We now focus on the selection of the optimal squared bandwidth T∈R>0 according to the relationship T=h2 between final iteration time T of the diffKDE and bandwidth parameter h . In the following we refer to this as the bandwidth selection for simplicity.

We stressed in Eq. () that the optimal choice of the bandwidth parameter depends on the true density f. In our setup of the diffKDE, the analytical solution for T of Eq. () depends not only on the true density f, but also on the parameter function p. It can be calculated as 17T∗=Ep(X)2Nπfp′′L2225, where ||⋅||L2 is the L2 norm and E⋅ the expected value. The proof of this equation is given in detail in . The role of the parameter function p is described in detail in Sect. .

In the simplified setup with p=1 as in Eq. (), the analytical optimal solution of Eq. () becomes 18T(p=1)∗=12Nπ|f′′|L2225.

Still, the smoothing parameter depends on the unknown density function f and its derivatives. So we will need to find a suitable approximation of f, which might again be dependent on f and p. This implicit dependency can be solved by so-called pilot estimation steps. Pilot estimates are generally rough estimates of f calculated in an initial step to use them for an approximation of Eqs. () and (), which later serve to calculate a more precise estimate of f. A more detailed introduction into pilot estimation and its specific benefit for diffusion-based KDEs is presented in Sect. .

used an iterative scheme to solve the implicit dependency of the bandwidth parameter on the true distribution f. This additional effort is avoided in our approach by directly approximating f with a simple data-based bandwidth approximation based on two pilot estimation steps described in detail in Sect. .

The possible difficulties in finding one single optimal bandwidth e.g., do not arise in the calculation of the diffKDE by default. This problem is solved by creating a family of estimates from different bandwidth parameters ranging from oversmoothed estimates to those with beginning oscillations . For the diffKDE, the progression of the time t up to a final iteration time T is equivalent to the creation of such a family of estimates. We thus only need to find a suitable optimal final iteration time T∗. Then the temporal solution of Eq. () provides solutions for the diffKDE for the whole sequence of the temporal discretization time steps smaller than T∗, which we can use as the requested family of estimates.

A crude first estimate of the true density f can serve as a pilot estimation step for several purposes . The most obvious purpose in our case is to obtain an estimate of f for the calculations of the optimal bandwidth in Eq. (). The second purpose is its usage for the definition of the parameter function p in Eq. (). Setting this as an estimate of the true density itself introduces locally adaptive smoothing properties . Since p appears in the denominator in the diffusion equation, it operates conversely to a classical diffusion coefficient. Choosing p to be a function allows for a spatially dependent influence on the smoothing intensity as follows: at points where the function p is small, the smoothing becomes more pronounced, whereas if p is larger, the smoothing is less intense. Low smoothing resolves more variability within areas with many similar values (high density), while the intensity of smoothing is increased where data values are more dispersed. Eventually, we calculate two pilot estimates – one for p and one for f – to support the calculation of the diffKDE. We set both pilot estimates to be the solution of Eq. () with an optimal smoothing parameter approximating Eq. (). This approach combines Gaussian KDE and diffKDE interchangeably to make best use of both of their benefits .

We start with the description of the discretization of the spatial domain Ω⊆R. This will reduce the partial differential equation in Eq. () into a system of linear ordinary differential equations. Let n∈N and (xi)i=0n⊆Ω¯, an equidistant discretization of Ω with xi-1<xi and spatial discretization step size R>0∋h=xi-xi-1 for all i∈{1,…,n}. For the following calculations, we set x-1=x0-h∈R and xn+1=xn+h∈R. Let u be the solution of the diffKDE and p its parameter function, both as defined in Sect. . We assume that u and p are both defined on x-1 and xn+1 and we set ui=u(xi) and pi=p(xi) for all i∈{-1,…,n+1}.

Let t∈R>0. We approximate Eq. () at x=x0 by applying a first-order central difference quotient as 0=∂∂xu(x0;t)p(x0)≈12hu1(t)p1-u-1(t)p-1.

This implies u-1(t)p-1=u1(t)p1.

