Zhaolin Xiao

Jinglei Shi

Xiaoran Jiang

Christine Guillemot

Ieee Fellow

Axial refocusing precision model with light fields

Keywords: Light field, Computational photography, Axial resolution, Sampling analysis, Digital refocusing

Axial refocusing precision model with light fields

I. INTRODUCTION

L IGHT field imaging has become popular in the last years, due to its potential for a variety of applications. In particular, post-capture digital refocusing can be easily implemented by shifting and adding the sub-aperture images [START_REF] Isaksen | Dynamically reparameterized light fields[END_REF], [START_REF] Wu | Light field image processing: An overview[END_REF], [START_REF] Ihrke | Principles of light field imaging: Briefly revisiting 25 years of research[END_REF], [START_REF] Wetzstein | Computational plenoptic imaging[END_REF], [START_REF] Levoy | Light fields and computational imaging[END_REF]. In theory, the light field focal plane can be moved continuously along the axial direction. However, in real systems, due to limited spatial resolution and baseline, the number of distinguishable focal planes is limited in the axial direction. In order to distinguish from the concept of NA in classical imaging, we use the baseline to denote the distance between two farthest separated angular views throughout this paper. In traditional imaging, the notion of depth of field (DoF) is a widely accepted metric for axial resolution, and for which a theoretical analysis can be found in [START_REF] Levoy | Light field microscopy[END_REF]. The definition of DoF concerns the acceptable sharpness within a given axial range, but we are instead interested in the focus precision, i.e.the feature separability between each focal plane and its adjacent planes when applying digital refocusing.

In this paper, we introduce a light field axial refocusing precision (LF-ARP) model within a general virtual camera. This model allows us to calculate the minimum spacing between two distinguishable adjacent refocusing positions in the axial dimension, given the fixed focal length, the baseline of the main lens (or aperture size), the sensor size and its resolution, and the object plane distance. Please note that the considered refocusing precision differs from the notion of DoF, it does not only consider the in-focus regions, but also the outof-focus blur. The out-of-focus blur is function of the distance between the objects and the current focal plane, which can affect the possibility to distinguish objects at different axial positions.

The refocusing precision can be essential for some applications, e.g. light field microscopy [START_REF] Prevedel | Simultaneous whole-animal 3d imaging of neuronal activity using light-field microscopy[END_REF], [START_REF] Palmieri | Robust depth estimation for light field microscopy[END_REF], and light field particle image velocimetry (LF-PIV) [START_REF] Belden | Threedimensional synthetic aperture particle image velocimetry[END_REF], [START_REF] Skupsch | Multiple-plane particle image velocimetry using a light-field camera[END_REF]. However, we show that the classical Fourier slicing refocusing method [START_REF] Ng | Fourier slice photography[END_REF] or the spatial shifting-adding re-sampling technique [START_REF] Isaksen | Dynamically reparameterized light fields[END_REF] produce refocused images do not allow differentiating features within the predicted precision, as shown in Figures 1(a) and (b) where we used the same focal plane setting. When using the shiftand-add refocusing method on the original light field, one can hardly see refocusing differences in Figure 1-(a). On the contrary, Figure 1-(b) shows that, by enhancing the refocusing precision, e.g. by increasing the baseline, we can better distinguish the objects at different axial positions. Please notice that, while the light field baseline often denotes the spacing between two adjacent views (or cameras), here the term baseline will refer to the spacing between two farthest views (or cameras). To extend the angular baseline, we introduce a learning-based view extrapolation algorithm in [START_REF] Xiao | A learning-based view extrapolation method for axial super-resolution[END_REF]. As shown in Figure 1, if the different objects need to be distinguished via refocusing, one can optimize the capture configuration parameters, such as the focal length, the aperture size or the baseline, the distance to the object plane. Here, we present a geometrical analysis of the axial precision model, which reveals the relations between the axial refocusing precision and the camera parameters. Quantitative and qualitative experiments are also conducted using both real and synthetic light field datasets.

II. RELATED WORK

A light field can be seen as a sampling of the plenoptic function introduced by Adelson and Bergen in [START_REF] Adelson | The plenoptic function and the elements of early vision[END_REF] to describe the radiance of light rays emitted by a scene along any direction, at any time and for every wavelength. By assuming that the light rays are not attenuated along a line, light fields are commonly represented as 4D functions with spatial and angular coordinates [START_REF] Levoy | Light field rendering[END_REF], [START_REF] Gortler | The lumigraph[END_REF].

Let LF (x, y, u, v) be the 4D representation of a light field, in which XY and UV stand for the spatial and angular planes respectively. For sake of simplicity, in the rest of the manuscript, we consider a 2D slice LF (x, u) of the light field. At capture time, physically changing the object distance or the focal length are two most straight forward operations to adjust image focus. Chang et al. [START_REF] Chang | Towards multifocal displays with dense focal stacks[END_REF] designed a focus-tunable lens with precise tracking of the focal length, which can scan 1600 focal planes per second.

Post-capture synthetic refocusing [START_REF] Isaksen | Dynamically reparameterized light fields[END_REF] can also produce a focal stack, i.e. a sequence of images each focused on a different plane, by re-sampling the input light field as

I α (x) = u LF 0 (x + αu, u) (1)
Therefore, one can generate different refocusings from the light field LF 0 by varying the parameter α, which implies the focal depth. Theoretically, one can set a focal plane Ω α 0 at a different axial position by tuning the parameter α. However, the number of focal stack images, or slices in the resulting volume, is limited by the axial resolution of the light field. Indeed, if α is small, then even for the outmost view u max , the disparity variation α * u max may be less than 1 pixel.

In this case, without sub-pixel interpolation, the re-focused images obtained by the re-sampled light field will appear the same as the re-focused images computed from the original light field. Similarly, if the disparity of the outmost view u max with respect to the central one is less than 1 pixel, then the shearing angle will be less than θ min (red box in Figure 2-(a)), and the re-focused images with the original or re-sampled light fields will not show any difference without sub-pixel interpolation. If the shearing angle is larger than θ min , it will be obviously resolvable because of the contribution of the new angular samples, as shown in Figure 2-(d) (marked in the slanted red line).

In classical imaging systems, increasing the numerical aperture (NA) decreases the DoF, but leads to a higher axial resolution (also called depth resolution) [START_REF] Murphy | Fundamentals of Light Microscopy and Electronic Imaging[END_REF], [START_REF] Martinez-Corral | Fundamentals of 3d imaging and displays: A tutorial on integral imaging, light-field, and plenoptic systems[END_REF]. Many solutions have been proposed to deal with the narrow DoF problem, such as image deblurring based on 3D PSF modeling and all-in-focus image fusion from multiple axial scans [START_REF] Martínez-Corral | Chapter 1 The Resolution Challenge in 3D Optical Microscopy[END_REF]. In the object space, Chen et al. [START_REF] Chen | Distance measurement based on light field geometry and ray tracing[END_REF] and Hahne et al. [START_REF] Hahne | Refocusing distance of a standard plenoptic camera[END_REF] proposed different optical models to accurately measure the distance of the object plane based on a geometric analysis of the standard plenoptic camera. Further, Hahne et al. [START_REF] Hahne | Refocusing distance of a standard plenoptic camera[END_REF] derived the distance to the refocused object plane and its corresponding DoF for different light fields, which has been experimentally verified by placing objects at the predicted distances.

Unlike the DoF distance in [START_REF] Hahne | Refocusing distance of a standard plenoptic camera[END_REF], the separability of LF-ARP refers to whether light field re-sampling will affect the integral imaging result, considering both the sharp focused regions and the blurred defocused regions. For each axial position, we derive the minimum light field re-sampling so that shifts of 1 pixel of the most extreme views, when constructing the focal stack with the shift-and-add procedure, yields distinguishable refocusing planes. The refocusing precision Arp is equivalent to the DoF only if the DoF of all the focal slices are nonoverlapping, which is not verified in most of cases.

III. AXIAL REFOCUSING PRECISION

We start by defining the refocusing precision in light field imaging and by recalling the notion of DoF. Then, we describe the proposed light field axial refocusing precision (LF-ARP) model.

A. Refocusing precision: definition

Let Ω 0 be a given focal plane on the object side, Ω α + 0 and Ω α - 0 are the nearest distinguishable planes around Ω 0 from the far side and the near side respectively. The refocusing precision is defined as the minimum distance between two distinguishable adjacent focal planes Ω α + 0 and Ω α - 0 in the object space, and can be expressed as

Arp (LF 0 , Ω 0 ) = d Ω α - 0 , d Ω α + 0 s.t. LF α 0 -LF 0 ∞ < ε, if α ∈ [α -, α + ] (2)
where d(Ω 0 ) stands for the distance between the target focal plane of Ω 0 (on the object side) and the camera plane (or the aperture plane) U V . Here, the U V plane is assumed to lay at the zero axial position. The parameter ε denotes a negligible difference between the original light field LF 0 and the resampled one LF α 0 . Figure 3 shows the difference between the axial refocusing precision and the DoF in the object space, and shows that the two adjacent distinguishable focus planes and the image planes located at distances d Ω α - 0 and d Ω α + 0 from the camera plane do not correspond to the borders of the DoF. The DOF is the distance (red and blue dash lines), within which the object is in-focus, i.e. without optical blur. The DOF has been exaggerated compared to the dimension of the imaging system for better visibility. Due to finite and discrete sampling of the 4D light field, the Arp (LF 0 , Ω 0 )) cannot be infinitely small with a fixed spatioangular resolution. Therefore, deriving the range [α -, α + ] for each possible focal plane Ω 0 is the main goal of this paper.

