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EDIT: I've included a VB.NET version of @Kirk's C# code. It's provided as is, but it should be a good starting point if it doesn't already completely satisfy its purpose.

    Public Shared Sub TestSumRaster()
        Dim path As String = "D:\topo_modeling\topo_modeling\gis\usgs\ned_10m.img"
        Dim rds = OpenRaster(path)
        Dim sum As Double = SumRaster(rds)
        Debug.Print("sum : {0}", sum)
    End Sub
    
    Public Shared Function SumRaster(rasDs As IRasterDataset) As Double
        ' sum up each band in a raster dataset
        Dim rasBandCol As IRasterBandCollection = DirectCast(rasDs, IRasterBandCollection)
        Dim sum As Double = 0.0
    
        ' Iterate through each band of the dataset.
        For i As Integer = 0 To rasBandCol.Count - 1
            sum += SumBand(rasBandCol.Item(i))
        Next
        Return sum
    End Function
    
    Private Shared Function SumBand(rBand As IRasterBand) As Double
        
' sum up each block in a band
        Dim sum As Double = 0.0
        ' QI to IRawBlocks from IRasterBandCollection
        Dim rawBlocks = DirectCast(rBand, IRawBlocks)
    
        ' Determine the tiling scheme for the raster dataset.
        Dim tile = New Tile(rawBlocks.RasterInfo)
    
        ' NoData is array, but usually just has one value in it.
        Debug.Print(rawBlocks.RasterInfo.NoData.[GetType]().ToString())
    
        ' Create the pixel block.
        Dim pb = rawBlocks.CreatePixelBlock()
    
        ' Iterate through the pixel blocks.
        For pbYcursor As Integer = tile.startY To tile.endY - 1
            For pbXcursor As Integer = tile.startX To tile.endX - 1
                ' Get the pixel block.
                rawBlocks.ReadBlock(pbXcursor, pbYcursor, 0, pb)
                sum += SumBlock(pb, pbXcursor, pbYcursor, rawBlocks.RasterInfo.NoData)
            Next
        Next
        Return sum
    End Function
    
    Private Shared Function SumBlock(pb As IPixelBlock, pbXcursor As Integer, pbYcursor As Integer, noDataValues As Object) As Double
        ' sum up each pixel in the pixelblock
        Dim sum As Double = 0.0
    
        ' Put the pixel block into a SafeArray for manipulation.
        Dim safeArray = TryCast(DirectCast(pb.get_SafeArray(0), System.Array), System.Array)
    
        Dim noDataList = New List(Of [Single])(DirectCast(noDataValues, [Single]()))
    
        ' Iterate through the pixels in the pixel block.
        For row As Integer = 0 To pb.Height - 1
            For col As Integer = 0 To pb.Width - 1
                Dim val As Object = safeArray.GetValue(row, col)
                Dim singVal = DirectCast(val, [Single])
    
                'Debug.Print(val.GetType().Name);
                If Not noDataList.Contains(singVal) Then
                    sum += singVal
                End If
            Next
        Next
        Return sum
    End Function