We approximate Eq. () at x=x0 by applying a second-order central difference quotient: 26u0′(t)≈121h2u1(t)p1-2u0(t)p0+u-1(t)p-1=121h22u1(t)p1-2u0(t)p0.

Analogously, we approximate Eqs. () and () at x=xn again by first- and second-order central difference quotients, respectively. This gives 27un′(t)≈121h2un+1(t)pn+1-2un(t)pn+un-1(t)pn-1=121h22un-1(t)pn-1-2un(t)pn.

Finally, we derive from Eq. () by applying a second-order central difference quotient for all i∈{1,…,n-1}: 28ui′(t)≈121h2ui+1(t)pi+1-2ui(t)pi+ui-1(t)pi-1.

Now, we identify p:=p0,…,pn∈Rn+1, u′(t):=u0′(t),…,un′(t)∈Rn+1, and u(t):=u0(t),…,un(t)∈Rn+1 with their spatial discretizations. Furthermore, we define vupper:=2,1,…,1∈Rn, vmain:=-2,…,-2∈Rn+1, and vlower:=1,…,1,2∈Rn to be the upper, main, and lower diagonal of the tridiagonal matrix V∈R(n+1)×(n+1). Now, we set 29121h2V1p=A∈R(n+1)×(n+1), where A∈R(n+1)×(n+1) means that A has real entries and n+1 rows and n+1 columns, and the division by p is applied column-wise. Then, Eqs. ()–() can be summarized as a linear system of ordinary differential equations: 30u′(t)≈121h2Vu(t)p=Au(t).

By these calculations the solution of the partial differential equation from Eq. () can also be approximated by solving the system of ordinary differential equations: 31u′(t)=Au(t),t∈R>0.

The time-stepping applied to solve the ordinary differential equation from Eqs. () and () is again built on equidistant steps forward in time. Let Δ∈R>0 be small and set t0=0 and tk=tk-1+Δ for all k∈N. Set uk,i=u(xi,tk)∈R for all i∈{0,…,n+1} and k∈N0 and identify uk=(uk,i)i=0n∈Rn+1 for all k∈N0 with their discretizations.

We use an implicit Euler method to approximate Eq. () for all k∈N0, that is, 32uk+1=ΔAuk+1+uk from which we obtain 33uk=In+1-ΔAuk+1 for all k∈N0.

The implicit Euler method is chosen at this place, since it is A-stable and by this ensures convergence of the solver.

Equation () together with the initial value Eq. () describes an implementation-ready time stepping procedure. The linear equation for uk+1 will be solved in every time step k∈N0.

The initial value in Eq. () depends on the δ distribution . The δ distribution is not a proper function but can be calculated as a limit of a suitable function sequence. A common approximation for the δ distribution is to use a Dirac sequence . Such is a sequence Φnn∈N of integrable functions that are non-negative and satisfy 34∫Φn(x)dx=1 for all n∈N and 35lim⁡n→∞∫R∖Bρ(0)Φndx=0 for all ρ∈R>0, where Bρ(0)={x∈R;|x-0|<ρ}=-ρ,ρ is the open subset of R centered around R with radius ρ. For our implementation we define a Dirac sequence Φhh∈R>0 depending on the spatial discretization step size h∈R>0 as an approximation of δ in Eq. (). The spatial discretization step size h∈R>0 equals the length of the domain Ω divided by the number of spatial discretization points n, namely h=Ωn. This relationship provides the dependency of Φh on n∈N and the equivalence of the limits n→∞ and h→0 in this framework. In the following we give the specific definition of our function sequence of choice and proof that this indeed defines a proper Dirac sequence.

We assume 0∈Ω. Then there exists an i∈{0,…,n} with 0∈xi-1,xi. If not readily defined, we set xi-2=xi-1-h∈R and xi+1=xi+h∈R, and we define the following (see also Fig. ): 36Φh(x)=xih3x+xixi-2h3,x∈xi-2,xi-1xi+xi-1h3x+xixi+xi-1h3-xi-1h2,x∈xi-1,xixi-1h3x+xi+1xi-1h3,x∈xi,xi+10,else, where x,Φh(x)∈R.

