Pseudocapacitances such as the hydrogen adsorption on platinum (HAoPt) are associated with faradaic chemical processes that appear as capacitive in their potentiodynamic response, which was reported to result from the kinetics of adsorption processes. This study discusses an alternative interpretation of the partly capacitive response of the HAoPt that is based on the proton transport of ad- or desorbed hydrogen in the double layer. Potentiodynamic perturbations of equilibrated surface states of the HAoPt lead to typical double layer responses with the characteristic resistive–capacitive relaxations that overshadow the fast adsorption kinetics. A potential-dependent double layer representation by a dynamic transmission line model incorporates the HAoPt in terms of capacitive contributions and can computationally reconstruct the charge exchanged in full range cyclic voltammetry data. The coupling of charge transfer with double layer dynamics displays a novel physicochemical theory to explain the phenomenon of pseudocapacitance and the mechanisms in thereon based supercapacitors. Dating back to 1941, Graham7 introduced the “pseudo-capacity” as a capacitive process that is based on the reversible electro-reduction of ions and that is clearly separated from the mechanism of the charge separation in the double layer. The double layer describes the ion arrangement at the electrode–electrolyte interface10. A change of the electrode potential rearranges the ions in the double layer13 and the related resistive-capacitive contributions represent part of the potentiodynamic response of every electrochemical system. In aqueous solutions, the response of the double layer to a potential variation is typically parameterized by a constant phase element19, which is a frequency domain based representation of the more complex transmission line20. The transmission line directly represents the potential and current distribution in the double layer and the related ion transport22.Pseudocapacitance is a phenomenon in electrochemistry that finds applications in supercapacitors23, covering a broad range of different types of electrochemical processes25. The hydrogen adsorption on platinum (HAoPt) appears as mainly capacitive in impedance investigations26. The HAoPt as a form of pseudocapacitance was discussed by Conway et al.27 in 1980, separating the capacitance of the double layer from that of the hydrogen adsorption. On the basis of this study, further analyses of the HAoPt were carried out using the same equivalent circuit and assumptions on the mechanisms of the pseudocapacitance30. Based on Conway’s kinetic theory on pseudocapacitance31, the understanding of physicochemical mechanisms of these phenomena today is that a linear dependence of the heat of adsorption on the surface coverage of the electrodeposited species leads to a potentiodynamic response in the form of a capacitance24. Methods to distinguish between battery-type redox reactions and pseudo capacitances are mainly based on the different shapes of these contributions to cyclic voltammetry (CV) data35.In today’s science, a pseudocapacitor is understood as an electrode that stores charge indirectly through faradaic chemical processes with an electrical behavior of a capacitor38. In contrast, electrochemical impedance spectroscopy40 (EIS) typically employs small amplitudes (< 0.1 V) in the form of sinusoidal potential variations to probe perturbations of stationary states. The phase angle and amplitude of the resulting sinusoidal current is measured in the frequency domain. The intrinsic potential dependencies of electrochemical processes that arise at large amplitude probing cannot be resolved by EIS in detail21, however, the related impedance calculus and a simple equivalent circuit parameterization display powerful features of this method. With their different scopes and strengths, EIS and CV complement each other to probe the potentiodynamic response of electrochemical systems21.CV is a standard method in electrochemistry that is typically operated at large amplitudes (> 0.1 V) with a triangular potential variation to characterize the transient change of surface states, electrochemical reactions and kinetics in the form of a current in the time domain41, while each surface orientation shows an individual potential dependence in the form of a characteristic CV profile44. From a quantum mechanical point of view the HAoPt can be described with density functional theory (DFT) on an atomic scale49. Such simulations of the surface states are typically conducted for equilibrated systems, excluding the dynamics that come into play during the potentiodynamic response. Modeled CV responses of the HAoPt that are based on differences of the DFT-determined equilibrium surface coverages are reported51. Despite these approaches neglect potentiodynamic relaxation phenomena (by the double layer and the kinetic adsorption) they show in full range CV data similar transient responses of the HAoPt to those measured41.The HAoPt mainly takes place at potentials below 0.4 VThe aim of this study is to present a new theory of pseudocapacitance that is based on the coupling of double layer dynamics with potentiodynamic charge transfer reactions. Hereto, EIS and CV data are collected on a polished and electro-oxidized (cycled) polycrystalline platinum sample. The potentiodynamic responses of the HAoPt to perturbations of equilibrated surface states show the typical resistive–capacitive relaxation of the double layer dynamics. EIS data at a stepwise potential variation are used to parameterize a dynamic transition line model that represents the potential dependence of the double layer. This model describes the mainly capacitive CV response at small amplitudes as well as the transient charge transfer features of the HAoPt that is examined with full range potential scans. On the basis of these results, the pseudocapacitance of the HAoPt is here discussed as a combination double layer dynamics with an instantaneous charge transfer, whereas the kinetics of the adsorption processes play a minor role in the potentiodynamic response.21. In this in-house made cell, a geometric area of 0.79 cm2 of the working electrode is exposed to the electrolyte. A porous glass frit with fine pores is used to purge the electrolyte with argon, so that the amount of dissolved oxygen in the electrolyte is reduced. The platinum specimen is polished with 4000 grid SiC sandpaper and water as a lubricant. Further polishing of the sample with pastes is avoided, as these typically contain organics that contaminate the surface. Solutions of 0.01, 0.1 and 1 M HClO4 (Alfa Aesar) are used as the electrolytes, as perchloric acid is a superacid with an anion valence of one, displaying an ideal model electrolyte for electrochemical studies. An Ag/AgCl reference electrode (Metrohm) with a 3 M KCl electrolyte is employed, from which the potentials vs the reversible hydrogen electrode (RHE) are calculated by adding 0.197 V  and In this study, a freshly polished polycrystalline platinum plate in the form of a commercially-purchased sputter target  is used as working electrode in a previously reported three-electrode cell20, which enables a time domain response modeling with differential equations22. Double layers typically show a potential-dependent response in impedance spectroscopy11. By dynamically changing the parameterization of the transmission line, the potential dependence of the double layer is implemented22. In comparison to a previously reported dynamic transmission model for CV data22, this study uses the following improvements .The measurements presented in this study are all conducted with one polycrystalline platinum specimen without intermediate polishing. After initially polishing this specimen, EIS and CV data were recorded within a potential window between 0.05 and 0.6 V vs RHE under electrolyte variation of 0.01, 0.1 and 1 M HClO4 electrolyte. The mean potential was applied 1 min prior to the measurements to obtain an equilibrated surface state. Besides the value of the impedance 11. In addition, the fits of the equivalent circuit (electrolyte resistance and