurement channels (located downstream of each street outlet). The water depth measurements were performed at least three times for each model run. As detailed in Supplement 5, comparing the repeated measurements demonstrates an excellent repeatability of the outflow discharge estimates (Figs. S12–S15). Hence, the main source of uncertainty in the outflow discharge estimates is assumed to stem from the accuracy of water depth measurement (1 mm) over the measurement weirs (Araud, 2012). This leads to about 2 %–6 % uncertainty in the outflow discharges, as shown in Fig. S16.

Figure 3Upscaled water depth profiles in prototype in street 4 of Araud (2012) predictions, colour shade represents the upscaled measurement uncertainty.

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For the second dataset, the accuracy of the probes, as described by Velickovic et al. (2017), is 0.1 mm. Velickovic et al. (2017) also reported the standard deviation of the recorded water depths at each probe location. We used this as a proxy for estimating the flow variability. Figure S9 in Supplement 3 shows the cumulative distribution function of this standard deviation over all measured data in the three layouts and for each experimental run. For 90 % of the measurement points, the standard deviation remains below 2–6 mm (depending on the experimental run), which reflects considerable fluctuations in the free surface, particularly for the higher flow rates.

Figure 4Spatial distributions (a, c, e) and scatter plots (b, d, f) of the differences between the upscaled water depths in Run 3 (d=7.1), 2 (d=8.6) and 1 (d=10), and those derived from Run 4 (d=3.4) from Araud (2012). The dashed purple lines indicate the range containing 90 % of the data.

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4.2 Effect of model distortion on predicted water depths

4.2.1 Dataset of Araud (2012)

All measured water depths were upscaled to the prototype scale using the vertical scale factors defined in Table 2. As exemplified in Fig. 3 for street 4 (and in Figs. S17 and S18 in Supplement 6 for streets A, B, and C), the tests conducted with the strongest distortion (d=8.6 and d=10) lead to the lowest estimates of water depths at the prototype scale. Conversely, the highest predicted water depths correspond systematically to the upscaled measurements from the tests involving a weaker distortion (d=3.4), except nearby the downstream boundary conditions where all predicted water depths are close to each other. The tests conducted with d=7.1 lead to intermediate estimates of water depths at the prototype scale. The reasons for these differences are discussed in Sect. 5.1. Here, the uncertainty associated to the upscaled water depths is larger when the model distortion is limited, since the absolute value of the measurement uncertainty remains the same in all model runs, but a larger vertical scale factor is applied.

Table 4Differences between upscaled water depths derived from various runs of Araud (2012).

^∗ MD stands for mean difference.

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Figure 5Cumulative distribution function (CDF) of the differences between upscaled water depths from the various model runs of Araud (2012); the grey lines represent the upscaled measurement uncertainties with various e_V (the larger the e_V, the deeper the line colour).

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In Figs. 4 and 5 and Table 4, we quantitatively compare the upscaled water depths obtained from the four different model runs listed in Table 2. The model run with the weakest distortion is used as a reference for comparisons. The following observations can be made:

• The scatter plots in Fig. 4 confirm that the water depths derived from the model with the lowest distortion are generally higher than those derived from the other model runs.

• The dashed purple lines (and corresponding labels; Fig. 4b, d, f) indicate the range containing 90 % of the data. This shows that, despite the uncertainties in the measurements, the differences between the model runs increase very consistently as the model distortion increases. This consistent trend is also emphasized by the evolution of the mean difference (MD in Table 4) from −0.058 to −0.107 as d varies from 7.1 to 10 as well as by the cumulative distribution of the differences in upscaled water depths derived from the various model runs (Fig. 5).

• The 5th percentile of the differences between the models with varying distortions is of the order of −20 to −25 cm (Table 4). When compared to the order of magnitude of the absolute value of water depths (∼ 2 m), the effect of model distortion in the tested configurations is of the order of 10 % of the upscaled water depths.