Then Φh is non-negative for all h∈R>0 and as a composition of integrable functions integrable with ∫Φh(x)dx=1 (see Appendix ) and Φh∈L1R={f:R→R;f integrable and ∫|f(x)|dx<∞}.

Now, let ρ∈R>0 and set h=ρ2∈R>0. Then we have by Eq. () ∫R∖Bρ(0)Φhdx=∫-∞ρΦhdx+∫ρ∞Φhdx=∫-∞ρ0dx+∫ρ∞0dx=0, and it follows that 37lim⁡h→0∫R∖Bρ(0)Φhdx=0 for all ρ∈R>0.

Hence Eq. () defines a Dirac sequence. We use Φh for the approximation of the δ distribution in our implementation of Eq. ().

The concept of the Dirac sequence also provides the justification to generally rely on the δ distribution in the construction of the initial value of the diffKDE. The Gaussian kernel defined in Eq. () that solves the diffusion equation as a fundamental solution is again a Dirac sequence . This link connects the diffKDE directly back to the δ distribution.

The implementation was conducted in Python and its concept is shown in Algorithm . We used the Python libraries Numpy and SciPy and the Python Math module for data preprocessing, calculation of the bandwidths, as well as setup of the differential equations and their solution. The algorithm iteratively calculates three KDEs: first, the two for the approximations of p and f as the pilot estimation steps described in Sect. , and the last one being u, the solution of the diffKDE built on the prior two. All three KDEs are calculated by solving the diffusion equation up to the respective final iteration time. The solution is realized in while loops solving Eq. (). The two pilot estimation steps can be calculated simultaneously, since they are independent of each other and only differ in their final iteration times Tp and Tf. All input variables are displayed in Table  and the return values are listed in Table .

The spatial grid discretizing Ω is set up according to the description in Sect.  in lines 1 and 2 of Algorithm . It consists of n∈N intervals, where n can be set by the user. The boundary values are xmin=min⁡X∈R and xmax=max⁡X∈R by default but can also be chosen individually. Setting the boundary values to an individually chosen interval in the function call results in a clipping of the used data to this smaller one before KDE calculation. Outside the interval boundaries, the diffKDE adds two additional discretization points to make it applicable for the case of a data point Xj, j∈{1,…,N} being directly located at one of the boundaries. This way it is possible to construct the initial value defined in Eq. (), which takes into account the two neighboring discretization points in each direction. This leads to a full set of n+1 equidistant discretization points (xi)i=0n saved in a vector variable denoted in Algorithm  as Ω. The spatial discretization (xi)i=0n includes an inner discretization between the handed in (or default set) interval endpoints xmin and xmax of n-4 equally sized inner discretization intervals.

The Dirac sequence Φh for the implementation of the initial value is defined in Eq. () and we use the same for initialization of all three approximations of the PDF (p,f,u) in line 3 of Algorithm . In its calculation, the algorithm searches for each j∈{1,…,N} for the i∈{1,…,n+1} with xi being the closest right neighbor of Xj. Then the initial value is constructed by assigning the values 1hxi-Xjxi-xi-1 and 1hXj-xi-1xi-xi-1 at grid point xi and xi-1, respectively, and zero elsewhere. These values correspond to the weighted heights Hi and Hi-1 displayed in Fig. . The final initial value is the normalized sum of all these individual approximations of the δ distribution. All three used KDEs (p,f,u) are initialized with this initial value.