CPE) to the measured spectra are shown. At 0.1 V, platinum adsorbs hydrogen and the related pseudocapacitance can be observed. Between 0.4 and 0.6 V the CV response of platinum is reported as featureless, for which it is stated to be dominated by the double layer capacitance41.The measurements on the polished specimen presented in the following were limited to the potential range between 0.05 and 0.6 V to avoid a potential-induced surface change in the form of an oxidation. Figure 11. In brief, when A detailed discussion of the impedance spectra of the double layer on gold, which can be transferred to the presented data of the platinum electrode, is given elsewhere11, higher electrolyte concentrations decrease the resistance and with reference to Eq.  potential perturbations of equilibrated surface states. Figure 21. At both mean potentials, the measured CV response displays the typical shape of the double layer response which is also observed for gold electrodes21. On the basis of the impedance data, the CV response is modeled with the above described dynamic transition line model, showing a good representation of the measured data.To further analyze the response in the HAoPt and double layer regime, the CV response is considered in the following, as the time domain analysis can reveal potential dependent features that are not visible in the frequency domain21. To summarize, the perturbations of equilibrated surface states thus far indicated that the double layer dynamics dominate the responses, while the HAoPt at 0.1 V increases the measured capacitances by more than a factor of ten compared to the values at 0.5 V.The capacitance measured with EIS and CV is at 0.1 V more than tenfold higher than that at 0.5 V, which can be attributed to the contributions of HAoPt. At both potentials, the responses show the typical shape related to the double layer dynamics that resembles previously reported results on the double layer dynamics of a gold electrodeFigure 22). Large amplitudes change the ion arrangement in the double layer of the probed system and lead to asymmetric ion movement21, which is not probed by the small amplitude impedance parameterization presented in Fig. On the basis of the parameterization in Fig. 53, nanoparticles55 and single crystals58 of platinum were reported in the literature. These works excessively examined the influence of sample preparation of platinum electrodes on the measured CV response, including flame annealing and electrochemical cleaning in the form of potential cycling41. Full range potential cycling leads to reproducible full range CVs. However, the surface atom arrangement is changed by oxidation59, platinum dissolution and redeposition61, while different surface sites of single crystals come with different dissolution rates62. The surface atom rearrangement due to oxidation and dissolution also changes the hydrogen adsorption profile63. In agreement with these literature, the CV data changes after electro-oxidation of the polycrystalline platinum specimen with cycling to 1.3 V. Reproducible CV data with a semi-stable surface state is obtained after approximately 50 cycles. The oxidation and dissolution of the surface atoms lead to their rearrangement that results in an increase of the surface orientation with the most stable configuration62.Figure Figure By applying larger upper potential limits than the 0.6 V, Fig. 51, which differs from a typical electrochemical reaction with a distinct equilibrium potential (based on the Nernst equation) that separates reducing or oxidizing conditions. Moreover, unlike a typical electrocatalytic redox-reaction where ions from the electrolyte are converted and the catalyst comes back to its initial condition, the adsorption changes the surface state of the electrocatalyst itself.The hydrogen adsorption displays a continuous variation of different adsorption states as a function of the potential64 they dominate the charge transport in the double layer. As a result, the processes of double layer charging and hydrogen adsorption are both expected to be affected by similar ion transport processes in the double layer.When one hydrogen atom is adsorbed at the platinum electrode, one proton is removed from the double layer. To maintain the charge balance in the double layer, a proton from the solution migrates to the double layer and is thus replacing the proton that was adsorbed. Accordingly, the hydrogen adsorption interacts with the shielding of the electrode in the double layer, for which it is directly coupled to the transport processes in the double layer. The presented data on small amplitude EIS and CV measurement at 0.1 and 0.5 V Figs.  showed t51) dominate the response. A simplified analogism can be found when a constant potential change The resistive charge transfer is overshadowed by the ion transport in the double layer, however, it is incorporated in the form of capacitive contributions in the double layer parameterization of the dynamic transmission line model. In the case of the full range CV response, the relaxation of the double layer plays a minor role, leading to a quasi-instantaneous response to resistive-capacitive contributions of the dynamic transmission line model. Accordingly, the charge transfer of the adsorption process that is related to the differences of the surface coverage  are sluggish, so that the potential perturbations that are used in the impedance measurements just partly cause a change of the surface state , examined by electrochemical impedance spectroscopy (EIS) and cyclic voltammetry (CV) in combination with the computational approach of a dynamic transmission line model. First, potential perturbations of equilibrated surface states of a polycrystalline and polished platinum specimen were examined with amplitudes of 0.02 V for EIS and 0.05 V for CV. At 0.1 V, the HAoPt led to more than tenfold higher capacitances than those at 0.5 V where hydrogen and oxygen adsorption play a minor role. However, in both potential regimes, the potentiodynamic responses were dominated by the double layer with its characteristic resistive-capacitive relaxation that is described by the transmission line model. Whether ions are captured in the double layer itself or protons are adsorbed at the electrode’s surface, in both bases the ion transport must proceed through the double layer. Thereby, the double layer dynamics overshadow the kinetics of the quasi-instantaneous charge transfer of the HAoPt for which the actual charge transfer appears as pseudo-capacitive. Second, the potential-dependence of the double layer was parameterized by impedance measurements under incremental potential variation. On the basis of this parameterization, full range CV data of the platinum specimen were computationally reconstructed with the dynamic transmission line model. In contrast to the potential perturbations with small amplitudes, in the framework of the slow and large potential variations during full range CV, the transient charge transfer by the HAoPt dominates the response whereas the double layer dynamics play a minor role. By precisely describing CV responses under different amplitudes and scan rates, the dynamic transmission line model builds a bridge between the pseudo-capacitive appearance of the HAoPt at small amplitudes and its transient charge transfer character at large amplitude. Based on this detailed analysis, changes of the hydrogen surface coverages and the related charge transfer reactions are found to be inevitably connected with the ion transport in the double layer. This coupling is ultimately responsible for the dualism of the pseudo-capacitive and transient charge transfer character of the HAoPt. The discussed pseudocapacitance due to the ion transport in the double layer is expected to appear in every pseudocapacitor and supercapacitor, which  can mix with the mechanisms described by the established kinetic theory of pseudocapacitance. To understand the combination of both mechanisms and its relation to the employed electrode materials in pseudocapacitors and supercapacitors, detailed experimental and computational studies have to follow.Supplementary Information.


