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Geoscience applications generate large datasets; thus, compression is necessary to facilitate the storage and transmission of geoscience data. One focus is on the coding of hyperspectral imagery and the prominent JPEG2000 standard. Certain aspects of the encoder, such as rate-allocation between bands and spectral decorrelation, are not covered by the JPEG2000 standard. This thesis investigates the performance of several JPEG2000 encoding strategies. Additionally, a relatively low-complexity 3D embedded wavelet-based coder, 3D-tarp, is proposed for the compression of geoscience data. 3D-tarp employs an explicit estimate of the probability of coefficient significance to drive a nonadaptive arithmetic coder, resulting in a simple implementation suited to vectorized hardware acceleration. Finally, an embedded wavelet-based coder is proposed for the shapeaptive coding of ocean-temperature data. 3D binary set-splitting with $k$-d trees, 3D-BISK, replaces the octree splitting structure of other shapeaptive coders with $k$-d trees, a simpler set partitioning structure that is well-suited to shapeaptive coding.