In the pilot estimation steps, we calculate the KDEs for p and for f required for the setup of the final iteration time T∗ for the diffKDE. The final iteration times Tp and Tf for p and f, respectively, are calculated based on the input data X in lines 4 and 5 of Algorithm  as described in Sect. . Then the KDEs are calculated by solving a linear ordinary differential equation by an implicit Euler approach in the first while loop in lines 7–9 of Algorithm . For the pilot estimation steps calculating p and f the matrix A defined in Eq. () does not incorporate a parameter function and reduces to a matrix denoted as Apilot, that is 38121h2V=Apilot∈R(n+1)×(n+1), where Apilot∈R(n+1)×(n+1) means that Apilot has real entries and n+1 rows and n+1 columns. Apart from this, the solutions for the pilot KDEs are the same as for the final diffKDE. The two pilot KDEs can be solved simultaneously, since they share their matrix Apilot and have independent pre-computed final iteration times. The difference in their bandwidths is implemented in different time step sizes Δp and Δf for p and f, respectively, which are initialized in line 6 of Algorithm  directly before this first while loop. The temporal solutions are calculated time steps∈N times in equidistantly increasing time steps until each individual final iteration time. Since we solve implicitly, there is no restriction to the time step size. But a larger time steps parameter reduces the numerical error proportional to the step size parameters Δp and Δf. In this temporal solution we rely on the fact that the involved matrices are sparsely covered. The applied solver is part of the SciPy Python library and designed for efficient solution of linear systems including sparse matrices .

The final iteration time T for the diffKDE solution u is calculated after the calculations of p and f, using them both as described in Sect.  in lines 12–14 of Algorithm . For the diffKDE u, the differential equation is given in Eq. () and the solution uses the implicit Euler approach in Eq. (). This is implemented in a second while loop described in lines 16–18 in Algorithm , which is separated from the final iteration time T∗ and the matrix A identical to the calculations in the pilot step.

The return value is a tuple providing the user the diffKDE and the spatial discretization as displayed in Table .

Possible problems are caught in assert and if clauses. Initially, the data are reshaped to a Numpy array for the case of a list handed in, and it is made sure that this is non-empty. For the case of numerical issues leading to a pilot estimate including zero values, the whole pilot is set back equal to 1 to ensure numerical convergence. Accordingly for the case of NaN value being delivered for the optimal bandwidth for the diffKDE, in which case this is also set to the bandwidth chosen for f in Eq. ().

Besides the standard use to calculate a diffKDE at an approximated optimal final iteration time for direct usage, we also included three possibilities to generate a direct visual output, one of them being interactive. Matplotlib provides the software measures for creating the plots. Most methods are part of the submodule Pyplot, whereas the interactive plot is based on the submodule Slider.

The function call evol_plot opens a plot showing the time evolution of the diffKDE (e.g., see Fig.  for an example output). The plot includes drawings of the data points on the x axis. In the background the initial values are drawn. The y axis range is cut off at 20 % above the global maximum of the diffKDE to preserve focus of the graphic on the diffKDE and evolution. The evolution is presented by drawings of the individual time evolution stages using the sequential color map Viridis. The diffKDE is drawn in a bold blue line. This visualization of the evolution provides the user with insight into the data distribution and their respective influence on the final form of the diffKDE.

The function call pilot_plot opens which shows the diffKDE together with its pilot estimate p, showing the intensity of local smoothing (e.g., see Fig.  for an example output). With this the user has the possibility to gain insight to the influence of this pilot estimator on the performance of the diffKDE. This plot also includes the data points on the x axis.

The function call custom_plot opens an interactive plot, allowing the user to slide through different approximation stages of the diffKDE (e.g., see Fig.  for an example output). This feature is based on the Slider module from the Matplotlib library and opens a plot showing the diffKDE. At the bottom of this plot is a scale that shows the time, initially being set to the optimal iteration time derived from Eq. () in the middle of the scale. By clicking on the scale, the user can display the evolution stages at the respective (closest) iteration time. This reaches down to the initial value and up to the doubled optimal iteration time. This interactive tool provides the user a simple tool to follow the estimate at different bandwidths. With the help of such a plot it is possible to decide on whether the diffKDE is desired to be applied with a final iteration time that is different from the default.

As described in Sect. , we included three plot functions in the diffKDE implementation. All of them open pre-implemented plots to give an impression of the special features that come with the diffKDE. An overview of the three possible direct visual outputs of the diffKDE software is described below.