The difference in the shear strength and other characteristics of the cohesive soil containing calcareous nodules (CSCN) between samples with large size and corresponding scaling size, which is called scaling effect, is significantly affected by its calcareous nodule content (CNC) of the gradation composition. However, current researches rarely reveal the influence of the CNC on the scaling effect in shear strength of samples. In this study, how and why the CNC affects the scaling effect in shear strength were explored. Then a method to reduce the scaling effect based on the reason for influence was proposed. Results show that the correlation between the scaling effect in shear strength and the CNC presents a step curve. This is attributed to that it is easier to form a skeleton effect in samples with scaling size for the same CNC. Considering the skeleton effect, a calculation model for the shear strength parameters of CSCN samples with large size is proposed to reduce the scaling effect. This paper demonstrates that the proposed calculation model provides an access to obtain calculated shear strength parameters of CSCN samples with large size by using measured results of samples with corresponding scaling size. In lab direct shear tests, since the particle size of calcareous nodules is generally between 1 and 80 mm3, a large-scale direct shear apparatus of which the ring with 300 mm in diameter and 150 mm in height5 is preferred to be used for CSCN samples. These CSCN samples with the same size as that of the above ring are called samples with large size6. During the process of the sample preparation, the time of the preparation for a reconstituted sample with large size is long and the cost of that is high7, which incurs the huge economic and time costs to carry out multiple groups of tests at the same time for some research institutes. In order to shorten the sample preparation time and cut down the test cost, samples with the diameter of 61.8 mm and the height of 20 mm, which are called samples with scaling size, are considered to replace samples with large size to study the shear properties of the CSCN8. Furthermore, calcareous nodules with the size of more than 5 mm  in samples with scaling size should be processed to meet the requirement that the maximum size of soil particles in samples is 1/5 ~ 1/6 of the sample diameter6.Cohesive soil containing calcareous nodules (CSCN) is a special soil-rock mixture9. For example, the oversize elimination method that sieving oversize calcareous nodules from CSCN samples decreases the calcareous nodule content (CNC) to change the gradation composition of the sample10, decreasing the shear strength of CSCN samples with scaling size11. In other words, the difference of the gradation composition results in the difference in the shear strength and other characteristics of the CSCN between samples with large size and corresponding scaling size, which is called scaling effect12.At present, the “oversize calcareous nodules” processing methods that are commonly adopted in lab tests, including the oversize elimination, equal quantity substitution and similar gradation methods, inevitably affect the macroscopic deformation and failure mechanisms of CSCN samples13. Many researches focus on the influence of the maximum particle size on the scaling effect of samples15. For example, the cohesion and internal friction angle16, maximum dry density17 of samples with scaling size which are prepared by the similar gradation method decrease with the decrease of the maximum particle size in gradation composition. On the other hand, many gradation equations based on the gradation composition have been proposed, which study the influence of gradation composition on the scaling effect by changing the equation fitting parameters19. However, these equations cannot directly reveal the influence law of grain group content of calcareous nodules, i.e., the CNC  on the scaling effect in soil strength. The current research on how and why the scaling effect in shear strength would be affected by the CNC is still superficial or even absent.According to previous studies, the difference of gradation composition is mainly reflected in the difference of maximum particle size and content of grain group of calcareous nodulesIn this paper, a series of direct shear tests were carried out on CSCN samples with large size and corresponding scaling size with different CNCs, respectively. The influence of the CNC on the scaling effect in shear strength of CSCN samples was studied systematically. From an understanding of the influence, a calculation model for the cohesion and internal friction angle of CSCN samples was proposed to reduce the scaling effect. The proposed model was then verified with data obtained from the published dissertations.Ф 300 × H 270 mm) and corresponding scaling size (Ф 61.8 × H 20 mm) were carried out, respectively. Six groups of samples with 0%, 10%, 20%, 30%, 40% and 50% in CNC were set for each type of test, respectively. In addition, the particle breakage of calcareous nodules can be ignored at the small normal stress9. Accordingly, the maximum normal stress of 200 kPa was applied in this test to reduce the influence of the particle breakage of calcareous nodules on the test results.Table Figure In this study, the CSCN was taken from Suqian, China and its typical constituents and structural characteristics are shown in Fig. 20.Samples with large size and corresponding scaling size were prepared from the natural CSCN. First, the calcareous nodules were isolated from the natural CSCN and sieved to obtain the grain group with 1–2 mm, 2–5 mm, 5–10 mm, 10–20 mm and more than 20 mm. A predetermined quantity of calcareous nodules was added to the soil and then mixed through a mixing machine to obtain different samples with CNC of 0%, 10%, 20%, 30%, 40% and 50%. The proportion of each grain group of calcareous nodules in each mixture was consistent with that in natural CSCN, as shown in Fig. 13. In order to control the same shear properties of soil, the particle size of soil in the CSCN was not scaled. Then the gradation distributions of samples with scaling size are shown in Fig. Differently, oversize calcareous nodules in samples with scaling size were first processed to within 2 mm applying the similar gradation mixed with equal mass substitution method, based on ASTM. This method can well control the content ratio of each grain group of calcareous nodules in samples with scaling size similar to that in samples with large size for the same CNCФ 300 × H 270 mm) and normal-scale (Ф 61.8 × H 20 mm) direct shear apparatus, respectively, in accordance with ASTM D3080/D3080M-11. Normal load applied were 353 kg, 707 kg, 1060 kg and 1413 kg for samples with large size. Before shearing, samples were consolidated for 24 h under 50 kPa, 100 kPa, 150 kPa and 200 kPa of normal stress, respectively. Strain controlled test was the type that was carried out with the sample sets and the apparatus applied strain at the rate of 0.8 mm/min.Direct shear tests for CSCN samples were carried out using the large-scale  of the cohesion of samples with large size and corresponding scaling size for the same CNC is used to express the scaling effect in the cohesion. It is worth noting that, the ratio is around 1.0 when the CNC is 0%–10% and 50% but which rises to around 1.5 when the CNC is 20%–30%. It suggests that there is an inflection point in the correlation curve of the scaling effect in the cohesion and CNC, which presents a step curve. In other words, the cohesion of samples with large size decreases at the CNC of 20%–30% while that for samples with corresponding scaling size at the CNC of 10%–20%. It is attributed to the significant role of resistance to shear stress a low CNC played in samples with corresponding scaling size, which is equivalent to that of a high CNC in samples with large size.Figure 23. Similar to the development trend of the cohesion, the difference of the internal friction angle of samples with 20% in CNC is the largest, approximately 8.7º. This suggests that the scaling effect in the internal friction angle is insignificant for samples with a low (0%–10%) or high (more than 30%) CNC. Accordingly, at this time, it is applicable to adopt samples with scaling size to study the shear properties of the CSCN without considering the scaling effect. As expected, there is also an inflection point in the correlation curve of the scaling effect in the internal friction angle with CNC. To be specific, the ratio (φL/φS) of the internal friction angle of samples