B. Depth of Field (DoF)

The axial re-focusing precision has some relationship with the concept of DoF. In traditional imaging, the DOF limit is given by

DoF = λn NA 2 + n M NA e (3)
where λ stands for the light wave length, n represents the refraction index of medium, and NA is the numerical aperture of the entire imaging system. The variable e is the minimum resolvable distance in the image plane of the main lens, whose lateral magnification from object to image plane is M . Theoretically, the total DoF is determined primarily by wave optics (first term of Equation ( 3)) if the pixels are small enough to not limit resolution. But, in case of lower numerical apertures, the DoF is dominated by the geometrical optical circle of confusion (CoC) represented by the second term of Equation ( 3). Please note that, if assuming the DoFs of all the focal slices are non-overlapping in a focal stack, only the in-focus regions are concerned, then the refocusing precision Arp is equivalent to the DoF . However, this assumption usually does not hold in real light field imaging systems for the two following reasons. First, since the pixel size may not be small enough to satisfy the theoretical wave DoF limit, the actual DoF of the different slices of the focal stack may be larger and overlapping in most practical systems, as shown in Figure 3. Second, the refocused pixel size is N u ×N v times larger than the sensor pixel size or the diffraction-limited spot size. Here, N u × N v is the angular resolution. Therefore, in the case of real light field imaging systems, and assuming N u = N v , the DoF of refocused image is given by [START_REF] Levoy | Light field microscopy[END_REF]

DoF ≈ (2 + N u ) λn 2NA 2 (4)
where the geometrical optical CoC, which is N u /2 times larger than the wave optics term, dominates. As a consequence, the DoF of light field imaging is significantly larger than the DoF of traditional imaging with the same resolution sensor. Therefore, in order to obtain a better spatial or lateral resolution, Broxton et al. [START_REF] Broxton | Wave optics theory and 3-D deconvolution for the light field microscope[END_REF] suggest to apply 3D-deconvolution on the defocused angular samples. The DoF resolution can also be enhanced by applying image based super-resolution, e.g. the state-of-art learning based light field super-resolution [START_REF] Jin | Light field spatial superresolution via deep combinatorial geometry embedding and structural consistency regularization[END_REF].

While the DOF only measures the range of in-focus area, our motivation here is to provide a model that measures the distance between adjacent distinguishable focus planes. Unlike the discussion on the distance of refocused object plane and the corresponding DoF in [START_REF] Hahne | Refocusing distance of a standard plenoptic camera[END_REF], the proposed model reveals the refocusing or re-sampling precision given a constant image plane and fixed field of view (FoV).

C. Axial refocusing precision model

In this section, we propose a refocusing precision Arp model, in which both the image plane and the FoV are assumed to be fixed. Assuming the image plane distance V to be constant when re-focusing with light field imaging, the following thin lens equation gives the relationship between the object distance d and the virtual focal length f after re-focusing

1 d + 1 V = 1 f (5)
where the focal length f is a variable, and d denotes the distance between the virtual focal plane and the physical main lens.

Let us now consider re-focusing with the shift-and-add procedure. Let r x and r y represent the horizontal and vertical resolutions of the sensor, and b stands for the baseline width.

Considering a 1 pixel shift, by geometrically analyzing Figure 4, one can deduce that the light field refocusing precision Arp, for any possible focal plane Ω 0 , can be expressed as

Arp(LF 0 , Ω 0 ) = φ(d, f, b, s) = [d -d(α -), d + d(α + )] = [d(1 - M e b/2 + M e ), d(1 + M e b/2 -M e )] (6)
where

e = s(d -f ) f r x 2 + r y 2 (7)
is the minimum distinguishable re-sampling distance in millimeters (mm), in the object side, corresponding to the diffraction limit λ 2N A on the image side. In most of practical imaging systems, the pixel size is larger than this diffraction limit, so that the pixel size corresponds to the minimum resampling unit e on the focal plane, as shown in Figure 2(d). The above LF-ARP model is a function of the object distance d, the focal length f , the width of baseline b, and the sensor size s. In traditional imaging, the sensor size s is constant, hence changing the focal length f implies changing the field of view (FoV). However, our model shows that, in the case of light field imaging, computational refocusing, i.e. changing the focus plane in which the object of interest is assumed to be, is equivalent to changing both the focal length f and the sensor size s, while keeping the physical distance between the image plane and the aperture plane constant, i.e.V is a constant. Figure 4 shows that a 1 pixel shift changes the convergence of the corresponding angular sample, as illustrated by the slight shifting orientations of the outmost view in purple, blue and green lines.

The concept of DoF can be seen as a metric based on focus and blur separation, in which the CoC varies with the shift of the image plane. Unlike the DoF, the proposed Arp model is derived assuming a given constant image plane V . More precisely, the FoV of P 0 and P (i.e. the physical width of the virtual images on Ω α+ 0 and Ω 0 respectively in the object space) stays the same when refocusing on different axial planes Ω α+ 0 and Ω 0 , i.e. the image of the object, remains constant, when re-focusing.

Please note that it assumes a thin lens model and does not account for possible off-axis aberrations. All the variables of our LF-ARP model correspond to physical measures. If the parameters of equation ( 6) and equation ( 7) are known, one can easily compute the axial indistinguishable refocusing ranges for each object plane, for a given minimum possible re-sampling limit, e.g. 1 pixel size. The refocusing separability not only depends on the axial depth, but also on the scene texture. But, only the minimum refocusing separability distance is derived by considering the characteristic of camera part, the object scene texture frequency is assumed high enough to satisfy the resolvable requirements in this paper.

Equations ( 6) and [START_REF] Prevedel | Simultaneous whole-animal 3d imaging of neuronal activity using light-field microscopy[END_REF] show that the Arp can be better with a close capture, or when using a lens with a long focal length. Figure 5-(a) shows how the distances d(α -) and d(α + ) on the axial direction, and the Arp vary when re-focusing with a shift of the outmost view of multiple e units. We can indeed see that the variation of the distance d(α -) contributes more to the Arp decrease than the one of d(α + ). We can indeed see that the blue curve is always above the green curve, which indicates that the distance d(α + ) contributes more to the Arp decrease than the one of d(α -). This means that the physical distance between the current focal plane and the closest distinguishable adjacent focal plane in the axial direction will be larger when refocusing further, than when refocusing on a plane closer to the main lens.

Since the Arp decrease is non-linear as u increases, axial separability can be improved via spatial super-resolution [START_REF] Broxton | Wave optics theory and 3-D deconvolution for the light field microscope[END_REF], however the authors in [START_REF] Broxton | Wave optics theory and 3-D deconvolution for the light field microscope[END_REF] give depth-dependent band limits for spatial super-resolution, indicating that one can not recover the spatial resolution up to the band limit using sharp focused angular samples. Instead of applying spatial super-resolution, computationally expanding the baseline b can be another option to increase the Arp. As we show in Figure 5-(b), there is a correlation between the Arp and the baseline width b that following the equations (6).

IV. EXPERIMENTS

In this section, we test the refocus precision model considering synthetic scenes generated using the light field plugin of the Blender software [START_REF] Honauer | A dataset and evaluation methodology for depth estimation on 4d light fields[END_REF]. The quantitative and qualitative comparisons are made using multiple synthetic light fields, and real light fields captured by 1st generation Lytro cameras, as well as on light field microscopy datasets.

A. Testing of the refocus precision model

To meet the resolvable requirements, ideally, we would need to capture a 3D scene with highly textured objects full filling the FoV, and then axially shift the object plane. Instead, for this first experiment, we created a synthetic experimental scene assuming a capture using a 9 * 9 camera array, with parameters ideally set as shown in Fig. 6, i.e.focal length f = 100.0mm, baseline b = 42.65mm, multiply factor M = 1.0, sensor size s = 35.0mm and its resolution r x = r y = 512.

For a first validation of the proposed Arp model, we created a synthetic scene using Blender. The scene consists of two clipped standard resolution cards (RCs) placed at different Fig. 6. Refocusing visual comparison when two boards are separated by a distance that is lower or higher than the LF-Arp value using Blender simulations. In the experiment, the estimated d(α -) is equal to 6053 mm with a refocusing made at 7000 mm. For this reason, the resolution board is first placed at 6500 mm (within the Arp range, in the top row), and then at 5500 mm (out of the Arp range, in the bottom row). We show the zoomed refocusing results when the resolution boards are in focus, or out-of-focus respectively. From 8.0x zoomed views (the green boxes), we can see clearly the separability changing between the two different separations, especially for the defocused case. positions in the axial direction (see the left part of Fig. 6). We placed a reference RC at a distance of 5.0mm, 6.0mm and 7.0m from the U V plane (i.e., the aperture plane) respectively, while the second RC is placed at neighbouring positions. The placement of the second RC at different neighboring positions allows us to measure refocus differences between focus planes in the same focal stack. Assuming a fixed image plane V , we calculated the ideal Arp range by varying only f, s.

For validating the model, we performed re-focusing at different planes, and then we compared the refocusing results of the regions corresponding to the two RCs, at five different axial positions, closer and further away from the reference focus plane, as shown in table I. We computed the focus/sharpness measure of the image regions of the two RCs. These measures should be very close when the re-focus at the two positions in the axial direction is not distinguishable. In order to measure the separability of the two RCs, we employed the local phase coherence-based sharpness index [START_REF] Hassen | Image sharpness assessment based on local phase coherence[END_REF] (LPC-SI). Instead of PSNR or SSIM, we use LPC-SI because it needs no reference image. Another reason is that the LPC-SI is less sensitive to the variation of scale and FoV as moving the second RC. In a same focal stack, let F S ref be the image region of the reference RC, which is placed at the original object plane at d 0 , and F S m be the image region of the compared movable RC, which can be placed at a neighbouring axial position d k . As in equation 8, we consider a new quantitive measure of the focal stack based refocusing separability RDif f (•), which measures sharpness and blur of multiple focal slices as where

RDif f (d k ) = SI(F S m (d k ), F S ref ) SI(F S m (d 0 ), F S ref ) , (8)
SI(F S a , F S b ) = 1 2N + 1 N i=-N (SI(F S i a ) -SI(F S i b )) 2 (9)
and where SI(•) is the LPC-SI sharpness measure as defined in [START_REF] Hassen | Image sharpness assessment based on local phase coherence[END_REF] (LPC-SI), F S i a and F S i b are two different image regions in the i -th focal slices respectively.

