Setting raster image coordinate system with known lat/long range

[LamaZip]

I have downloaded a SST image of the United Kingdom. It is in PNG format with no coordinate system.

I know the dimensions of my image and its lat/long range. However, i'm not sure what approach to take in Arcmap to set the co-ordinate system.

There is a web-page describing how to convert image co-ordinates to latitude/longitude but it is beyond my scope link

The image is apparently produced in Mercator projection

Note, this will need to be carried out on multiple rasters, so a automated/batch approach, which isn't interactive or manual, is required

Details of Image

Dimension   1596 x 1617 

BPP 8 

Latitude range  47 — 62.999108
Longditude range    -15 — 13
Pixel dimensions    1.100149 x 1.100149

[AbdullahAnter]

Do you can using Georeference?

http://desktop.arcgis.com/en/arcmap/latest/manage-data/raster-and-images/entering-specific-x-y-coord... 

[LamaZip]

Could you give an example of how this would be achieved? 

[AbdullahAnter]

• Display georeference toolbar in arc map.
• Add your image (Add Data).
• Click the Add Control Points tool .
• Click a known location in the unreferenced image to add the first coordinate in the link. (the corner of image)
• Right-click the image and click Input Lat and Long.
• Type the reference coordinates on the Enter Coordinates dialog box
• Click Ok.
• Do this for two corners of image.