with large size and corresponding scaling size for the same CNC climbs to the largest, approximately 1.3 at the CNC of 20%. Hence, a great degree of bite force is associated with a high CNC in samples with large size while that with a low CNC in samples with corresponding scaling size.Figure 9 pointed out that the primary reason for the influence of the CNC on the shear strength of samples was due to a skeleton effect formed by calcareous nodules. The skeleton effect here mainly refers to the influence of the framework formed by the contacts of calcareous nodules in the sample on the deformation and strength. A large number of calcareous nodules gather together, contact each other and transfer interaction, which will easily form a strong skeleton effect24. Figure 25, as shown in Fig. 26 and bite force to increase the shear properties of samples, which is the common reflection of the interaction between calcareous nodules and soil27. When the CNC is more than a certain content, the skeleton structure in samples is almost completely built, which results in that the "pseudo cohesion" and bite force provided by calcareous nodules gradually reaches to a limit state28. Hence, the increase of CNC creates an insignificant difference in the macro-mechanics between samples with large size and corresponding scaling size.Xu and Zhang29. In samples with large size, the grain groups of calcareous nodules are widely distributed from 1 to 40 mm. In order to make the gradation curves with continuous gradation, there is a certain content of calcareous nodules in each grain group19. During the preparation for samples with corresponding scaling size, the grain group distribution range of calcareous nodules is reduced to between 1 and 2 mm. For the same CNC, more calcareous nodules with scaling particle size are arranged into the downsized grain group, which increases the content in this grain group. In other words, this behavior makes the content of grain group of 1–2 mm the largest, which affects significantly the soil structure. As the similar gradation composition, compared with that for samples with large size, the ratio of the particle size of calcareous nodules with the largest content in samples with corresponding scaling size to the sample diameter is larger. To be specific, for samples with scaling size, the content of calcareous nodules with 1–2 mm in particle size is the largest and the ratio of this particle size to the sample diameter is 1/60–1/30, as shown in Fig. The great difference of scaling effect in shear properties of samples is associated with the skeleton effect caused by soil structure in CSCN samples during different levels of CNC. The skeleton effect works effectively in samples with scaling size at the CNC of 10%–20% but it does not work in samples with large size until the CNC is 20%–30%. This behavior is attributed to the larger ratio of the particle size of calcareous nodules with the largest content in samples with scaling size to the corresponding sample diameter30. For example, Li et al.6 proposed a modification method for shear properties of samples with large size using the modifying coefficient, which is expressed by Eqs. .2\do.2\do30. 31 pointed out that the skeleton contacts can only transfer elastic interaction but cannot resist tension and rotation. Based on this, contacts of calcareous nodules can be obtained based on the following two assumptions. First, all the calcareous nodules are assumed to be rigid spheres with same particle size, which is generally the particle size of calcareous nodules with the largest content in samples. Second, there is a same interaction assumed between each contact of calcareous nodules. Then, steps to obtain contacts are as follows:However, these calculation methods only consider the law of the difference in the shear properties between samples with large size and corresponding scaling size, which is the result caused by the scaling effect, ignoring the skeleton effect which is the reason for the scaling effect. To overcome this problem, a calculation model for shear strength parameters of CSCN samples considering the skeleton effect is investigated. First of all, from an understanding of the reason for the influence on the scaling effect in shear strength, the relationship between contacts of calcareous nodules which can express the skeleton effect and shear properties of samples is revealed. Shi et al.Step 1, the total volumes of calcareous nodules and fine soil in a sample with small size are calculated according to the CNC, as expressed by Eqs.  and 8)..8). ThenStep 2, the volumes of a single calcareous nodule sphere and a real sample are obtained according to the calculation formula of the ball and cylinder, respectively, as expressed by Eqs.  and 11)11).10\doStep 3, the coefficient of sample reduction can be expressed by Eq.  accordinr in radius in the sample with a small size.Step 4, the number of calcareous nodules in the sample according to the total volume of calcareous nodules [see Eq. ] and theStep 5, contacts of calcareous nodules in the real sample is expressed by Eq. .14\docum24 and suggests that the prediction results of contacts of calcareous nodules can be applied in the calculation model. It is worth noting that, the correlation between the cohesion, internal friction angle and contacts of CSCN samples can be expressed as a line, which can be expressed by Eqs. 24 and su33 was used to validate the proposed calculation model. The degree of accuracy of the proposed calculation model was evaluated through a comparison of the calculated and measured values. Figure . The error bounds for the calculated cohesion and internal friction angle are ± 18.6% and ± 12.0%, respectively, which are lower than the average error (about 25%) evaluated using the Li’s method6. This suggests that calculated values are in a good agreement with measured values and it is reasonable to propose to adopt the calculation model for estimation of the shear properties of CSCN samples. It is also believed that the calculation model provides an access to obtain calculated shear strength parameters of CSCN samples with large size by using measured results of samples with corresponding scaling size.Some data taken form the published dissertationsIn this paper, direct shear tests were performed to investigate the influence of the CNC on the scaling effect in shear strength of CSCN samples. From the understanding of the reason for the influence, a calculation model for the shear strength parameters of CSCN samples was proposed. The conclusions are as follows.The correlation between the scaling effect in shear strength and the CNC presents a step curve and the inflection point locates at the CNC of 20%–30%. This suggests that the scaling effect in the cohesion and internal friction angle is obvious for samples with 20%–30% in CNC while that is negligible for samples with a lower or higher CNC. Therefore, when the CNC is low (0%–10%) or high (more than 30%), it is applicable to adopt samples with scaling size to study the shear properties of the CSCN without considering the scaling effect.The arrangement of scaling particles into downsized grain groups makes the soil structure be affected significantly by the particle size of calcareous nodules with the largest content. As the similar gradation composition, the ratio of the particle size of calcareous nodules with the largest content to the sample diameter for samples with corresponding scaling size is larger than that for samples with large size. This performance makes it is easier for calcareous nodules in samples with scaling size to form the skeleton structure to resist the compression stress during the consolidation, resulting in the skeleton effect working more easily to increase the scaling effect when shearing.The calculation model proposed in this paper calculated the cohesion and internal friction angle of CSCN samples reasonably well. The error bound for the calculated cohesion and internal friction angle is approximately ± 18.6% and ± 12.0% of the measured values, respectively. It is believed that the calculation model outlined in this paper can provide an access to obtain calculated compression characteristic parameters of CSCN samples with large size by using measured results of samples with corresponding scaling size. Furthermore, the numerical analysis mixed with statistical analysis can be used as an attempt to simulate contacts of calcareous nodules in samples with different sizes in the further study, which can build a precise calculation model to provide guidance for construction.