First we demonstrate how to display the diffKDE's evolution. By calling the evol_plot function, a plot opens that shows all temporal evolution stages of the solution of Eq. (). The temporal progress is visualized by a sequential color scheme progressing from light yellow over different shades of green to dark blue. On the x axis, all used data points are drawn. In the background, a cut-off part of the initial value as the beginning of the temporal evolution is depicted in light yellow. The final diffKDE is plotted as a bold blue line in front of the evolution process. This gives the user insight into the distribution of the initial data and their influence on the shape of the estimate. As an example of the default setting, we created an evolution plot from 100 random samples of Eq. () visualized in Fig. . The second example shows the possibility of displaying the diffKDE together with the pilot estimate p by the function pilot_plot. This is the parameter function in Eq. () responsible for the adaptive smoothing. Where this function is larger, the smoothing is less intense and allows more structure in the estimate of the diffKDE. Contrarily, where it is smaller the smoothing becomes more pronounced and data gaps are better smoothed out. The result of the diffKDE is shown together with its parameter function p in figure Fig.  on the same random sample of the distribution from Eq. () as before.

Lastly, we illustrate example snapshots of the interactive option to investigate different smoothing stages of the diffKDE by the function. We chose simpler and smaller example data for this demonstration, because these are better suited for visualization of this tool's possibilities. The function custom_plot opens an interactive graphic, starting with a plot of the approximated optimal default solution of the diffKDE at T∗. In this graphic the user is able to individually choose, by a slider, the iteration time at which the desired approximation stage of the diffKDE can be seen. The time can be chosen from 0, where the initial value is shown, up until the doubled approximated optimal time (2×T∗). A reset button sets the graphic back to its initial stage of the diffKDE at T∗. Four snapshots of this interactive experience are shown in Fig. .

In this section we present results obtained by random samples of the trimodal distribution from Eq. () and lognormal distributions with differing parameters. Wherever suitable, the results are compared with other commonly used KDEs. These include the most common Gaussian KDE with the kernel function from Eq. () in an implementation from SciPy , the Epanechnikov KDE with the kernel function from Eq. () in an implementation from Scikit-learn , and an improved implementation of the Gaussian KDE by in a Python implementation by . We begin with an example of how the user may choose individually different smoothing grades of the diffKDE, then compare the different KDEs with the true distribution, followed by investigating the influence of noise on different KDEs, and finally show the convergence of different KDEs to the true distribution with increasing sample size.

We start with an individual selection of the approximation stages. This is one of the main benefits of the diffKDE compared with standard KDEs by providing naturally a family of approximations. This family can be observed by the function custom_plot. Individual members can be produced by setting the bandwidth parameter T in the function call of the diffKDE. This gives the user the chance to choose among more and less smooth approximations. A selection of such approximations along with the default solution are shown in Fig.  on a random sample of 50 data points from the trimodal distribution in Eq. (). The plot shows how smaller iteration times resolve more structure in the estimate, while a substantially larger iteration time has only little influence on the increased smoothing of the diffKDE.

In the following we only work with the default solution of the diffKDE at T∗. We start with comparisons of the diffKDE and the three other popular KDEs directly to the underlying true distribution.

We use differently sized random samples of the known distribution from Eq. () and the standard lognormal distribution both over -1,12 for a direct comparison of the accuracy of the KDEs. The random samples are 50, 100, and 1000 data points of each distribution and all four KDEs are calculated and plotted together in Fig. . The underlying true distribution is plotted in the background to visually assess the approximation accuracy. In general, the diffKDE resolves more of the details of the structure of the true distribution while not being too sensitive to patterns introduced by the selection of the random sample and individual outliers. For the 50 random samples test of the trimodal distribution, all KDEs do not detect the third mode and only the diffKDE and the Epanechnikov KDE detect the second. The magnitude of the main mode is also best resolved by these two. In the 100 random samples test of the trimodal distribution, the diffKDE and the Botev KDE are able to detect all three modes. The main mode is best resolved by the diffKDE, whereas the third mode is best resolved by the Botev KDE. In both test cases for the trimodal distribution, the Gaussian KDE is the smoothest and the Epanechnikov KDE provides the least smooth graph. In the 1000 random samples test the diffKDE best detects the left mode and the Botev KDE best detects the two others. Generally, diffKDE and Botev KDE are closely aligned in this case. The Gaussian and Epanechnikov KDEs are also closely aligned but with a worse fit of all structures of the true distribution. The steep decline to 0 is best reproduced by the diffKDE particularly with low random sample sizes. The Gaussian KDE always performs the worst. The Botev KDE is generally also close to the diffKDE but resolves in the tail of the distribution too much influence of individual outliers. In the 1000 random samples test with the lognormal distribution are again diffKDE and Botev KDE closely aligned as well as Gaussian and Epanechnikov KDE. The first two are very close to the true distribution but resolve too much structure of the random sample. The diffKDE resolves more structure in the area close to 0 and becomes smoother towards the tail of the distribution. The Botev KDE performs the other way around and provides a smoother estimate close to 0 and more structure of the random sample towards higher data values. An analysis of the integral of the KDEs over the observed domain is presented in Table  and reveals that the diffKDE is the only one that integrates to 1 in all test cases.