• The spatial distributions provided in Fig. 4 also reveal that the influence of model distortion tends to increase in the upstream part of the urban district (i.e. closer to the north and west faces). Two reasons contribute to explain this: (i) the water depths in the downstream part are mainly controlled by the free outflow boundaries and remain therefore more similar whatever the distortion; and (ii) the cumulated effect of friction (expected to be underestimated in the more distorted models compared to the less distorted or undistorted models, as detailed in Sect. 5.1) is stronger in the upstream part of the flow.

Table 5Differences between upscaled water depths derived from various runs of Velickovic et al. (2017).

^∗ MD stands for mean difference.

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4.2.2 Dataset of Velickovic et al. (2017)

Figure S19 in Supplement 7 displays the upscaled water depth profiles based on the dataset of Velickovic et al. (2017) and the scale factors defined in Table 3 for each of the three layouts. The upscaled water depths in the three model layouts differ substantially as the same discharge Q_p is imposed (h_p in narrow street: ∼ 5 m; median street: ∼ 3.5 m; wide street: ∼ 2.5 m). Now, for each model layout separately, we compare the observations corresponding to various inflow discharges in the same way as in the dataset of Araud (2012). It is found that the upscaled water depths in the urban area become systematically lower as the model distortion increases. This is observed consistently for the three layouts of obstacles (narrow, intermediate and wide streets), and the differences arising from the change in model distortion greatly exceed the estimated experimental uncertainty. This result is also confirmed by the values of mean differences (MDs) reported in Table 5 as well as by the cumulative distribution functions of the differences in water depths (Fig. S20 in Supplement 7). In each layout, the model characterized by the lowest value of d was taken as a reference for comparison.

Figure 6(a) Upscaled outflow discharge in each street compared to the corresponding value deduced from the observations in the less distorted model (d=3.4), and (b) ratio of the associated water depths from Araud (2012) and Finaud-Guyot et al. (2018). Red dashed lines represent the ±10 % range.

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4.3 Effect of model distortion on outflow discharges

For the dataset of Araud (2012), the upscaled values of the discharge at the outlet of each street are shown in Supplement 8 (Fig. S21). Although the differences in the outflow discharge appear to be of the same order of magnitude as the experimental uncertainties, the variation in the outflow discharge partition with d shows a consistent monotonous trend (increasing, constant or decreasing) in all outlet streets. Since the mass balance remains unchanged at the level of the whole district, the direction of change in the outlet discharge varies from one street to the other. In Fig. 6a, we compare the upscaled outflow discharge in each street to the corresponding value derived from the less distorted model (d=3.4). The changes in the estimated outflow discharge when d is varied are in the range of ±10 % in 11 streets out of 14 (and are in the range of ±12 % in 13 streets out of 14, with a maximum change of +18 % in just a single street).

The magnitude of model distortion effects may differ from one flow variable to the other (Heller, 2011). To be able to appreciate the relative influence of model distortion on the outflow discharge and on the water depths, for each street outlet (noted i), we estimated the change in upscaled water depth (h_i∕h_ref) “associated” with the observed change in the outflow discharge (Q_i∕Q_ref) when the model distortion is varied (from d_ref to d_i). To do so, using Froude similarity leads to $h_i/h_{\mathrm{ref}}=(Q_i/Q_{\mathrm{ref}}{)}^{\mathrm{2}/\mathrm{3}}{e}_{\mathrm{V},i}/e_{\mathrm{V},\mathrm{ref}}$. As shown in Fig. 6, the results indicate that, in the present case, the magnitude of model distortion effects appears to be relatively comparable for the water depths and the outflow discharges (of the order of ±10 %).

5.1 Relative importance of the main causes of distortion effects

The differences in the upscaled water depths as a function of the model distortion d may result from differences in localized flow features (e.g. local head losses at street intersections) and from more distributed effects, which in the present case are likely to dominate only along the mainly one-dimensional flow regions (typically within a street). The latter effects may be categorized into three types:

• differences in the relative importance of the bottom roughness,

• differences in the relative importance of viscosity effects,

• differences in the aspect ratio (ratio between water depth and street width) of the flow section, notably affecting the secondary currents and velocity profiles of streamline velocity.