Many real-world systems change their parameters during the operation. Thus, before the analysis of the data, there is a need to divide the raw signal into parts that can be considered as homogeneous segments. In this paper, we propose a segmentation procedure that can be applied for the signal with time-varying characteristics. Moreover, we assume that the examined signal exhibits impulsive behavior, thus it corresponds to the so-called heavy-tailed class of distributions. Due to the specific behavior of the data, classical algorithms known from the literature cannot be used directly in the segmentation procedure. In the considered case, the transition between parts corresponding to homogeneous segments is smooth and non-linear. This causes that the segmentation algorithm is more complex than in the classical case. We propose to apply the divergence measures that are based on the distance between the probability density functions for the two examined distributions. The novel segmentation algorithm is applied to real acoustic signals acquired during coffee grinding. Justification of the methodology has been performed experimentally and using Monte-Carlo simulations for data from the model with heavy-tailed distribution (here the stable distribution) with time-varying parameters. Although the methodology is demonstrated for a specific case, it can be extended to any process with time-changing characteristics. Many real-world systems change their parameters during the operation. It could be a continuous progressing change (like start up of the machine) or switching from regime A to another regime B . Analysis of such time-varying processes (using acquired data) is difficult. If the analyzed data have a complex structure, then before further analysis they should be divided into simpler subsignals. That requires the use of different methods. Such approaches are commonly called signal segmentation ,2,3.The task of segmentation considered in this paper is more general than the classical segmentation, where one is looking for a moment where characteristics of the signal have changed. Usually basic statistics  or more advanced features are used as criteria for splitting the signal into two or more homogeneous parts.As mentioned, the reasons for using segmentation may be very different. It is commonly used as pre-processing for a non-stationary signal which consists of stationary segments. In real data, there are many situations related to this problem. A good example is a machine that may change the regime of operation or a car with a manual gearbox where changing gear may be associated with a change of internal conditions in the system. Thus, it requires separate treatment.Signal segmentation is very advanced in speech processing where a single word or even a vowel should be extracted and recognized. Detection of the event in the signal  to very advanced techniques for a more complicated process. Interesting approaches used for the problem related to the changing of the location parameter (like mean) can be found in ,38,39,40In this paper, contrary to the classical segmentation problems, we consider a generalized case by assuming that the parameters of the model behave according to certain functions that stabilize at some level with time. Therefore, the change of regime in the data occurs smoothly. Moreover, the mentioned parameters corresponding to the scale and impulsiveness of the data, change not only according to some deterministic function, but we also observe certain random fluctuations in their values. Therefore, we consider the problem of segmenting the independent observations derived from the distribution with randomized parameters.In the theory of probability and in statistics, there is a class of distributions resulting from the assumption that a random variable is distributed according to a certain parametrized distribution for which some or all parameters are random. Such distributions are called compound probability distributions. If a scale parameter is randomized, the resulting distribution is also called a scale mixture. In the literature, there are examples of well-known probability distributions derived in this way, for instance, the nonstandardized t-Student distribution (Gaussian distribution with inverse gamma distributed variance), the Laplace distribution , the negative binomial distribution (Poisson distribution with gamma distributed rate parameter), or the beta-binomial distribution . Lee et al.  providedThe systems with randomized coefficients are considered also in the analysis of the stochastic processes. We mention here the autoregressive time series with random coefficients ,60,61,62Segmenting the signals with a smooth transition from one process to another seems to be a more complicated task than the classical case of segmentation mentioned earlier. The problem is even more complex when two parameters of the distribution change. This work proposes a segmentation method for such signals which relies on comparing the probability density functions (pdfs) calculated in moving windows. It is based on the assumption that when the distribution of the data stabilizes, the pdfs are similar. It is important to emphasize that this procedure does not split the entire signal only into homogeneous parts due to the nature of the signal. Nevertheless, the method allows indicating homogeneous fragments of time series that correspond to the parts of the signal with stabilized parameters.In the proposed procedure, we divide the original signal into segments with a priori predefined length. For each segment, we calculate its empirical pdf and utilise the divergence  to measure the distance between the calculated empirical densities. Finally, we differentiate the Jeffreys distance and use log-likelihood ratio to detect the change point. The introduced segmentation algorithm is applied to real acoustic signals acquired during coffee grinding. The proposed procedure can be summarized in three steps: the estimation of the pdf for data corresponding to the selected segments, calculation of the divergence measure, and identification of the change points for data representing the empirical divergence using one of the classical segmentation algorithm. Justification of the methodology has been performed experimentally and using Monte-Carlo simulations for data from the model with heavy-tailed distribution (here the stable distribution) with time-varying parameters. Although the methodology is demonstrated for a specific case, we hope it is universal and can be extended to any process with time-changing characteristics. Example of such a signal could be seismic signal (vibrations with damping), speech signal, or vibrations in mechanical systems with processes as cutting, compressing or crushing.The rest of the paper is organized as follows. In In this paper, we consider a non-stationary, highly impulsive, and energetic signal with the distribution stabilizing close to Gaussian. To confirm such an assumption, some preliminary analysis of real data has been performed, see t is more A than B or vice versa. However, as the transition is nonlinear and at some point the transition stabilize, it may be use as a criterion for segmentation.The transition between the process with property A  and the process with property B (process B) is smooth, as there is no sharp boundary. It makes change recognition difficult. We may say that the process at time As an illustration, we will use the acoustic signals captured during a grinding of coffee beans in a grinder. At the beginning, due to the