We refined the test cases from Fig.  by investigating a lognormal distribution with different parameters and a restriction to the interval 0,12 in Fig. . We varied mean and variances of the normal distribution and used two different means and three different variances resulting in six test cases. All of them are run with 300 random samples and again with all four KDEs. The larger the variance becomes, the more structure of individual data points is resolved by the Botev KDE. The Gaussian KDE fails for increasing variance too, resulting in intense oversmoothing. The Epanechnikov KDE performs well for smaller variances and larger means, but it also oversmoothes in the other cases. The diffKDE is generally one of the closest to the true distribution while not resolving too much of the structure introduced by the choice of the random sample, especially for increased variances. But this too tends to resolve too much structure in the vicinity of the mode for smaller variances. The integrals of the KDEs are also presented in Table  and our implementation is again always exactly 1.

Now we show the performance of the diffKDE on increasingly large data sets. We still use the trimodal distribution from Eq. (). We start with four larger random data samples ranging from 100 to 10 million data points of the trimodal distribution and then restrict ourselves to our core area of interest [-1,12]. We calculate the diffKDE from all of them as well as the respective runtime on a consumer laptop from 2020. We compare the results again to the true distribution in Fig. . All of the estimates could be calculated in less than 1 min. For 100 data points there is still an offset to the true distribution visible in the estimate. For the larger data samples the estimate only shows some minor uneven areas, which smooth out until the largest test case.

Furthermore, we investigated the convergence of the diffKDE to the true distribution, again in comparison with the three other KDEs. The error between the respective KDE and the true distribution is calculated by the Wasserstein distance with p=1 by a SciPy function and the MISE defined in Eq. (). For the approximation of the expected value in Eq. () we applied an averaging of the integral value for 100 different random samples for each observed sample size. We used increasingly large random samples from the trimodal distribution starting with 10 and reaching up to 1 million. The errors calculated for each of the KDEs on each of the random samples are listed in Table . The values from Table  are visualized in Fig.  on a log-scale. The diffKDE, the Gaussian, and the Botev show a similar steep decline, whereas the Epanechnikov KDE decreases its error much slower with increased sample size. The diffKDE and the Botev KDE generally show similar error values, the diffKDE relatively smaller ones on smaller data samples and the Botev KDE relatively smaller ones on data samples larger than around 5000.

Finally, we investigated the noise sensitivity of the diffKDE compared with the three other KDEs on data containing artificially introduced noise. We again used the trimodal distribution from Eq. () and 1000 random samples. From this, we created noised data Xθ∈RN by 40(Xθ)i=(X)i+(-1)τrand10-2θσ for all i∈{1,…,1000}, where θ∈{0,1,5,15,30} defines the percentage of noise with respect to the standard deviation σ∈R. τ∈{1,2} was chosen randomly as well as rand∈[0,1]. The error is again expressed by the Wasserstein distance between the original probability density and the respective KDE. The results are visualized in Fig.  with an individual panel for each KDE. The error of the Epanechnikov KDE is overall the largest and also increases to the largest. The Gaussian KDE produces the second largest error, but this even decreases with increased noise. The Botev KDE produces the smallest errors, but for increased noise this increases and approaches the magnitude of the one from the diffKDE. The error of the diffKDE only minimally responds to increased noise in the data. Visually, all four KDEs follow a similar pattern of a shift to the left of the graph. The Botev KDE additionally resolves more structure of the noised data as the noise increases.