It is possible that the third of these effects has a substantial influence on the flow. However, the available datasets, involving only observed water depths and discharges at the model outlets, do not enable investigating these more localized effects in detail. This could be achieved by means of new experiments with detailed velocity measurements within the urban district. 2-D and 3-D computational modelling may also be useful in this respect, the former enabling us to account for the spatial variations of flow characteristics and the latter giving us full access to the complex flow fields developing at the street intersections.

Nonetheless, we appreciate here the influence of the first two types of effects, which are of relevance to mainly one-dimensional flow regions. To do so, we estimate the energy slope S_f,p at the prototype scale in the various model configurations based on the Darcy–Weisbach equation.

In this equation, the friction coefficient f_m can be computed as a function of the Reynolds number R_m and the relative roughness height $k_{\mathrm{s},\mathrm{m}}/R_{\mathrm{H},\mathrm{m}}$ using the explicit approximation of the Colebrook–White formula given by Yen (2002). The roughness height k_s,m was taken to be equal to 10⁻⁵ m to represent the smooth bottom and walls of the experimental set-ups.

Figure 7Distortion effect on energy slope of (a) scale models and (b) the model in prototype (dataset of Araud, 2012).

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Figure S22 shows the values of parameters R_m, $k_{\mathrm{s},\mathrm{m}}/R_{\mathrm{H},\mathrm{m}}$, f_m, $h_{\mathrm{m}}/R_{\mathrm{H},\mathrm{m}}$, and F, which are representative of the flow conditions at the street inlets in the various experimental runs of Araud (2012). Given the values of R_m and $k_{\mathrm{s},\mathrm{m}}/R_{\mathrm{H},\mathrm{m}}$, the flow is in a transitional regime in the Moody diagram. All these parameters change substantially when the model distortion d is varied (Fig. S23). As d is increased, the friction coefficient f_m decreases systematically ($f_{i,\mathrm{m}}/f_{\mathrm{ref},\mathrm{m}}\sim \mathrm{0.8}$) due to the joint effect of a higher Reynolds number and a lower relative roughness height ($k_{\mathrm{s},\mathrm{m}}/R_{\mathrm{H},\mathrm{m}}$). Similarly, the Froude number tends to slightly increase with the model distortion. In contrast, parameter $h_{\mathrm{m}}/R_{\mathrm{H},\mathrm{m}}$ shows a considerable systematic increase (1.6–2.3) as d increases due to the change in the aspect ratio of the flow section. The resulting energy slope S_f,m in the inlet streets in each model run and the corresponding upscaled values S_f,p are presented in Fig. 7. The energy slope is lower in major streets (4, C, F) and in the straight streets (A, B). The energy slope S_f,m in the model increases monotonously with the model distortion (mainly due to the increased wetted perimeter), whereas the energy slope S_f,p at the prototype scale declines as d is increased (Fig. 7b). This results from a “competition” between the increase in S_f,m and a decrease in the ratio e_V∕e_H, as d increases. This effect appears dominant in the mainly one-dimensional flow regions.

5.2 Reference used for comparison

Here, we used the “less distorted” models as a reference for our comparisons since the aim is to assess the influence of model distortion. However, given that we simply reanalysed already existing datasets obtained in experiments which were not designed for the sake of investigating the effect of model distortion in the first place, these less distorted models are also those characterized by the largest vertical scale factors, hence enhancing other artefacts such as stronger viscosity effects (see asterisks in Fig. 1). Also, the upscaled measurement uncertainties are at a maximum in this case, as can be seen in Fig. 3. Therefore, the present study needs to be complemented by new tailored experiments involving a model series (Heller, 2011) with various levels of distortion and including a valid reference model, i.e. at least one model without distortion and that is characterized by sufficiently small scale factors so that viscosity effects and experimental uncertainties are kept to a minimum. This may be achieved by considering a model series in which the vertical scale factor e_V is kept constant in all models, while only the horizontal scale factor e_H is varied. This approach contrasts with the procedure followed here in which e_H was constant and e_V was varied to take benefit of existing datasets.