cutting of beans by sharp spinning knives in the grinder, the process is very noisy and cutting is an impulsive process. Once the coffee beans are ground, the material in the grinder is more powdery than grainy. Thus, the acoustic signature of the grinding process is no more impulsive nor energetic. The key issue is how much time we should perform the grinding to achieve a satisfactory structure of the ground coffee. To validate our procedure, many experiments have been performed with various durations of grinding and after each of them a photo of the structure of coffee was taken and analyzed. A discussion on that is presented in From the mathematical point of view, the problem can be formulated in the following way. Let us imagine that the given observations correspond to some theoretical time series  The case when the To illustrate the problem and to validate our procedure using experimental data, we perform several experiments related to coffee bean grinding. Using a popular coffee grinder for domestic use, we prepared dozens of coffee samples with comparable volume and quality parameters. For each sample, the acoustic signal has been registered using a mobile phone, see The acoustic signals analyzed in the paper, obtained through the experiment described in the previous section, show some special properties. It should be emphasized that the mentioned experiment was conducted many times and the nature of the data is repetitive. For illustration purposes, we selected several realizations, which are subject to preliminary investigation in the next sections. Then, on the basis of the observations made during the initial analysis, a method of data segmentation is proposed.In the paper, we examine eight signals denoted as Signals 1–8 presented in the subsequent panels of Following the above-mentioned remarks, we can conclude that the distribution behind the data presented in the subsequent panels of The main goal of the paper is to introduce a procedure leading to the segmentation of the signals presented in Z, which provides the parameters of the distribution,The stable distribution  is not as strong as the notion of distance. The divergences do not have to be symmetric in the arguments nor satisfy the triangle inequality. An essential class of contrast functions are the  follows ,80,81J(p(xEquation . We mentJ(p(xEquaquation (J^(p(x),gof thumb . The kerM segments of length equal to L.Divide the signal into Estimate the pdf in each segment, L samples in the signal, Estimate the pdf corresponding to the last L samples,Calculate the Jeffreys distance between the pdfs in subsequent segments and the pdf of the last Calculate the increments (differences) of Jeffreys distance,Use the LLR method to find the index Consider the sub-samples In this section, we describe the proposed segmentation algorithm. The methodology relies on comparing the pdfs using the Jeffreys distance presented in L, then we move by L samples and as the second segment we take the samples from L, then we move by l samples and as the second segment we take the samples from l affects the accuracy with which we indicate the moment of regime change. In l is called step.The procedure described above leads to detecting the indexes Algorithm 1: Segmentation procedure based on Jeffreys distanceData: Input dataM segments of length equal to LDivide data into : Segment 1, Segment 2, …, Segment MEstimate the pdf in each segment:1 fortoMdo2 L samples in the signalEstimate the pdf of the last : Calculate the Jeffreys distance:3 fortoMdo4 Calculate the increments of Jeffreys distance:5 fortodo6 Use the LLR method to find change in scale in : Use the LLR method to find change in scale in : Use the LLR method to find change in scale in : In items 6 and 7, to segment the increments of Jeffreys distance, we use the log-likelihood ratio method due to the fact that the scale parameter in Equation  changes In this section, we present the results of applying the procedure described in To illustrate how the distribution changes over time, for the signals presented in In the following part, we estimate the parameters of the stable distribution, described in According to the segmentation procedure presented in As mentioned, the procedure is repeated for all signals presented in In this section, we apply the introduced methodology to the simulated signals constructed based on the assumptions mentioned in As an illustration, in In the following part of this section, we examine the simulated signals consisting of, similarly to the real signal mentioned above, 575 segments with 2500 independent symmetric stable random variables in each of them. The parameters of the distribution in the subsequent segments change according to the deterministic functions fitted to the parameters Signal 1  disturbeThe construction of the sample simulated signal is presented in In In In the last step, we verify the efficiency of the proposed methodology by conducting a Monte Carlo simulation study for a number of signals generated analogously to the one with the trajectory presented in panel (c) of To validate the obtained results, we repeated the experiment of coffee grinding several times. With each repetition, we extended the grinding time and took photos of the ground coffee beans after the experiment was completed. The same amount of coffee was ground for about 5 s, 10 s, 15 s, 20 s, 25 s, and 30 s sequentially. Pictures of the product obtained after grinding are shown in The preliminary evaluation of the results based on the photos is also supported by the analytical results. We remind here that the method proposed in As mentioned in In this paper, an original signal segmentation procedure for a random process with time-varying characteristics is proposed. Typically, the signal segmentation process is related to the detection of a moment in time when the process A switches to process B. The situation is relatively simple when the segments contain data described by different distributions or the corresponding distribution is the same but the parameters are different in those segments. The rule is simple here: the bigger the difference between the segments, the easier the segmentation. In that case, the segmentation algorithm divides the data into homogeneous parts.The case considered in this paper is much more complicated. The analyzed input data correspond to the model given by Equation . In thatStable distribution with varying parameters has been proposed to describe the data. For some combination of values of its parameters . The final step is to find (using LogLikelihood ratio LLR) the change point in the differentiated distance time series. Note that there is just one important parameter (segment size). For longer segments, we will improve the quality of pdf estimation, but we will lose the resolution in time. To minimize this effect, overlapping is used. The method is pretty intuitive and universal.To illustrate the problem and provide evidence of the effectiveness of the proposed method, two approaches have been proposed. A model of the signal has been proposed using the mentioned Both approaches (photos and simulations) provide similar information, so we assume that the method is appropriate and effective.We believe that the problem discussed in the paper may be important in many engineering applications . One may assume also that other specific parameters of the process can vary simultaneously. Therefore, further work might be related to validation on other real cases as well as generalization of segmentation to any processes with smooth transition.