For each pair of the reference and compared RCs at < d 0 , d k >, we first draw the LPC-SI curves when both RCs are placed at the original object plane d 0 , e.g. the 5000l (the reference one) curve and 5000r (the compared one) curve in Fig. 7 (a). From Fig. 7, we can see that the two curves are highly consistent when the two RCs are placed at the reference object plane. The consistency implies that the refocusing of the two RCs is indistinguishable for all focal slices, in both the sharp focused and blurred defocused regions. When placing the compared RCs away from the reference position and outside the Arp range, their refocusing becomes more distinguishable, especially when it is severely defocused (see the red and blue curves in Fig. 7). When keeping the compared RCs inside the Arp, the curves are similar with the reference one. A visual comparison is shown in Fig. 6.

The refocus precision is decreasing when the reference object plane moves away from the U V plane. We can indeed see that the curves become closer to each other, as shown in Fig. 7 (b) and(c). Fig. 7 (d) shows that the mean of RDif f is 3 to 4 times higher, compared to the positions outside and inside the Arp range.

In order to further test the refocusing precision, we simulate a natural scene, which is composed of multiple pieces of complex textures. Different from the two RCs scene, as shown in the top right sub-images of Fig. 8, the scene is assumed to be captured at the distance d 0 = 2000mm, with the focal length f = 30.0mm, baseline b = 113.14mm and sensor size s = 28.0mm. These parameters have been ideally set according to the camera viewpoint. On the top right, the synthetic scene with ideal camera settings is generated using the Blender light field plugin [START_REF] Honauer | A dataset and evaluation methodology for depth estimation on 4d light fields[END_REF] with the parameters shown in Table II. is very small (green curves), while it becomes much larger (red color lines) when the spacing between two refocus planes is larger than the Arp interval. The dashed and solid lines indicate the LPC-SI variation when increasing and decreasing the refocus distance respectively. From Fig. 8, we can see that the axial separability or resolvability is significantly higher when the refocus plane spacing is larger than the Arp limit, which indicates the correctness of the proposed Arp model. Please note that, there is no standard metric for the measurement of axial separability or resolvability, which not only depends on the characteristics of the camera, but also on the scene texture.

Fig. 9 shows our experiment setup for testing the refocus precision of the 1st generation Lytro camera. The reference RC is placed in front of the camera, at distances 500mm, 600mm, 700mm, 800mm, 900mm, 1000mm from the aperture plane respectively. The second RC is then placed at neighbouring positions using a translation device, which can be precisely controlled in steps of ±0.01mm by a digital controller. The RCs are placed perpendicular to the optical axis of the camera, so that the axial distance between the camera aperture plane and the RCs can be simply measured using a laser rangefinder. In order to keep the illumination constant, we use a 200 W LED photographic light at a color temperature of 6500 K. The Arp limit will be less than 1mm at a distance of 200mm or even closer. Therefore, we choose to measure the Arp limit at a range from 500mm to 1000mm. The movable RC is placed at reference positions from 10 to 20 times a step of 1mm on both sides. The setting of the Lytro camera is manually fixed while capturing a same reference position, i.e.the reference RC is constant. In real scene experiments, we employ the same refocusing precision metric, and set RDif f (•) = 2.5. In the experiments, we use a focal length f = 35.0mm, a baseline b = 17.5mm and sensor size s = 7.67mm, r x = 3280, ry = 3936, M = 1.0 as the ideal parameters of the 1st generation Lytro camera. Due to the non-ideal illumination and sensor noise, the captured real scenes are more blurred and noisy than the simulated ones. Therefore, we consider smaller clipped regions of both RCs for computing the LPC-SI. Fig. 10 (a)-(c) shows that the peak of the red curve (outside the Arp range) is obviously different from the other curves. Due to the noise and blur of the sampling stage, the LPC-SI values are within a smaller range than for the simulated scene, which leads to a lower separability in refocusing. The predicted and measured Arp values of the Lytro camera along the axial direction are shown in Fig. 10 (d). Due to the presence of noise and distortion, and the limited number of tested discrete axial positions, there is an obvious difference between the measured and predicted values of Arp, e.g. at the positions of 600mm, 800mm, 900mm and 1000mm. Nevertheless, we can still see the high similarity in trend and values from the curves, which indicates the correctness of the proposed Arp model. The proposed Arp model is only validated with discrete re-focus planes positions. Validating the model for a continuum of positions on the optical axis would require designing a scene with carefully chosen distribution of objects, texture for each object, camera exposure settings, etc.

B. Evolution of refocusing precision with the baseline

By extending the baseline (or the aperture), the digital refocusing can produce a shallower DoF, thus leading to more accurate refocusing. To validate the axial refocusing enhancement, we further tested the resolvability or separability of the adjacent refocusing results. Different from the LPC-SI metric, the SSIM is employed to measure the differences between two adjacent refocus images. In Figure 11, for each axial focal plane (corresponding to different disparities), we give the mean SSIM of 7 Stanford microscopy datasets. A small value of mean SSIM indicates high separability. The black curve stands for separability for each axial focal plane with the ground truth 1.0X baseline light field, while the other curves marked with cross and triangles represent the axial separability of refocused results with 2.0X and 4.0X baselines respectively. We can see that the curves follow the same trend as those in Figure 5 (b), in the sense that the Arp value is lower when increasing the baseline, i.e. that enables a more precise refocusing. And the refocusing separability is decreasing in a non-linear manner when moving the focal plane further away from the original conjugate focal plane Ω 0 at axial position d 0 on the object side, as also shown in Figure 5 (a).

Fig. 11 shows the curves of the extrapolation method proposed in [START_REF] Mildenhall | Local light field fusion: Practical view synthesis with prescriptive sampling guidelines[END_REF] based on multi-plane images (MPI) and of the shearing and extrapolation network (SENet) we proposed in [START_REF] Xiao | A learning-based view extrapolation method for axial super-resolution[END_REF], in green color and red color respectively, from which we can see that even the computational baseline extension can give better performances than the 1.0X baseline in the axial separability. The vertical peak in Figure 11 can be explained by the non-linear characteristics of the disparity space. When away from the zero disparity plane, the distance between the distinguishable adjacent refocus planes increases, for the same disparity value. A larger physical separation will lead to high separability (smaller SSIM value), which yields the central peak of SSIM curves. The left and right wings of the curves are due to the fact that the entire specimen is out of focus, the blurriness indeed leads to a high SSIM value. Finally, Figure 12 visually compares refocused images with 1.0X and 4.0X baseline. The detail changes can hardly be recognized with 1.0X baseline original refocusing, e.g. , those areas pointed by the green arrow. But, one can easily distinguish the sharp focused plane of these details on the refocused images with 4.0X baseline. To extend the baseline to more than 4.0X, we suggest interested readers to use other state-of-art learning based solutions, e.g. . the solutions in [START_REF] Shi | Learning fused pixel and featurebased view reconstructions for light fields[END_REF], [START_REF] Mildenhall | Nerf: Representing scenes as neural radiance fields for view synthesis[END_REF].

C. Limitations

The proposed refocus precision analysis assumes a thin lens model, and does not account for possible off-axis deformations or distortions. The proposed Arp model is validated only at discrete and limited axial positions. Although the LPC-SI-based refocus sharpness metric does not need a known reference like PSNR or SSIM, it is not a standard metric by considering only the characteristics of the refocus difference. Therefore, a potential better metric will be still helpful, especial as in presence of noise.

V. CONCLUSION

In this paper, we introduced a light field refocusing precision model, which establishes the minimal resolvable or distinguishable spacing in the axial direction, given the fixed focal length, baseline, sensor size and resolution, and object plane distance. While the concept of DoF can be seen as a metric based on focus and blur separation, the proposed geometrical analysis is derived in the spatio-angular sampling space, and measures the distance between adjacent distinguishable focus planes. The model also reveals the relationship between the refocusing precision and camera settings. The proposed model can be used to optimize some configuration parameters of the capturing settings, such as the distance to the object plane, the aperture size or the baseline. It can benefit potential light field applications with specific axial refocusing precision demands, e.g. microscopy, particle velocimetry etc.

VI. APPENDIX

Let P 0 be the width of the conjugate virtual sensor in the object focal plane ω 0 . Figure 13 shows the proportional relation between P 0 and the sensor size s. Therefore, e can be derived under the pin-hole camera model assumption. We Fig. 13. The relation between P 0 , s and the theoretical shifting limit e of a pin-hole camera with a fixed image plane. Please note that the image plane distance V is a constant. Otherwise, the FoV will vary with different focal lengths f , so that P 0 can not be derived using Equation12 below. start the derivation from the simple geometry in Figure 13,

P 0 d = s V (10)
As mentioned in Section III C, we assume that the image plane distance V is a constant and that the FoV is fixed. Then V can be simply inferred from the thin lens equation (Equation 5) as

V = (d -f ) f d . ( 11
)
By replacing V in Equation 10 by its expression in Equation 11, P 0 can be calculated as

P 0 = sd V = s(d -f ) f (12)
Then we can obtain e as

When using a magnifying lens, from image side to object side, the limit e will be multiplied by M (the reciprocal of lateral magnification factor M ), as

M e (b/2) = d(α + ) d + d(α + ) (15)
One can easily derive d(α + ) from Equation 14as

d(α + ) = dM e b/2 -(M e) (16)
In a similar way, d(α -) can be derived as

d(α -) = dM e b/2 + (M e) (17)
Finally, the Arp model can be expressed as

which is the same as in Equation 6in Section III C.

Fig. 1 .

 1 Fig. 1. Light field refocusing precision illustration. (a) a low refocusing precision case. A given disparity corresponds to a specific focal plane when applying digital refocusing, b represents the baseline in the angular plane. Using a 1.0X baseline, the difference between refocusing at disparity = 0.0 and disparity = +0.5 can be hardly perceivable. (b) High precision refocusing with the same focus setting as in (a), from which we can see clear differences with a computational 2.0X enlarged baseline. (c) On the right zoomed views, stamens and leaves can be clearly found lying on different depth layers with high refocusing precision.

  Figure 2(a) illustrates the effect of refocusing on a 2D slice of the light field, where X and U denote spatial axis and angular axis respectively. The disparity variation αu is 0 on the original focal plane. The shifting factor α ∈ [-∞, +∞], is positive or negative when we focus closer or further respectively from the original focus plane. This leads to different shearing angles in the sampling space, as shown in Figures 2(b) and 2(c).