The δ13CPOC data were collected to serve for direct data analyses as well as for future model assessments . We show here the Gaussian KDE as it was used in the data publication in a direct comparison with the diffKDE. Furthermore, we added the Epanechnikov and the Botev KDEs. Since in this case no true known PDF is available, we have to compare the four estimates and subjectively judge their usefulness. In Fig.  we show the KDEs on four different subsets of the δ12CPOC data: (a) the full data set, (b) a restriction to the core data interval of [-35,-15], where 98.65 % of the data are located, and then even further restricted to (c) the euphotic zone and (d) only data sampled in the 1990s. The euphotic zone describes the upper ocean layer with sufficient light to enable photosynthesis that produces organic matter . While its depth can vary in nature , here we pragmatically selected included data in the upper 130 m consistent with the analysis in the data set description (Verwega et al. 2021). In all three cases that involve deep ocean measurements, the Botev KDE produces strong oscillations while the Gaussian KDE strongly smoothes the dip between the modes at around δ13CPOC=-24 and δ13CPOC=-22 and mostly the one between around δ13CPOC=-28 and δ13CPOC=-24. The Epanechnikov KDE resolves more structure than the Gaussian, but is less pronounced than the diffKDE. Especially in the full data analysis, the diffKDE reveals the most structure while not resolving smaller data features of individual data points. The KDEs from the euphotic zone data are all reasonably smooth. The Gaussian KDE is again the smoothest and missing the mode at δ13CPOC=-22 completely. The other three KDEs resolve a similar amount of data structure. The Botev KDE reveals a better distinction between the modes at around δ13CPOC=-24 and δ12CPOC=-22 while the diffKDE shows the first one more pronounced. These observations are consistent with those from the experiments from Figs.  and , where especially the Gaussian and the Botev KDEs struggle with the resolution of data with increasing variances or noise. Of the four observed δ12CPOC data sets here, the euphotic zone data shown in Fig. c have the smallest standard deviation, 7.78. The other shown data have variances of 13.91, 10.96, and 9.61 in Fig. a, b, and d, respectively.

Another example demonstrates the performance of the diffKDE if applied to plankton size data . The data of size and abundance of protist plankton was originally collected for resolving changes in plankton community size structure, providing complementary insight for investigations of plankton dynamics and organic matter flux e.g.,. In the study of a KDE was applied for the derivation of continuous size spectra of phytoplankton and microzooplankton that can potentially be used for the calibration and assessment of size-based plankton ecosystem models. In their study they used a Gaussian KDE, as proposed in , but with two different approaches for generating plankton size spectra. Uncertainties, also with respect to optimal bandwidth selection, were accounted for in both approaches by analyzing ensembles of pseudo-data resampled from original microscopic measurements. Smooth plankton spectra were obtained using the combined approach, where all phytoplankton and all zooplankton data were lumped together, respectively, and single bandwidths were calculated for every ensemble member (set of resampled data). This procedure avoided overfitting but was also prone to oversmoothing, which can mask details such as troughs in specific size ranges. More details in the size spectra were resolved with a composite approach, where individual size spectra, calculated for each species or genus, were assembled. Since the variance within species or genus groups is smaller than within the large groups “phytoplankton” or “zooplankton”, the individual bandwidths, and therefore the degree of smoothing, were considerably smaller than obtained in the combined approach. This computationally expensive method revealed many details in the spectra but at the same time tended to resolve narrow peaks that were either clearly insignificant or remained difficult to interpret (see supplemental material in ). The here proposed diffKDE was tested with resampled data used for the simpler combined approach. The objective was to identify details in the size spectra that remained previously unresolved while insignificant peaks, as found in the composite approach, became smoothed out. Figure  shows the performance of the diffKDE in comparison with the original combined and composite spectra that were derived as ensemble means of estimates obtained with a Gaussian KDE. The spatial discretization of the diffKDE was set to n=1000 to be comparable to the other already published KDEs in this case. The diffKDE seems to meaningfully combine the advantages of the two Gaussian KDE approaches in both spectra, of the phytoplankton and microzooplankton, respectively. With the diffKDE it is possible to generate estimates that display more detailed structure of the composite KDE. This becomes obvious in some size ranges, such as between 5 and 50 μm, in particular in the microzooplankton spectrum. Concurrently, detailed variations, as caused by overfitting in the composite spectra, become suppressed for cell sizes larger than 10 μm. Thus, with the diffKDE it is possible to generate a single robust estimate that otherwise is only achieved by analyzing a composite of a series of individual estimates of a Gaussian KDE. The application of the diffKDE for analyzing details in plankton size spectra, or generally in particle size spectra, reduces computational efforts considerably.