This study aims to illustrate the landscape of passing opportunities of a football team across a set of competitive matches. To do so positional data of 5 competitive matches was used to create polygons of pass availability. Passes were divided into three types depending on the hypothetical threat they may pose to the opposing defense . These categories were used to create three heatmaps per match. Moreover, the mean time of passing opportunities was calculated and compared across matches and for the three categories of passes. Due to the specificity of player’s interactive behavior, results showed heatmaps with a variety of patterns. Specifically the fifth match was very dissimilar to the other four. However, characterizing a football match in terms of passing opportunities with a single heatmap dismisses the variety of dynamics that occur throughout a match. Therefore, three temporal heatmaps over windows of 10 min were presented highlighting on-going dynamical changes in pass availability. Results also display that penetrative passes were available over shorter periods of time than backward passes that were available shorter than support passes. The results highlight the sensibility of the model to different task constrains that emerge within football matches. We assume that a team with greater hegemony will be more threatening and this threat can be characterized by how each team explores the space that is temporarily free of opponents; allowing to either keep the ball possession, progress towards the goal or create a goal scoring opportunity4. To do so, the players’ decisions and actions are highly influenced by prospective information that emerges in the course of the players’ interactive behavior, i.e. a temporary open gap in the opponent defense5. Thus, this information continuously suggests not only what to do, but also when, where, and how to do it, highlighting ongoing possibilities of action8.During a football match, players adjust their decisions and actions to each other’s as they play over the hegemony of the game9, as it is influenced by the changing positioning of the players 10, as well as the players’ capability to perceive what the others can do12. Thus, passing opportunities emerge, persist, and dissolve within a limited space–time window13, and greatly depend on the players’ interactive behavior. For instance, the interaction between the ball-carrier and the off-ball players is determinant to create passing channels at a given moment. On one hand, the ball carrier’s attunement to relevant information  influences how open an environment is for his/her teammates. On the other hand, the presence of several potential pass receivers, increases a player unpredictability regarding the actions to perform, creating additional difficulties to the opposing team14.The rich set of individual and environmental constraints that can arise in a football match influences the availability of a diverse set of potential actions (some menacing whereas others are not), which can be conceptualized as a landscape of potential actions15. Previous research addressed this issue and created a landscape of passing opportunities in football which was based on the analysis of a single match17). Results revealed the key areas on field providing most opportunities for penetrative passes16 and also support and backward passes17 on a single competitive football match. Moreover, a heterogeneous space–time spread of passing opportunities across the entire field was displayed, which highlighted the specificity of player’s interactive behaviour during the course of a match. However, these previous research works has some limitations some of them we aim to address now: (1) instead of a single match analysis, on the current research work it was analysed the dynamics of the created landscapes, within and between a set of five competitive football matches; (2) the current research work drove us to the need of analyse the images created by the heatmaps beyond visual inspection. Therefore, to allow a more objective analysis we used a technique called Earth Mover’s Distance (EMD), which quantifies the differences between the heatmaps of the five matches under analysis; (3) concerning the dynamics of the landscapes within each match, temporal heatmaps were created which allows to identify how the threat (for the defence) caused by the different types of passing opportunities changed during the course of a match.This flow of prospective information is dynamically evolving over timeDespite a relevant initial contribution of the previous research work, to the authors’ knowledge, there is still a gap in the literature concerning the depiction of off-ball players’ actions, and how they are affected by specific constrains . Beyond the influence of these specific constraints is it possible to depict a similarity on player’s interactive behaviour along a set of matches? Aiming to answer this question, the main goal of this study was to identify the landscapes of passing opportunities as a result of spatio-temporal windows that emerge due to the interactive behaviour between off-ball players, defenders, and the ball carrier during different football matches.x and y) coordinates of each player of both teams recorded at 25 fps by an opto-tracking system. The data was recorded and provided by Footovision , complemented by game events . This study received institutional ethics approval from Faculdade de Motricidade Humana, University of Lisbon. It is worth noting that we analyzed performance that did not require identification of individual performers.The data analyzed corresponds to five matches of the Spanish second division respective to the 2017/18 season, focusing on one team and five of its competitors. Over these matches, the team under scope had 20 participating players. The data corresponds to bi-dimensional  the players’ current position and (2) to the players’ estimated position a second later (taking into account the player current velocity) as long as the velocity was higher than 2 m/s. This analysis was only carried on when the ball was in play. Based on the ball carrier and support player’s current and estimated positions, two potential passing lines were created18 (see19 for a similar approach to the one taken to create these areas). Using the velocity of each player, a matrix was created that corresponded to each turning capability in such a way that each player velocity on percentile 90 was set as the point with the lowest turning capability (40º) and a velocity of 0 m/s was stablish as the highest turning capability (360º)18. These turning capabilities were then used to create coverage areas  that were used to calculate the time each defender will take to arrive to each point of the potential passing line assuming that the player will take a second to arrive to the end of the area. When the velocity of a player was under 1.5 m/s we used this value instead of the actual velocity of the player. With this, unrealistic situations were defenders did not cover any space were avoided.Afterwards, aiming to evaluate if these passing opportunities were actually available, defensive coverage areas were defined based on the defending players’ running line velocitiesIf the player could arrive to any point of the line before the ball, the pass was consider to be intercepted (see Algorithm 1). If the pass was available, a polygon was defined as the area that is temporarily available to perform a pass, with the ball carrier, the receiver and the two nearest defenders as its vertexes. In the absence of any defensive player, the sideline was used as an alternative. Based on player’s positional data this polygon was updated every 0.2 s. In this study the ball was considered to move at a constant 10 m/s speed. While this value was chosen for illustrative purposes, it is important to acknowledge that ball speed is a dynamic parameter  and that it can, and should be, fine-tuned by the end-user.20. By ‘outplay’ we mean the defending player’s that after a pass were further away from the own goal than the ball. Therefore, this outplayed principle is proposed in order to categorize the different types of passes that could be used to create a landscape model of passing opportunities in football. This principle expresses the number of opponents a ball carrier has in between her/him and their own goal22. Passes that outplay more opponents have been shown to be correlated with the number of goals scored24, are therefore, more threatening but are also generally more risky25.Some actions taken by attacking players in football are more threatening than others. For instance, to increase the chances to shot at goal, the ball carrier must perform passes that outplay as many opponents as possiblepenetrative pass, a passing opportunity to a player that outplays more opponents than the ball carrier; (2) support pass, a passing opportunity to a player that outplays the same opponents than the ball carrier; and (3) backward pass, a passing opportunity to a teammate which outplays less opponents than the ball carrier   penetrat24) may create different landscapes of passing opportunities, and due to specific task constraints that characterize each match, they could vary across different matches.Moreover due to the level of threat that might cause to the opposing team, it was hypothesized that different passes (based on the outplay principlepenetrative passes (as they will receive a higher reward for them) and these passes are more threatening and risky than the other two, we hypothesized that penetrative passes opportunities were available for shorter periods than support or backward passes.Finally, following the rationale that players are more willing to perform https://github.com/luisjordana/landscape-passing-opportunities] including 3 min of data, as well as Matlab routines, including algorithms 1 and 2, and others that can be used to load and process the data, run the algorithm and create similar videos and heatmaps as the ones presented in this article.Additionally, we created a repository [To define the landscape of passing opportunities, heatmaps were built by overlapping the different polygons available throughout a match. For the heatmaps to express how long a given area was under potential passing opportunities, the obtained count values were divided by the data set’s sampling rate (or frequency). Thus, and