Fig. 2 .

 2 Fig. 2. Differences between original and re-sampled light fields. (a) The original 2D light field, (b) The positive shifting corresponds to a re-sampling of the original light field in (a), in order to move the focus plane closer in the axial direction. (c) The negative shifting is opposite to (b). (d) The shifting leads to a resolvable re-sampling even without sub-pixel interpolation.

Fig. 3 .

 3 Fig. 3. Difference between light field DoF and refocusing precision. The pink and blue shadowed regions correspond to the definition of traditional DoF. The refocusing precision is defined as the minimum distance Arp between two distinguishable adjacent focal planes in the object space.

Fig. 4 .

 4 Fig. 4. Illustration of axial refocusing precision. The axial refocusing precision is a function of the object distance d, the focal length f , the angular baseline of the main lens b, and the sensor size s. Since a 1 pixel shift of the outmost view corresponds to the minimum possible re-sampling of the light field, for any possible object side focal plane Ω 0 , we can derive the distances to nearest resolvable adjacent refocusing positions Ω α+ 0 and Ω α- 0 in both directions.

Figure 3

 3 indeed shows that the 1 pixel shift of outmost views corresponds to the minimum resolvable unit distance e on the focal plane Ω 0 . As shown in Figure 4, d(α -) and d(α + ) represent the distances between the backward and forward refocus planes and the focal plane, which are marked as the green, blue and red planes respectively. Please note that, d(α -) and d(α + ) are different from d Ω α - 0 and d Ω α + 0 , which are measured with respect to the coordinate origin of the optical center. The term M is the reciprocal of lateral magnification factor M (i.e.from image side to object side) for the imaging system having a magnifying lens (e.g. a light field microscope). Please see more details of the model derivation in the AppendixVI.

Fig. 5 .

 5 Fig. 5. LF-ARP for different parameter settings. (a) d(α -), d(α + ) and Arp when re-focusing corresponding to a shift of multiple e units of the outmost view. Here, we set f = 30mm, s = 28mm, M = 1.0, rx = ry = 512, b = 10mm, d = 500mm. The Arp range is the sum of the blue curve and the green one, which correspond to d(α -) and d(α + ) respectively. (b) Arp as a function of the object distance d for different values of the baseline b. The setting of f , s, M and rx, ry is the same as in (a).

Fig. 7 .

 7 Fig. 7. Quantitative focus measurements when placing the resolution board inside and outside the Arp at different axial positions. (a)-(c) are focus measurements using LPC-SI. In each subplot, the red line and blue line show the distinguishable differences of refocusing when putting the resolution board outside the LF-Arp. The green lines are reference refocusings when setting the resolution board at 5000 mm, 6000 mm, and 7000 mm. If putting the resolution board inside the LF-Arp, the refocusing difference is hardly distinguishable, as shown by the other lines. (d) shows the mean difference between the reference refocusing and the refocusing inside or outside LF-Arp.

Fig. 8 .

 8 Fig. 8. Validation of the proposed refocus precision model. We show the LPC-SI between images refocused on adjacent planes within and outside the refocus precision limit Arp = [d -d(α -), d + d(α + )] at different axial positions.On the top right, the synthetic scene with ideal camera settings is generated using the Blender light field plugin[START_REF] Honauer | A dataset and evaluation methodology for depth estimation on 4d light fields[END_REF] with the parameters shown in TableII.

Fig. 8

 8 Fig. 8 shows that, within the refocus precision limit Arp = [d -d(α -), d + d(α + )], with interval near and far boundaries denoted d -d(α -) and d + d(α + ) respectively, the LPC-SI difference between two refocused images on adjacent planes

Fig. 9 .

 9 Fig. 9. Experimental setup of refocus precision evaluation for the 1st generation Lytro camera at different axial positions.

Fig. 10 .

 10 Fig. 10. Quantitative evaluations of refocus precision of Lytro camera along axial direction. (a)-(c) show the quantitative focus measurements when changing the spacing between the reference resolution board and the moving one. The red line corresponds to the case where refocusing is outside Arp, while the refocusing inside Arp corresponds to the remaining lines. (d) By testing Arp on discrete and limited positions, we show a comparison between ideal and measured Arp of Lytro camera within the range of 500mm-1000mm.

Fig. 11 .

 11 Fig. 11. SSIM between two adjacent refocus images with 1.0X, 2.0X, 4.0X baseline extension factors for two methods, the MPI-based method[START_REF] Mildenhall | Local light field fusion: Practical view synthesis with prescriptive sampling guidelines[END_REF] and the proposed one. Lower is the SSIM, more distinguishable (i.e. showing more structural differences) are the two adjacent refocus planes.

Fig. 12 .

 12 Fig. 12. Comparison of axial refocusing results with 1.0X, 4.0X baselines, where different disparity values correspond to different focal planes in axial direction. (a) Refocused results on same axial focus planes, with 1.0X and 4.0X baselines respectively. (b) Zoomed views, the green arrows are pointing at details on different refocusing planes that are more distinguishable with a 4.0X baseline and less resolvable with the 1.0X baseline.

e = P 0 r x 2 +

 2 r y 2 = s(d -f ) f r x 2 + r y 2(13)Further, the relation between d(α + ) and d + d(α + ) can be found by solving triangles in Figure 4, i.e., e (b/2) = d(α + ) d + d(α + )

Arp(LF 0 ,

 0 Ω 0 ) = φ(d, f, b, s) = [d -d(α -), d + d(α + )] = [d(1 -M e b/2 + M e ), d(1 + M e b/2 -M e )]

TABLE I PARAMETERS

 I OF THE IDEAL LIGHT FIELD CAMERA ARRAY FOR TESTING LF-ARP USING A SYNTHETIC SCENE MADE OF TWO STANDARD RESOLUTION CARDS (UNIT MM)

	Group	d	f	s	e	d -d(α -)	d + d(α + )
	Ω0 = 5.0m	4250.00 4750.00 5000.00 100.00 35.00 2.37 99.65 34.88 2.01 99.89 34.96 2.25 5300.00 100.11 35.04 2.51 5800.00 100.28 35.10 2.76	3884.58 4297.10 4500.02 4741.21 5136.44	4691.31 5309.62 5624.68 6008.11 6660.45
	Ω0 = 6.0m	5050.00 5650.00 6000.00 100.00 35.00 2.85 99.69 34.89 2.39 99.90 34.96 2.68 6450.00 100.12 35.04 3.07 7100.00 100.26 35.09 3.38	4540.54 5018.67 5292.28 5638.49 6127.80	5688.23 6463.02 6926.22 7534.38 8438.85
	Ω0 = 7.0m	5500.00 6500.00 7000.00 100.00 35.00 3.34 99.61 34.86 2.61 99.89 34.96 3.09 7600.00 100.11 35.04 3.63 8800.00 100.29 35.10 4.21	4900.20 5676.55 6053.31 6495.79 7350.58	6267.11 7602.89 8297.70 9156.51 10961.41

TABLE II PARAMETERS

 II OF THE IDEAL LIGHT FIELD CAMERA ARRAY CORRESPONDING TO THE EXPERIMENT OF FIGURE 8 (UNIT MM)

	Array size Refocus planes	d	f	s	e	d -d(α -) d + d(α + )
	9 × 9	-d5	1633.79 29.90 27.91 2.07	1579.19	1695.77
	V	-d4	1695.81 29.92 27.92 2.15	1633.79	1762.72
	30.46	-d3	1762.77 29.94 27.94 2.23	1695.81	1835.24
	rx(pix.)	-d2

ACKNOWLEDGMENT This work has been funded in part by the National Natural Science Foundation of China (No. 61871319 and No. 62031023), in part by the Natural Science Basic Research Plan of Shaanxi Province(No.2019JM-221), in part by the China Scholarship Council (CSC) under grant No.201808610055, and in part by the EU H2020 Research and Innovation Programme under grant agreement No 694122 (ERC advanced grant CLIM) DISCLOSURES The authors declare no conflicts of interest.


Frédéric Labat
email: frederic.labat@aquabio-conseil.com

Gabrielle Thiébaut

Christophe Piscart

A new method for monitoring macrophyte communities in small shallow lakes and ponds

Keywords: standing waterbodies, depressional wetlands, floristic relevés, floristic richness, conservation value

Macrophytes are key to the functioning of small shallow lakes (SSLs) because they maintain a clear water state through numerous positive feedbacks. The composition of macrophyte communities changes under anthropogenic pressures; as a result, tools designed to easily and rapidly assess their structure and composition are increasingly requested. We tested three sampling methods (the S3m sampling method, a stratified method, and a mapped inventory) to monitor macrophytes in 26 SSLs. The effect of each method was evaluated on seven descriptors of macrophyte communities, including the median conservation value. The results were comparable for the three methods, but the stratified method failed to accurately monitor the median conservation value and the number of species present at a low frequency, including exotic and patrimonial species, hence serious consequences for management decisions. S3m was applied to 262 SSLs ranging from 1 m² to 43 ha in surface area. Generalised additive models were used to investigate the environmental factors correlated with four conservation value or ecosystem functioning descriptors. The S3m method showed that surface area, distance from the source, elevation, and bank verticality were determinants of macrophyte richness. Invasive crayfish impacted the macrophyte richness and the coverage of submerged macrophytes, whereas fish presence increased the macrophyte richness and the percentage of exotic macrophytes and reduced patrimonial interest. S3m was successfully applied to a wide diversity of SSLs in France. It proved to be rapid, reproducible, and representative for monitoring macrophytes in SSLs. Therefore, it should be applied for SSL management.