Our last example refers to PDFs that reflect temporal changes (monthly means) of surface chlorophyll a concentration within an off-shore ocean region (approximately 350 × 330 km) that exhibits substantial mesoscale and submesoscale variability. The selected area is part of the of the Mauritanian upwelling system, located at the Moroccan coast of North Africa. This eastern boundary upwelling is known for the formation and spread of filaments, with some distinct characteristics in terms of the spatial variability in temperature and the distribution of plankton e.g.,. For this example we use remote sensing (satellite) data of monthly mean chlorophyll a concentration from the year 2019, as processed and made available through the Ocean-Colour Climate Change Initiative (OC-CCI) . We deliberately chose 3 months in which an eddy-like filamentous structure of elevated chlorophyll a concentration developed after February and then evolved during a 2-month period (March and April). Such development reveals specific spatial patterns, of low and increased chlorophyll a, which leave a clear imprint in the corresponding PDFs, as depicted in Fig. . In Fig.  we find multiple modes, as well as a shift towards low chlorophyll a concentrations that are much better resolved when the diffKDE is applied. Such details derived with the diffKDE could be compared to complementary PDFs, for example, as obtained from remote sensing sea surface temperature data within the same time period, and we might gain further insight into the underlying processes involved in generating distinctive spatial and temporal patterns. Also, in cases of simulations of mesoscale and submesoscale processes, we should not expect to obtain a model solution with filamentous structures at identical times at the same places. However, we may regard a model's performance as credible if the model's solution yields similar spatial structures visually within the same region, perhaps at some different time, and the associated PDFs may then be directly assessed against the PDFs obtained from the remote sensing data.

In geoscientific research, the derivation and comparison of well-resolved PDFs can be useful, as demonstrated in our selected examples. Yet, the significance of resolving details in non-parametric PDFs remains unclear. However, having high resolution PDFs available, as obtained with the diffKDE, is readily of value and will likely guide further research. An obvious benefit of the diffKDE is its lesser dependence on the specification of a single, albeit optimal, bandwidth. Its application is likely more robust for the assessment of simulation results, either against data or results of other models (e.g., multimodel ensembles), which is particularly relevant for evaluations of future climate projections obtained with earth system models e.g.,.

Comparison of model and field data requires additional processing to account for spatial–temporal differences between collected samples and model resolution. Typically, simulation results are available at every single spatial grid point and in every time step. In comparison, field data are usually sparsely available only. Interpolating such sparse field data can introduce high uncertainty e.g.,. PDFs provide a useful approach to investigate data independent of the number of data points available . A comparison of two such functions can easily resolve the issue of non-equal field observations and simulation results. Histograms are commonly used as an approach to compare and ultimately constrain the distribution of model data to observations. However, many issues arise including the subjective selection of intervals and histograms not being proper PDFs themselves.

The presented diffKDE provides a non-parametric approach to estimate PDFs with typical features of geoscientific data. Being able to resolve typical patterns, such as multiple or boundary-close modes, while being insensitive to noise and individual outliers makes the diffKDE a suitable tool for future work in the calibration and optimization of earth system models.