considering the color scale used, dark blue means that no passing opportunities were available, whereas dark red means that passing opportunities were available for more than 100 s per match.penetrative, support and backward). To allow comparisons through the matches, the field dimensions were normalized to 105 m × 70 m.A heatmap was built for each of the five matches and each of the three types of potential passes previously defined . This method provides a measure of similarity between two images27, with lower values meaning high similarity and vice versa. As an EMD output we created a colored matrix where the coldest colors  display a high similarity between heatmaps, whereas the warmest colors  display a low similarity between heatmaps.Additionally to go beyond a visual inspection, the heatmaps of the five matches for each type of pass were compared using the Earth Mover’s Distance , a two-way ANOVA was performed using Rstudio . It is important to note that since the distribution of passing availability is not normal , a requirement for properly implementing an ANOVA, a logarithmic transformation was implemented. After this transformation, q-q plots and residual plots where consistent with a normal distribution with homogeneity of variances. Furthermore, the factors were found not to be collinear.2) statistic, with values near 0.01 meaning the effect was small, 0.06 medium, and 0.14 high28. To asses uncertainty the 95% confidence intervals regarding these power of the effects were provided. In case there was a significant effect of a factor and/or the interaction, a Tukey post hoc analysis was run to analyze the main effects and/or interactions between the two factors, providing information on which specific matches and/or type of passes differ. For all tests performed, the level of significance was p < 0.05.The power of the effect is quantified by the eta squared  and own side of the field . Backward passing opportunities  = 376.80, p < 0.001; Penetrative: 1.00 ± 0.98 and Support: 1.52 ± 1.39, Backward: 1.13 ± 0.97), with a medium effect power. Post-hoc results showed that on average the possibilities to perform a penetrative pass had less time available than the opportunities to perform a backward pass, which in turn had less time available than the opportunities to perform a support pass.The ANOVA revealed a significant effect for the factor TypeOfPass  = 28.07, p < 0.001; 1st: 1.34 ± 1.13 s, 2nd: 1.15 ± 1.04 s, 3rd: 1.21 ± 1.18 s, 4th: 1.17 ± 1.17 s and 5th: 1.16 ± 1.15 s), with a low power effect , with the post-hoc analysis revealing that the first match had potential passes available for longer periods, while the remaining matches did not differ between them.The ANOVA analysis highlighted a significant effect of the factor Game on the mean passing availability  = 5.06, p < 0.001), with a low power of the effect . A post hoc analysis showed that penetrative and backward passing opportunities were available for longer periods in the first match than in the other four matches. Furthermore, the difference between backward and penetrative passes was only significant over the three first matches. Finally, support passing opportunities were similar between all five matches. Table Additionally, there was a significant effect of the interaction between the two factors  are not homogeneously spread across the space.Visually inspecting the obtained landscapes enables the interpretation of game patterns that may characterize a team’s performance. For example, we may suggest that the team under analysis had difficulties to create opportunities to pass within the opposing midfield, since most of passing opportunities were created in its own midfield. Specifically, backward passing opportunities displayed the highest similarity between the five matches under analysis. A similar result was achieved for the first four matches regarding the penetrative and support passes opportunities. However, the fifth match demands a more detailed explanation. On the 5th match the EMD results displayed a low similarity of penetrative and support passing opportunities when compared with the remaining four . The analysis of the heatmaps with shorter temporal windows may enable to identify changes in the landscapes of passing opportunities that emerge during the course of a match and how they evolve with time. For example, in the 4th match it was possible to observe the transitions of rd minute, which means that the opponent team played with one player less for approximately half an hour, which demands an adjustment on his team spatio-temporal configuration. Consequently, these adjustments may create additional difficulties to the team to defend and close passing opportunities but also to regain ball possession and create passing opportunities. Thus, this landscape model reinforces the notion that disturbances in a team initial configuration  alter players interactive behavior which consequently lead to changes in the time that a passing opportunities were available. It is noteworthy that this difference only happens for backward and penetrative passes and not for support passes, which suggests that these type of passing opportunities were less affected by the team defense capabilities and strategies.Regarding the mean time that passing opportunities were available, there were differences over the matches and for the different type of passing opportunities. The first match had longer mean times for their potential passes than the other four (that did not differ between them). This is probably due to an opposing player receiving a red card on the 63penetrative passing opportunities been available for less time than support or backward passes (for the first three matches); these may be because penetrative passes will place the ball onto an off-ball player that is outplaying more opponents than the ball carrier, which is hypothetically more dangerous for the defending team than a support or a backward pass3. This theoretical threat may increase the defenders’ willingness to intercept passing opportunities, which consequently decreases the time that penetrative passing are available compared to backward or support passes. However, this should mean that support passing opportunities had less time available than backward passes. Nevertheless, the results revealed otherwise, which could be explain by a possible increased pressure to recover ball possession close to the opposing goal31. This could explain the results of the fourth and fifth match where there was no significant differences between the time that backward and penetrative passing opportunities were available, possibly due to a high defensive pressure of the opposing team.Concerning Differences in the time that passing opportunities were available as well as visual differences in the landscapes of passing opportunities accordingly with the different type of passes, led us to suggest that this landscape model is sensible to changes in game constrains, such as defensive pressure of the opposing team, a red card shown to an opposing player, or other events that may change players spatio-temporal configuration allowing to identify the most vulnerable areas of the field. From an applied perspective, this landscape model provides relevant information for performance analysis and team technical staff about the passing opportunities that have been created by both teams due to the interactive behaviour among ball carrier and off-ball players  as well as in which areas of the field and for how long those passing opportunities were available.Penetrative passes are not the only that can increase the probability of a goal as backward passes can also help in scoring goals32. This is especially relevant close to the opposing goal, where a pass backwards that creates a better angle to score could be a better decision than penetrative passes that leave the receivers in a worst angle to shoot to the goal. In this regard, it is important to keep in mind that previous research have shown that the location of the goal was a relevant task constrain in Futsal33; (3) currently a constant ball speed was assumed. As ‘real’ balls do not fly with constant speed, this could be leading to under or overestimation of the passing opportunities available and should therefore be reformulated .Regardless of the promising results, this model suggests three issues for further research: (1) so far the model assumed that all the players are technically and tactically equal. An issue for further research is to add individual technical characteristics to shape each player passing opportunities on a competitive match; (2) the outplay principle to categorize passes requires an accuracy improvement. support and penetrative passes. Hence we may conclude that this model is sensitive to task constrains and to the uncertainty related with the uniqueness of each football match, at least regarding the landscape of passing opportunities. Nevertheless, it was possible to show evidence that due to player’s interactive behaviour some matches exhibit more similarity than others, as displayed by the EMD results.Based on ball carrier and off-ball players spatiotemporal interactions this landscape model depict the passing opportunities that emerge in the course of a football match. The landscapes of passing opportunities varied through different matches, especially for penetrative and backward passing opportunities last less than support passing opportunities.Having the heatmaps as an output provides visual information regarding the areas of the field where more passing opportunities were available and for how long they last. Additionally by creating temporal heatmaps it was possible to identify how the threat (for the defense) caused by the different types of passing opportunities fluctuates in space and time during the course of a match. As expected, the different type of passing opportunities were available for different time windows. For different reasons Finally, we reinforce that this is a methodological driven research, thus we do not aim for a generalization of results, but rather a generalization of the method usability. The outputs of this landscape model provides information that could be relevant to a post-match report shortening the gap between performance analysis team and the technical staff.Supplementary Video 1.Supplementary Video 2.Supplementary Video 3.