Introduction

Macrophytes strongly influence environmental characteristics in many ways. They modify substrate and water chemistry -e.g., the dissolved oxygen concentration in the rhizosphere [START_REF] Rehman | Constructed Wetlands: Perspectives of the Oxygen Released in the Rhizosphere of Macrophytes: Water[END_REF]) -change biogeochemical cycles [START_REF] Wetzel | Role of Submerged Macrophytes for the Microbial Community and Dynamics of Dissolved Organic Carbon in Aquatic Ecosystems[END_REF], and contribute to primary and secondary productivity [START_REF] Peters | GE (ed) Lake ecosystem ecology: a global perspective: a derivative of Encyclopedia of inland waters[END_REF]. Macrophytes also provide a substrate for epiphytic algae and their grazers [START_REF] Wolters | Macrophyte-specific effects on epiphyton quality and quantity and resulting effects on grazing macroinvertebrates[END_REF], act as a food source for fish and birds [START_REF] Jeppesen | The Structuring Role of Submerged Macrophytes in Lakes[END_REF], and offer a physical refuge that buffers interactions between fish and zooplankton [START_REF] Scheffer | Alternative Attractors of Shallow Lakes[END_REF]. They are involved in various feedback mechanisms that tend to maintain a clear-water state [START_REF] Scheffer | Catastrophic regime shifts in ecosystems: linking theory to observation[END_REF]. They also provide numerous ecosystem services [START_REF] Hilt | Translating Regime Shifts in Shallow Lakes into Changes in Ecosystem Functions and Services[END_REF]. Macrophyte composition and abundance recently changed in most waterbodies because of various anthropogenic pressures [START_REF] Körner | Loss of Submerged Macrophytes in Shallow Lakes in North-Eastern Germany[END_REF]) such as eutrophication [START_REF] Rosset | Is eutrophication really a major impairment for small waterbody biodiversity?[END_REF], fish stocking [START_REF] Williams | Fish induced macrophyte loss in shallow lakes: top-down and bottom-up processes in mesocosm experiments[END_REF], or biological invasions (Stiers et al. 2011). Furthermore, fish have a positive influence on floristic richness [START_REF] Hassall | Environmental correlates of plant and invertebrate species richness in ponds[END_REF], whereas invasive crayfish have a negative influence on macrophyte communities [START_REF] Linton | Reductions in Submersed Macrophyte Biomass and Species Richness by the Crayfish Orconectes rusticus[END_REF].

Macrophyte monitoring programs have been widely implemented in the running waters and lakes of European states following the Water Framework Directive (WFD) [START_REF] Birk | Three hundred ways to assess Europe's surface waters: An almost complete overview of biological methods to implement the Water Framework Directive[END_REF]), but they have largely failed to include shallow lakes < 50 ha in surface area (Oertli et al. 2005b) and had little influence on their biological conservation [START_REF] Hassall | The Ecology and Management of Urban Pondscapes[END_REF]. Some methods -e.g., the "Biological Macrophyte Index in Lakes" (IBML) in France [START_REF] Boutry | Méthode d'évaluation de la qualité écologique des plans d'eau basée sur les communautés de macrophytes Indice Biologique Macrophytique en Lac (IBML) -Rapport d'avancement[END_REF]) and the "Ecological State Macrophyte Index" (ESMI) in Poland [START_REF] Ciecierska | ESMI: a macrophyte index for assessing the ecological status of lakes[END_REF] -could be applied for monitoring aquatic plant communities in large shallow lakes so as to evaluate their ecological status. However, they were not designed to evaluate the conservation value of the communities. Small shallow lakes and ponds (SSLs) are major contributors to biodiversity conservation in freshwater ecosystems [START_REF] Lukács | Plant diversity and conservation value of continental temporary pools[END_REF][START_REF] Panzeca | Aquatic Macrophytes Occurrence in Mediterranean Farm Ponds: Preliminary Investigations in North-Western Sicily (Italy)[END_REF]. Moreover, the biological monitoring of aquatic plant communities is still in its infancy in European SSLs despite an urgent need to protect them and to preserve ecosystem services [START_REF] Hill | New policy directions for global pond conservation[END_REF]. This implies monitoring the biodiversity of aquatic plant communities with adapted and efficient methodologies. Apart from Great Britain and Switzerland, most European countries still lack ambitious SSL monitoring tools and programs. The British Predictive System for Multimetrics (PSYM) for ponds [START_REF] Biggs | Biological Techniques of Still Water Quality Assessment Phase 3. Method Development[END_REF] and the Swiss "Indice Biologique Etangs et Mares" (IBEM) [START_REF] Indermuehle | The pond biodiversity index "IBEM": a new tool for the rapid assessment of biodiversity in ponds from Switzerland. Part 2. Method description and examples of application[END_REF] provide contemporary examples of SSL monitoring. PSYM is based on the assessment of the richness of invertebrates and macrophytes and associated trophic ranking scores [START_REF] Palmer | A botanical classification of standing waters in Great Britain and a method for the use of macrophyte flora in assessing changes in water quality. incorporing a reworking data[END_REF]. It can be used to assess ponds with surface areas ranging from 1 m² to 5 ha [START_REF] Biggs | Biological Techniques of Still Water Quality Assessment Phase 3. Method Development[END_REF]. Macrophyte samplings correspond to a simple qualitative inventory (lists of species present in the lake). The IBEM index is valid in Switzerland and its border regions for waterbodies ranging from 300 to 1,000 m above sea level (asl). It can be used to assess the floristic richness of SSLs 50 m² to 6 ha in surface area and 0.30 m to 9 m mean depth. Neither method can be used to monitor SSLs neglected by the WFD (a few m² to 50 ha in surface area). As a consequence, a new method called S3m (Sampling of Small Shallow lake macrophytes) derived from the PSYM method was proposed and compared with a stratified method derived from the IBEM method [START_REF] Indermuehle | The pond biodiversity index "IBEM": a new tool for the rapid assessment of biodiversity in ponds from Switzerland. Part 2. Method description and examples of application[END_REF]) and with a mapped inventory method ("mapped inventories" hereafter) [START_REF] Simpson | Impact of three aquatic invasive species on native plants and macroinvertebrates in temperate ponds[END_REF].

Our goal was to test the efficiency of the S3m method for conducting surveys of plant species composition and abundance in SSLs and calculating diversity and conservation indices. To be applied by a broad range of users, this method should be cost-effective and (1) representative, (2) rapid, (3) reproducible, and (4) flexible. As the choice of the sampling method influences monitoring results and diversity or conservation indices, we compared the S3m method with two other sampling methods: (1) a sampling strategy inspired by the IBEM method [START_REF] Indermuehle | The pond biodiversity index "IBEM": a new tool for the rapid assessment of biodiversity in ponds from Switzerland. Part 2. Method description and examples of application[END_REF]), ( 2) mapped inventories with coverages computed with the SIG tool (e.g., Hutorowicz 2020). The comparison was performed across a panel of 26 SSLs. Our first hypothesis was that S3m would provide a representative picture of macrophyte richness and of the conservation value of plant communities, and be less time consuming than the stratified and the mapped methods. We investigated the key biotic and abiotic factors of the macrophyte communities of 262 SSLs using the S3m method. Our second hypothesis was that a lower floristic richness and a higher conservation value in high-altitude SSLs would be observed due to harsh climatic conditions.

Our third hypothesis was that a high distance from a source -a proxy of SSL connectivity and watershed sizewould decrease the conservation value of aquatic plant communities and floristic richness because a high connectivity facilitates biological dispersal and increases nutrient inputs. Our last hypothesis was that the presence of fish and crayfish would reduce the floristic richness and the conservation value of aquatic plant communities.

Materials and Methods

Site selection and macrophyte sampling methods

Twenty-six SSLs were selected in southwest France. They represented a large panel of morphometric conditions and considerable variations in water quality (Table 1): surface area from 258 to 95,000 m², mean depth from 0.1 to 5 m, shoreline index D [START_REF] Hutchinson | A Treatise on limnology: Vol-1 Part-1: Geography and physics of lakes[END_REF]) from 1.04 to 1.88, and various bank slopes ranging from sites with less than 5% of the shoreline with vertical banks (15 sites) to sites with more than 75% of the shoreline with vertical banks (one site). All field data were collected in the summers of 2013-2014. The S3m method was applied using a meander method, which is very efficient in detecting rare species [START_REF] Huebner | Detection and Monitoring of Invasive Exotic Plants: A Comparison of Four Sampling Methods[END_REF]. Macrophytes were surveyed in a zigzag pattern, regardless of the depth. Deeper water zones were point-sampled. The IBML littoral five-scale from the French norm XPT90-328 classes (AFNOR 2010) was chosen to estimate plant abundance (class 1: a few individuals; 2: isolated small patches; 3: numerous small patches; 4: large discontinuous patches; 5: large continuous patches).

The stratified sampling method used quadrats and transects. The number of quadrats increased with the surface area of the SSL. A grid pattern was fixed with regularly spaced transects, perpendicular or parallel to the longest axis of the SSL. For each transect, two quadrats were located at each end of the transect, directly against the shoreline, at the usual SSL limit. The SSL shore is usually richer in plant species than the profundal zone (Oertli et al. 2005a), and a higher sampling pressure in this zone determines the stratified strategy. Other quadrats were positioned at each transect intersection. A formula adapted from the IBEM method was used to evaluate the number of quadrats:

𝑁 = 0.3652 × 𝑎𝑟𝑒𝑎 .
Square-metre quadrats, best fitted for sampling macrophyte communities in lakes, were selected (Ling and Jacobs 2010). In the field, transects were identified with pegs. Only the macrophyte species observed in the quadrats were recorded.

For the mapped inventory, coverages of patches and isolated plants were estimated visually as accurately as possible and drawn directly using a recent aerial photograph of each SSL as a guide [START_REF] Simpson | Impact of three aquatic invasive species on native plants and macroinvertebrates in temperate ponds[END_REF]. Drawings were converted into SIG shape files using QGIS software (QGIS Development Team 2021).

All surveys were conducted by walking on the littoral and shallow zones. Vegetation in the deeper zones was surveyed with a grapnel or a rake from a small boat with an electric motor when the depth was beyond 1.5 m or by free diving. Species such as Characeae, Ranunculus subg. Batrachium, and mosses were kept in alcohol or dried for identification in the laboratory. All macrophytes (spermatophytes, bryophytes, ferns, and Characeae)

were identified to the species level, when possible.

Comparison of the sampling designs

All data were converted into percent coverages. The relative coverage of each taxon from the mapped inventory, expressed as a percentage of the total area, was considered further as the reference data (the most complete and representative one). It was computed with R software (R Core Team 2020) and the sf R package (Pebesma 2018).

