*3.4. Comparison of Integer Encoders for k*2*-Raster*

Experiments were conducted to determine whether other variable-length encoders of integers might serve as a better substitute for DACs, which were the original choice in the *k*2-raster structure initially proposed by Ladra et al. [33]. The performance of DACs was compared to that of other encoders such as Rice, Simple9, PForDelta, Simple16 codes, and gzip. In these experiments, the *Lmax* and *Lmin* arrays were encoded using these codes, and the results are shown in Table 13. For Rice codes, the *l* value, as explained in Section 2.4, produced different results depending on the mean of the raster's elements, and only the ones with the best *l* value are shown.


**Table 12.** The sizes of AIRS Granule (AG9) produced by 3D to 2D mapping from *k* = 2 to *k* = 4. The individual band sizes ranging from 1481 to 1500 are also shown. Sizes for individual bands are in bits per pixel (bpp), while the ones for all bands are in bits per pixel per band (bpppb).

**Table 13.** A comparison of the storage size (in bpppb) using different integer encoders on *Lmax* and *Lmin* from the *k*2-raster built from our datasets. The combined entropies for *Lmax* and *Lmin* are listed as a reference. The *l* value that was used in Rice codes is enclosed in brackets. The best and optimal *k* values for DACs are also enclosed in brackets. Except for the entropy, the best rates for each scene's data are highlighted in blue.



**Table 13.** *Cont.*

The results showed that, in most cases, DACs still provided the best storage size compared to other encoders for our datasets. They also had the added advantage of direct random access to individual elements of the matrix whilst the other encoders would need to decompress each raster in order to retrieve the element, thus requiring much longer access time. When DACs did not yield the best performance, DACs results were usually only less than 0.1 bpppb worse. In the worst cases, DACs results lagged behind by, at most, 0.4 bpppb.

#### **4. Conclusions**

In this research, we examined the possibility of using different integer coding methods for *k*2-raster and concluded that this compact data structure worked best when it was used in tandem with DACs encoding. The other variable-length encoders, though having competitive compression ratios, lacked the ability to provide users with direct access to the data. We also studied a method whereby we could obtain a *k* value that gave a competitive storage size and, in most cases, also a suitable access time.

For future work, we are interested in investigating the feasibility of modifying elements in a *k*2-raster structure, facilitating data replacements without having to go through cycles of decompression and compression for the entire compact data structure.

**Author Contributions:** Conceptualization, K.C., D.E.O.T., M.H.-C., I.B., and J.S.-S.; methodology, K.C., D.E.O.T., M.H.-C., I.B., and J.S.-S.; software, K.C.; validation, K.C., I.B., and J.S.-S.; formal analysis, K.C., D.E.O.T., M.H., I.B., and J.S.-S.; investigation, K.C., D.E.O.T., M.H.-C., I.B., and J.S.-S.; resources, K.C., D.E.O.T., M.H.-C., I.B., and J.S.-S.; data curation, K.C., I.B., and J.S.-S.; writing, original draft preparation, K.C., I.B., and J.S.-S.; writing, review and editing, K.C., M.H.-C., I.B., and J.S.-S.; visualization, K.C., I.B., and J.S.-S.; supervision, I.B. and J.S.-S.; project administration, I.B. and J.S.-S.; funding acquisition, M.H.-C., I.B., and J.S.-S. All authors read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund under Grants RTI2018-095287-B-I00 and TIN2015-71126-R (MINECO/FEDER, UE) and BES-2016-078369 (Programa Formación de Personal Investigador), by the Catalan Government under Grant 2017SGR-463, by the postdoctoral fellowship program Beatriu de Pinós, Reference 2018-BP-00008, funded by the Secretary of Universities and Research (Government of Catalonia), and by the Horizon 2020 program of research and innovation of the European Union under the Marie Skłodowska-Curie Grant Agreement #801370.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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