For the stratified method, the percentage of quadrats occupied by each species was treated as relative coverage.

For the S3m method, a conversion scale (1 = 0.0001, 2 = 0.001, 3 = 0.02, 4 = 0.2, 5 = 0.6) was used. To obtain this scale, each S3m value was converted into corresponding median values obtained with the mapped inventory from 9 randomly chosen SSLs out of the 26 SSLs. The values obtained with the other 17 SSLs were used for data validation tests. Then, 7 descriptors were selected to identify the effects of the sampling method on macrophyte communities: (1) total floristic richness -a synthetic but quantitative picture of biodiversity -, ( 2) the Shannon diversity index, (3) the median conservation value according to the national red list, (4) the M-NIP trophic index [START_REF] Sager | The M-NIP: a macrophyte-based Nutrient Index for Ponds[END_REF], ( 5) the percent coverage of submerged species (spermatophytes + Characeae + bryophytes + ferns), ( 6) the percentage of exotic species, and ( 7) the richness in 'infrequent species' (difficult to observe because their coverage is < 1% in mapped inventories). The median conservation value was calculated using the species rarity index (SRI) as follows:

(1) all species were assigned a score according to their status on the French national red list, with 32 = CR (Critically Endangered), 16 = EN (Endangered), 8 = VU (Vulnerable), 4 = NT (Near Threatened), 2 = LC (Least Concern), 1 = other, 0 = neophyte; (2) the scores of all species in each sample were summed to provide a species rarity score; and (3) the species rarity score was divided by the number of species recorded in the sample to provide the SRI score [START_REF] Foster | Classification of water beetle assemblages in arable fenland and ranking of sites in relation to conservation value[END_REF][START_REF] Rosset | Comparative assessment of scoring methods to evaluate the conservation value of pond and small lake biodiversity: ASSESSMENT OF CONSERVATION VALUE OF POND AND SMALL LAKE BIODIVERSITY[END_REF].

The sampling methods were compared using Wilcoxon's test and Spearman's correlation between the index values from the mapped inventories and the values from the other two methods, and illustrated with linear regression plots.

Finally, the duration of assessment time is key to the success of a monitoring method. The times needed to implement each of the three methods were compared by measuring the time spent monitoring each SSL (sampling + data entry).

Application of the S3m method to 262 SSLs: influence of environmental factors on aquatic plant communities In total, 262 SSLs were selected, differing by their geographical situation and physical features (Figure 1, Table 2). They were located in various geological bedrocks (acid, calcareous, mixed), under different climatic or altitudinal conditions, and in varied environments (forest, urban, agricultural). They were natural or man-made SSLs, with different hydrological regimes. Twenty-two percent of them dried up in summer. They underwent different biotic pressures (fish, invasive crayfish). Thirteen percent were invaded by exotic crayfish, and 40% harboured fish. Each SSL was sampled once in the summertime. Macrophyte taxa were monitored using S3m from 2013 to 2020. The influence of environmental factors (detailed in Table 2) was studied on 4 descriptors: (1) total richness, ( 2) the SRI score, (3) the percent coverage of submerged species, and (4) the percentage of exotic species.

A generalised additive model (GAM; [START_REF] Hastie | Generalized additive models[END_REF] between floristic richness and 11 environmental factors was used to identify which factors were the best predictors. The best predictors were identified according to the REML method combined with null space penalisation [START_REF] Marra | Practical variable selection for generalized additive models[END_REF]. All GAMs were computed using R (R Core Team 2020) and the mgcv package [START_REF] Wood | Generalized additive models: an introduction with R, Second edition[END_REF]. Factors were transformed when appropriate, with assessment for normality and homoscedasticity using Shapiro-Wilk tests and

Results

Comparison of the three sampling methods

A total of 148 species -97 helophytes, 33 submerged spermatophytes, 10 bryophytes, and 8 Characeaewere observed in the 26 SSLs used for the sampling comparison. Eight species were exotic, including six infrequent species. According to the French national red list, one species (Utricularia intermedia) was vulnerable and three (Hippuris vulgaris, Potamogeton acutifolius, and Utricularia minor) were near threatened. One species (Caropsis verticillato-inundata) was restricted to the southwest of France, and it is nationally protected and considered vulnerable on the IUCN red lists. Among these five species, three were infrequent (Table S1). The effects of each sampling method on the macrophyte community relevés are illustrated in Table 3 and Figure 2.

The coefficient of correlation of each descriptor (richness, Shannon diversity index, SRI, M-NIP, percentage of submerged vegetation, and percentage of exotic species) was higher in the mapped inventory and S3m results than in the stratified method results (Table 4). The very high correlations (r = 1) between S3m and the mapped inventory based on qualitative data (floristic richness, SRI, M-NIP, percentage of exotic species, richness in species present at low frequencies) confirmed that all species were inventoried by S3m. Other correlations between S3m and the mapped inventory remained high, as regards descriptors based on quantitative data such as coverage of submerged species (r = 0.95) and the Shannon diversity index (r = 0.85). S3m was well correlated with the stratified method (r ≥ 0.6, except SRI, r = 0.45, and exotic species r = 0.58). The correlation between the stratified method and the mapped inventory was weakest for qualitative descriptors such as SRI and the percentage of exotic species (r = 0.45 and 0.58, respectively), and was high for M-NIP (r = 0.84) and floristic richness (r = 0.81). No significant difference between the three methods was obtained for any of the descriptors, except for the Shannon diversity index and the number of species present at a low frequency (Wilcoxon test, p < 0.005). The stratified method significantly overestimated the Shannon diversity index compared with the mapped inventory and S3m (Table 3). The total number of species present at a low frequency was underestimated by the stratified method compared with the mapped inventory and S3m (Table 3). Consequently, three of the eight exotic species and two of the patrimonial species including vulnerable Utricularia intermedia were not found by the stratified method. Most of the other exotic or patrimonial species were missed in at least one site (Table S1). The monitoring time was lower with S3m (mean time = 80 ± 53 min) than with the stratified method (mean time = 180 ± 134 min).

The mapped inventory was the most time-consuming method (243 ± 162 min).

Influence of environmental factors on macrophyte communities in French SSLs

A list of 238 taxa -40 bryophytes, 20 Characeae, 107 helophytes, and 71 submerged or floating spermatophytes or fern species -was established in all 262 SSLs. According to the national red list, 3 species were considered vulnerable, 9 near threatened, 152 least concerned, 1 data deficient, and 13 exotic.

The mean floristic richness per site was 20 ± 14 taxa, with a minimum of 1 species and a maximum of 73 taxa. Sampling time was usually 30 min to 4 h, with a maximum of 8 h in a very unfavourable context (a 7.4 ha SSL, with most of the shoreline colonised by dense brambles, and the whole waterbody occupied by dense

Ceratophyllum stands that made it impossible to sail with an electric motor).

GAMs identified 7 key factors of total richness, 6 key factors of SRI, 3 key factors of the coverage of submerged species, and 5 key factors of the percentage of exotic species (Table 5). Fitted relationships are shown in Figure 3. Surface area was the strongest predictor of total richness, with a logarithmic linear influence. The influence of elevation on floristic richness was not clear from 0 to 1,000 m, whereas a clear decrease in floristic richness with increasing elevation was established above 1,000 m asl (Figure 3). DIS -a proxy of SSL connectivity and watershed size -had a positive influence on floristic richness when it was less than 5 km, and a negative effect when it was more than 5 km. The presence of fish favoured floristic richness, whereas the presence of invasive crayfish, a woodland context and bank verticality and instability had negative effects on floristic richness (Figure 3). SRI was well explained by environmental factors (R² = 0.23). Surface area and elevation tended to have a positive influence, whereas the presence of fish, DIS and woodland had a negative one. Mean depth had a positive influence on SRI from 0 to 1.8 m, and a negative one when it was more than 1.8 m. Three factors -invasive crayfish, shade, and mean depth -had a weak impact (R² = 0.12) on submerged species coverage (Figure 3). A decrease was observed in the submerged vegetation cover along an increasing shade gradient. The impact of mean depth on the submerged vegetation cover was measured when depth was > 1.8 m. At less than 1.8 m depth, the influence of mean depth was positive (Figure 3). The presence of invasive crayfish reduced the cover of submerged plants.

The percentage of exotic species was favoured by the distance from the source, the presence of fish, and mean depth (Table 5). Elevation and the presence of invasive crayfish tended to limit the percentage of exotic plant species (Figure 3, Table 5).

Discussion

Comparison of the efficiency of the three sampling methods SSLs are a matter of concern due to conservation issues, hence the importance of using a sampling method able to detect floristic biodiversity. Moreover, the accuracy and reproducibility of macrophyte coverage estimations are a prerequisite to effectively monitor biological integrity [START_REF] Karr | Biological monitoring and assessment: using multimetric indexes effectively[END_REF]. The S3m method is a rapid assessment method that provides a representative picture of macrophyte richness and structure within the aquatic plant communities of SSLs. These results validate our first hypothesis.

Comparable results were obtained with S3m and the mapped method to compute species coverage, and they were more accurate and complete than those obtained with the stratified method. Moreover, sampling with the S3m method required less than half the time required by the other two methods. Species aggregation in patches can strongly affect the accuracy of stratified methods [START_REF] May | mobsim: An r package for the simulation and measurement of biodiversity across spatial scales[END_REF]. Consequently, the stratified method failed to provide an accurate picture of the conservation value because the sampled richness was incomplete. Moreover, the stratified method failed to consider numerous low-frequency species, so that it underestimated the floristic richness and SRI and overestimated the percentage of exotic species. However, it was accurate for monitoring trophic levels, confirming the relevance of stratified methods for monitoring trophic alterations in various standing water bodies.

Our findings do not allow us to conclude that any one method was best for estimating species coverage.

Stratified sampling failed to identify small, isolated vegetation patches and rare species which may represent almost half of the species. However, it was effective in estimating the coverage of dominant species [START_REF] Pante | Getting to the Point: Accuracy of Point Count in Monitoring Ecosystem Change[END_REF]. The mapped and S3m methods did not include plant density in patches and were poorly fitted to report the coverage of dispersed or low-coverage species. A simple visual estimation can be more representative [START_REF] Dethier | Visual versus random-point percent cover estimations: "objective" is not always better[END_REF][START_REF] Bråkenhielm | Comparison of field methods in vegetation monitoring[END_REF] and less time-consuming [START_REF] Lillie | Development of a Biological Index and Classification System for Wisconsin Wetlands Using Macroinvertebrates and Plants[END_REF] than quadrats when the distribution is patchy, at least in small lakes. The accuracy of visual estimation can decrease with increasing surface area [START_REF] Traxler | Handbuch des vegetationsökologischen Monitorings. Methoden, Praxis, angewandte Projekte[END_REF]. The accuracy of the IBML 5-scale can limit this bias. All three methods were valuable for estimating species coverage or at least the coverage of functional groups and dominant species. However, the Shannon index results obtained with the stratified method suggested that stratified methods are less efficient for monitoring community diversity.

A comparison of our S3m data with literature data showed that S3m recorded a higher number of plant taxa than other sampling methods did in other studies (Table 6). A comparison of the number of plant species found in literature data with those found using S3m showed its efficiency for monitoring the macrophyte richness of SSLs. However, these differences are only indicative, and could be explained by different local conditions or geographical isolation [START_REF] Scheffer | Small habitat size and isolation can promote species richness: second-order effects on biodiversity in shallow lakes and ponds[END_REF][START_REF] Gledhill | Pond density as a determinant of aquatic species richness in an urban landscape[END_REF]. The accuracy of S3m is particularly useful to evaluate the conservation value or monitor low-frequency patrimonial taxa or exotic species. However, the inclusion or exclusion of infrequent species in biological monitoring is an old debate [START_REF] Van Sickle | Exclusion of rare taxa affects performance of the O/E index in bioassessments[END_REF]). Deleting infrequent species can alter the sensitivity of community-based methods when it comes to detecting ecological changes [START_REF] Cao | How important are rare species in aquatic community ecology and bioassessment?[END_REF]). These species can also play a key role in management decisions when they have a patrimonial value [START_REF] Thompson | Sampling Rare or Elusive Species Concepts, Designs, and Techniques for Estimating Population Parameters[END_REF].

S3m is also useful for detecting exotic species early, which is essential for implementing a rapid response and minimising the impact of potential invasive species [START_REF] Reaser | The early detection of and rapid response (EDRR) to invasive species: a conceptual framework and federal capacities assessment[END_REF]. For instance, two small stands of Myriophyllum aquaticum were observed, corresponding to 0.2% of the SSL surface area in one of the 26 SSLs used for comparison. Four years later, its coverage represented approximately half of the surface area. This dramatically increased the cost of its control or eradication and reduced the coverage of other macrophytes such as Luronium natans -a species protected at the national scale. Our results from the 26 SSLs highlight that numerous exotic or patrimonial species can be missed when a stratified method is used. Moreover, the exotic or patrimonial species inventoried in the 262 SSLs were mostly infrequent according to their mean coverage (Table S2). Consequently, the use of stratified methods induces serious consequences in management decisions. The S3m method is cost effective, rapid (less time consuming than the other methods), and efficient for assessing aquatic plant communities in SSLs.

Influence of environmental factors on aquatic plant communities

The higher richness found among the 262 sites vs. the 26 sites (238 vs. 148 taxa) can be explained by the diversity of climatic, elevational, and geological conditions (e.g., Figure . 1).

Our results, from lowly to highly impacted SSLs, confirmed the key role of surface area, elevation, and distance from the source previously established on French lowimpacted SSLs [START_REF] Labat | Principal Determinants of Aquatic Macrophyte Communities in Least-Impacted Small Shallow Lakes in France[END_REF]. The decrease in floristic richness above 1,000 m asl was probably due to an indirect effect of the short duration of the growing season, cold temperatures [START_REF] Jones | Area, Altitude and Aquatic Plant Diversity[END_REF], and maybe lower nutrient availability [START_REF] Lacoul | Physical and Chemical Limnology of 34 Lentic Waterbodies along a Tropical-to-Alpine Altitudinal Gradient in Nepal[END_REF].

Despite a lower floristic richness, altitude ponds are often occupied by transition mire and quaking bogs, usually combined with oligotrophic to mesotrophic vegetation in standing waters. This vegetation includes rare or specific species like Sparganium angustifolium or Callitriche palustris at the highest elevation. Consequently, the conservation value increases with elevation despite lower floristic richness. The influence of anthropogenic pressure cannot be excluded, even if it is usually lower with elevation [START_REF] Fernández-Aláez | Environmental drivers of aquatic macrophyte assemblages in ponds along an altitudinal gradient[END_REF]. Thus, our second hypothesis was validated.

The distance from the source had an influence on the macrophyte community composition [START_REF] Labat | Principal Determinants of Aquatic Macrophyte Communities in Least-Impacted Small Shallow Lakes in France[END_REF]. Moreover, higher DIS was often characterised by communities typical of natural eutrophic lakes including Magnopotamion or Hydrocharition alliances, whereas lower DIS was characterised by communities of oligotrophic to mesotrophic standing waters including Littorelletea uniflorae and/or the Isoeto-Nanojuncetea.

When DIS = 0 (isolated ponds), vegetation was usually typical of natural dystrophic lakes and ponds, or hard oligo-mesotrophic standing waters with benthic vegetation composed of Charophytes. DIS also had a positive influence on floristic richness up to 5 km, and a negative one above 5 km. Our third hypothesis stipulating that distance from the source would influence floristic richness was validated. Connectivity between SSLs and rivers facilitates the dispersal of plant propagules [START_REF] Vogt | Seed deposition in drift lines during an extreme flooding event -Evidence for hydrochorous dispersal?[END_REF], particularly from exotic species [START_REF] Parendes | Role of light availability and dispersal in exotic plant invasion along roads and streams in the H. Andrews Exp For Or Conserv Biol 14: Pebesma E[END_REF]. Above 5 km, the effect of DIS on total floristic richness became negative, perhaps because (1) colonisation by invasive species was higher with higher DIS (Figure 4), and ( 2) eutrophication was higher because of larger watersheds and increasing anthropogenic pressure. Vertical or unstable banks also impacted floristic richness by reducing the expression of the pool of species (cf. the colonisation gradient based on the littoral depth gradient (e.g., Pokorný and Björk 2010)).

Our hypothesis on the positive effect of fish on floristic richness was validated. This effect had already been found [START_REF] Linton | Reductions in Submersed Macrophyte Biomass and Species Richness by the Crayfish Orconectes rusticus[END_REF][START_REF] Hassall | Environmental correlates of plant and invertebrate species richness in ponds[END_REF]. However, the presence of fish could have an impact on the conservation value of the communities, as suggested by our SRI index results. The influence of fish, especially on submerged species, could depend on fish communities. Large cyprinids reduce hydrophyte coverage and composition (Pı ṕalová 2002), whereas their impact on macrophyte communities can be limited by piscivorous fish [START_REF] Tátrai | Changes in water clarity during fish manipulation and postmanipulation periods in a shallow eutrophic lake[END_REF]. [START_REF] Goulder | Angling and species richness of aquatic macrophytes in ponds[END_REF] suggested that higher floristic richness in fishponds can be explained by moderate trampling by anglers. Many anglers or other users can also carry native and exotic plant fragments or seeds on their footwear [START_REF] Ware | Humans introduce viable seeds to the Arctic on footwear[END_REF] or on their boats. Higher depths motivate boating or sailing for fish or sport, which is a major vector of native and exotic species dispersal [START_REF] Rothlisberger | Aquatic Invasive Species Transport via Trailered Boats: What Is Being Moved, Who Is Moving It, and What Can Be Done[END_REF][START_REF] Sytsma | Vectors for Spread of Invasive Freshwater Vascular Plants with a North American Analysis[END_REF] and can partly explain the positive influence of depth on the percentage of exotic species.

As hypothesised, exotic crayfish had a strong impact on floristic richness, especially on submerged macrophytes and could limit the abundance of exotic macrophyte species. Exotic crayfish can consume and fragment macrophytes, drastically reduce plant biomass and remove some plant species [START_REF] Carreira | How consumption and fragmentation of macrophytes by the invasive crayfish Procambarus clarkii shape the macrophyte communities of temporary ponds[END_REF]).

However, no impact of crayfish on the patrimonial value of SSLs was found by the SRI score, probably because the contribution of submerged species to SRI was generally weak (low richness, and usually a low conservation score). An indirect impact of exotic crayfish on the conservation value is not excluded because submerged species usually shelter numerous invertebrate and fish species [START_REF] Engel | The Role and Interactions of Submersed Macrophytes in a Shallow Wisconsin Lake[END_REF]) and strongly influence zooplankton and algal communities (Celewicz-Gołdyn and Kuczyńska-Kippen 2017).

Bank verticality and woodland were also key for plant richness, whereas shade impacted submerged plants. These findings are congruent with literature data [START_REF] Cowardin | Classification of Wetlands and Deepwater Habitats of the United States[END_REF][START_REF] Da | The prediction of macrophyte species occurrence in Swiss ponds[END_REF]Angélibert et al. 2010;[START_REF] Hassall | Environmental correlates of plant and invertebrate species richness in ponds[END_REF][START_REF] Sender | The effect of riparian forest shade on the structural characteristics of macrophytes in a midforest lake[END_REF].

The key factors of conservation interest identified from the SRI score should be viewed cautiously.

Furthermore, the reliability of red lists in conservation evaluation is a matter of discussion because the red list status is based on expert opinion or political or empirical decision (Le [START_REF] Berre | How to hierarchise species to determine priorities for conservation action? A critical analysis[END_REF]. It might be useful to define rarity as including functional rarity [START_REF] Loiseau | Global distribution and conservation status of ecologically rare mammal and bird species[END_REF] to better understand the implications of human and natural pressures in SSL functioning or to simply define species sensitivity to alterations, as suggested in numerous integrity indices in North American lakes [START_REF] Beck | Development of a macrophyte-based index of biotic integrity for Minnesota lakes[END_REF]) and for depressional wetlands (e.g., [START_REF] Reiss | Developing Biological Indicators for Isolated Depressional Forested Wetlands[END_REF].

The plants listed in red lists can strongly differ between neighbour countries or regions or according to the scale (regional, national, European). Differences can also be noticed as regards taxa. The French national red list used for SRI computing excludes mosses and Characeae and can miss key factors for these groups or overestimate the influence of factors key to groups included in this red list. Moreover, most of the species in our database were considered LC in the national red list, which tends to have less influence than exotic species in SRI trends. This can be observed through opposite trends for elevation, DIS, mean depth, and fish presence between the SRI and the percentage of exotic species. The SRI score was also influenced by surface area and woodland.

Conclusion

This study proposes and validates a new sampling method (S3m) adapted to SSLs. S3m provided similar or even better results than the stratified method and was less time consuming. Therefore, it constitutes an efficient tool for assessing the plant communities of permanent or temporary SSLs. The conservation value of SSLs still remains neglected because of the lack of adapted sampling methods. We tested S3m on a wide range of French SSLs differing by their locations, physical features, and human uses, and confirmed (1) the importance of biodiversity, including patrimonial species, in SSLs, and (2) that macrophyte richness and submerged macrophyte coverage can be predicted using physical or biological factors.

Large lowland SSLs defined by a short distance from the source, a low coverage of nearby woodlands and the presence of fish harboured potentially higher floristic richness than small mountainous SSLs. Therefore, heterogeneity of the key factors is a prerequisite for the conservation of various communities harboured by SSLs at the landscape level. Table S1 -French national status ("National red list" or "exotic"), frequency, mean coverage, and standard deviation of each macrophyte species from the 26 SSLs sampled with the mapped inventory method. Species without a status were omitted. DD = data deficient; LC = least concern; NT = near-threatened; VU = vulnerable.

Table S2 -French national status ("National red list" or "exotic"), frequency, mean coverage, and standard deviation of each macrophyte species from the 262 SSLs sampled with the S3m method. Total mean coverages and corresponding deviations for all plants for each status are also indicated. Species without a status were omitted. DD = data deficient; LC = least concern; NT = near threatened; VU = vulnerable.

  histograms. As suggested by[START_REF] Hassall | Environmental correlates of plant and invertebrate species richness in ponds[END_REF], the GAM was expected to disentangle non-linear relationships between macrophyte richness and biotic or abiotic factors.The factors were (1) elevation; (2) surface area; (3) distance from the source (DIS) as proposed in Labat et al. (2021); (4) shoreline development index[START_REF] Hutchinson | A Treatise on limnology: Vol-1 Part-1: Geography and physics of lakes[END_REF]); (5) shoreline influence estimated with a 5-scale estimation according to the percentage of perimeters with a slope > 50% or instable banks (0 = 0-5%; 1 = 5-25%; 2 = 26-49%; 3 = 50-75%, 4 > 75%); (6) mean depth; (7) 'shade'; (8) percentage of woodland in the SSL surroundings (in a 50-m buffer zone); (9) drying (0 = not known to dry up, 1 = known to dry up); (10) presence of fish; and (11) presence of invasive crayfish (0 = unknown presence of fish or crayfish, 1 = known presence of fish or crayfish). The factors were estimated using at least three field observations during various seasons at the time of macrophyte sampling or according to landowners' observations.

Figure 1 -

 1 Figure 1 -Localisation of the sampling sites. The number in the circle indicates the number of small shallow lakes sampled in each area.

Figure 2 -

 2 Figure2-Linear regression plots obtained for each method compared to mapped inventory results for seven descriptors: total floristic richness, Shannon diversity index, species rarity index according to the French national red list (SRI), trophic index for Swiss ponds (M-NIP), relative coverage of submerged vegetation, percentage of exotic species in relation to total richness, and number of infrequent species. The corresponding Spearman correlation (r) with mapped inventories is mentioned in each plot legend.

Figure 3 -

 3 Figure3-Fitted relationships between residuals of the generalised additive models (GAMs) and each significant predictor factor for floristic richness, coverage of submerged species, and species rarity index according to the French national red list (SRI), and percentage of exotic species in relation to the total richness of the 262 SSLs. Solid lines are fitted splines in the continuous factors and fitted values for each of the categorical factors. Dashed lines are standard errors.

Table 1 -

 1 Summary statistics of the physical features of the 26 selected SSLs. SD = Standard Deviation.

		Mean ± SD	Median	Max	Min
	Surface area (m²)	16,579 ± 27,508	4,220	95,000	258
	Mean depth (m)	1.2 ± 1.1	1	5	0.1
	Shoreline index	1.28 ± 0.23	1.21	1.88	1.04
		[0-5%] = 15 sites; [5-25%] = 4 sites;
	% Banks with slope > 50%	[25-50%] = 3 sites; [50-75%] = 2 sites; [>75%] = 1 site

Table 2 -

 2 Main features of the 262 sites. SD = Standard Deviation

	Environmental factors	Mean ± SD	Median	Max	Min
	Surface area (m²)	17,614 ± 46,172	734	414,100	1
	Mean depth (m)	1.0 ± 1.1	0.6	7	0.05
	Shoreline index D*	1.34 ± 0.33	1.24	2.96	0.99
	% Banks instable or with slope > 50%	0 = 127 sites; 1 = 59 sites; 2 = 34 sites; 3 = 28 sites; 4 = 14 sites
	Elevation (m a.s.l.)	456 ± 670	138	3,340	2
	Distance from source (km)	33.8 ± 126.2	0.1	950	0
	Woodland (%)	44 ± 45	20	100	0
	Shade (%)	11 ± 25	0	1	0
	Drying		22%
	Invasive crayfish presence		13%
	Fish presence		40%

* D = Perimeter/(2 * √(π * Surface area)

Table 3 -

 3 Mean values and standard deviations of the seven descriptors (n = 26 sites) according to the three sampling methods. SRI, species rarity score; M-NIP, macrophyte nutrient index for ponds. The corresponding values for rare species (mapped coverage ≤ 1%) are also indicated for discussion. Small letters, significant differences between the descriptors estimated by the stratified method and those calculated by the other two methods (*** p-value < 0.005, Wilcoxon tests).

	S3m	Mapped	Stratified

Table 4 -

 4 Correlations between the S3m, stratified and mapped-inventory sampling methods for the seven descriptors. SRI, species rarity score; M-NIP, macrophyte nutrient index for ponds. SIG = mapped inventories. Best correlations are in bold.

			Richness			Shannon diversity index		SRI			M-NIP
		SIG	Strat.	S3m	SIG	Strat.	S3m	SIG	Strat.	S3m	SIG	Strat.	S3m
	SIG	1	0.81	1
	Strat.		1	0.81
	S3m			1
	SIG				1	0.60	0.85
	Strat.					1	0.64
	S3m						1
	SIG							1	0.45	1
	Strat.								1	0.45
	S3m									1
	SIG										1	0.84	1
	Strat.											1	0.84
	S3m												1
			Submerged			Exotic		Infrequent Species
		SIG	Strat.	S3m	SIG	Strat.	S3m	SIG	Strat.	S3m
	SIG	1	0.83	0.96
	Strat.		1	0.84
	S3m			1
	SIG				1	0.58	1
	Strat.					1	0.58
	S3m						1
	SIG							1	0.85	1
	Strat.								1	0.85
	S3m									1

Table 5 -

 5 Results of the generalised additive model (GAM) describing the four descriptors of macrophyte communities in 262 SSLs in terms of biotic and abiotic factors. All values are F-statistics, apart from parametric factors (bank verticality & instability, invasive crayfish, and fish presence), which are t-statistics. Significance: *** P < 0.001; ** P < 0.01; * P < 0.05. Only factors and values with at least P < 0.05 are indicated.

		Total richness	SRI	Submerged	Exotic
	Surface area	19.28***	0.47*
	Distance from source (DIS)	5.57***	1.03**		3.80***
	Elevation	5.08***	2.54***		3.31***
	Bank verticality & instability	-3.57***
	Presence of invasive crayfish	-2.57*		-2.36*	-2.28*
	Presence of fish	2.11*	-2.34*		3.18**
	Woodland	1.50***	0.45*
	Shade			1.88***
	Mean depth		0.84*	0.74*	0.81*
	R²-adjusted	0.71	0.23	0.12	0.38
	Explained deviance	0.73	0.25	0.15	0.39

Table 6 -

 6 Differences in floristic richness observed between studies conducted in France or neighbouring countries.

	Localisation	Method	Richness	Number of SSLs Elevation-Surface area
	Région	Centre

Acknowledgments

We thank the Aquabio team, especially Joyce Lambert, David Serrette, and Nicolas Tarozzi, who assisted the first author in macrophyte sampling and identification. We also thank the Pyrenees National Park, the Regional Natural Parks of Bauges, Ballon des Vosges, Causses du Quercy, Perigord-Limousin, Plateau des Millevaches, Prealps Cotes d'Azur, Volcans d'Auvergne, and Vosges du Nord; the EPTB Seine Grands Lacs; the Regional Conservatories of Natural Spaces of Aquitaine, Bourgogne, Lorraine and Limousin; the French National Forestry Office; the French National Office of Hunting and Wildlife; the Chartered Fisheries and Aquatic Environment Protection Departmental Federations of Dordogne, Gironde, Puy de Dome and Vosges; the Gironde Hunting Federation; Pinail and Glomel Reserve Associations, and the communities of the cities of Andernos, Sage-Blavet, and Tregor-Lannion. Special thanks to the managers or site owners who have granted permission to carry out sampling. The BIOME project was labelled by the scientific council of DREAM Competitiveness Cluster.

Funding

This research followed the BIOME (BIOindication des Mares et Etangs) project, funded by Aquabio, and the call for proposals "IPME Biodiversité" launched by ADEME, grant number 1682C0129.

Availability of data and material

The datasets generated during and/or analysed during the current study are not publicly available due to private funding but are available from the corresponding author upon reasonable request.

Conflicts of interest

All authors declare that there is no conflict of interest.